Evolutionary Multitask Optimization: Advanced Algorithms and Transformative Applications in Computational Science

Abstract

This article provides a comprehensive exploration of Evolutionary Multitask Optimization (EMTO), an emerging paradigm in computational intelligence that enables the simultaneous solving of multiple optimization tasks through implicit parallelism and knowledge transfer. We examine the foundational principles of EMTO, starting from the pioneering Multifactorial Evolutionary Algorithm (MFEA) to contemporary hybrid models. The review systematically details innovative methodologies addressing critical challenges like negative transfer and premature convergence, including adaptive operator selection, domain adaptation techniques, and novel knowledge transfer mechanisms. Further, we present a rigorous analysis of validation frameworks and performance metrics used to benchmark EMTO algorithms, supplemented by compelling case studies of real-world applications spanning reservoir scheduling, combinatorial network optimization, and complex system design. This synthesis is tailored for researchers and computational professionals seeking to understand and apply EMTO to accelerate discovery in data-intensive domains including biomedical research and drug development.

The Foundations of Evolutionary Multitasking: Principles, Paradigms, and Potential

Evolutionary Multitask Optimization (EMTO) represents a paradigm shift in how computational optimization problems are approached. Inspired by the human ability to leverage experience from solving one problem to tackle another, EMTO is an emerging branch of computational intelligence that aims to solve multiple optimization tasks simultaneously within a single algorithmic run [1]. Unlike traditional evolutionary algorithms that address problems in isolation, EMTO explicitly exploits the complementarities between different, yet potentially related, tasks [2]. This is achieved by allowing the implicit transfer of knowledge—in the form of genetic material, search biases, or other encoded information—between the populations dedicated to each task [3]. The principal goal is to harness the latent synergies between tasks, thereby accelerating convergence, improving the quality of solutions, and preventing premature convergence on individual problems [4] [1].

The field has garnered significant interest from the evolutionary computation community since the introduction of the seminal Multifactorial Evolutionary Algorithm (MFEA) by Gupta et al. in 2015 [1] [5]. The relevance of EMTO is particularly pronounced in real-world domains where multiple interrelated optimization problems naturally coexist. One such domain is drug discovery, where the simultaneous optimization of compounds for multiple properties—such as efficacy, toxicity, and synthesizability—is paramount [6]. The ability of EMTO to efficiently manage such complex, multi-faceted optimization landscapes positions it as a powerful tool for researchers and professionals aiming to accelerate development cycles and enhance outcomes in fields like pharmaceuticals and bioinformatics [4] [6].

Core Principles and Definitions

Foundational Concepts

At its core, EMTO is built upon several key concepts that distinguish it from single-task evolutionary optimization:

• Task ((Tk)): An EMTO environment comprises (K) distinct optimization tasks, denoted as ({T1, T2, ..., TK}) [2]. Each task (Ti) has its own objective function (fi: Xi \rightarrow \mathbb{R}), defined over a unique search space (Xi) [7]. The goal is to find a set of solutions ({x1^*, x2^, ..., x_K^}) such that each (x_i^*) is the global optimum for its respective task [7].

• Knowledge Transfer: This is the fundamental mechanism that enables EMTO to outperform isolated searches. It involves the exchange of valuable genetic or algorithmic information between concurrently evolving tasks [2] [3]. The underlying assumption is that useful traits evolved for one task might be beneficial for another, especially if the tasks' fitness landscapes share common features [1].

• Implicit vs. Explicit Transfer: Knowledge transfer in EMTO is broadly categorized into two types. Implicit transfer, as seen in MFEA, maps different tasks to a unified search space and allows knowledge to be shared automatically through crossover operations between individuals from different tasks [7]. In contrast, explicit transfer employs dedicated mechanisms (e.g., mapping functions) to directly and controllably transfer knowledge, offering a more guided approach to inter-task information exchange [7].

• Negative Transfer: A significant challenge in EMTO is the risk of negative transfer, which occurs when the exchange of knowledge between two dissimilar or misaligned tasks hinders the optimization process, potentially leading to premature convergence or degraded performance [7] [2]. A primary cause is the curse of dimensionality in direct transfer mechanisms, where mappings learned from sparse data become unstable [7]. Mitigating negative transfer is a central focus of advanced EMTO research.

The Multifactorial Evolutionary Algorithm (MFEA)

The MFEA, often considered the cornerstone of EMTO, implements a single, unified population that evolves under the influence of multiple "cultural factors" (tasks) [1]. Its key innovations are:

• Skill Factor: Each individual in the population is assigned a skill factor ((\tau)), which identifies the single task on which that individual is most effective [1]. The population is thus implicitly divided into task-specific groups without physical segregation.

• Assortative Mating and Vertical Cultural Transmission: MFEA allows crossover between parents from different task groups with a defined probability. This encourages the creation of offspring that inherit genetic material from parents skilled at different tasks, thereby facilitating implicit knowledge transfer [1]. The offspring then inherits the skill factor of the parent on which it performs better.

• Selective Imitation: This operator enables an individual to potentially improve its performance on its assigned task by learning from a random individual that excels at a different task, further promoting cross-task knowledge exchange [1].

Table 1: Key Algorithmic Variants in EMTO and Their Contributions

Algorithm Name	Type of Transfer	Key Innovation	Primary Application Context
MFEA [1]	Implicit	Pioneering framework using skill factor and assortative mating.	General single- and multi-objective optimization.
MFEA-II [7]	Implicit	Introduces online transfer parameter estimation to adaptively control knowledge transfer.	Complex optimization with uncertain task relatedness.
MFEA-AKT [7]	Implicit	Adaptive knowledge transfer to dynamically manage inter-task interactions.	Tasks with fluctuating synergies.
MFEA-MDSGSS [7]	Explicit	Uses Multidimensional Scaling (MDS) for domain adaptation and Golden Section Search (GSS) to avoid local optima.	High-dimensional tasks and tasks with high risk of negative transfer.
EMT-PU [8]	Explicit (Bidirectional)	Formulates Positive and Unlabeled (PU) learning as a bi-task problem with specialized transfer.	Machine learning, specifically PU classification.

A Practical EMTO Framework for Drug Discovery

The drug discovery pipeline is inherently a multitask problem, involving the simultaneous optimization of a molecule for multiple, often competing, properties [6]. EMTO offers a powerful framework to streamline this process. The following workflow diagrams a typical EMTO application in this domain, from problem formulation to solution deployment.

Diagram 1: EMTO workflow for multi-property drug optimization.

Problem Formulation and Task Definition

The first step involves decomposing the overarching drug design goal into specific, concurrent optimization tasks. For a given molecular compound (x), common tasks include:

• Task 1 - Potency Optimization: Maximize binding affinity to a target protein: (f_1(x) = -\text{BindingAffinity}(x)).
• Task 2 - Toxicity Minimization: Minimize predicted toxicological endpoints: (f_2(x) = \text{ToxicityScore}(x)).
• Task 3 - Synthesizability Maximization: Maximize the likelihood of a feasible synthetic pathway: (f_3(x) = -\text{SyntheticComplexity}(x)).

These tasks are solved simultaneously: ( \text{find } x^* \text{ that minimizes } {f1(x), f2(x), f_3(x)} ) [7] [6].

Successfully implementing an EMTO pipeline for drug discovery relies on a suite of computational tools and data resources.

Table 2: Essential Research Reagent Solutions for EMTO in Drug Discovery

Item Name	Type	Function in the EMTO Protocol
Molecular Representation	Data Structure	Encodes a chemical compound into a format processable by the evolutionary algorithm (e.g., SMILES string, molecular graph, fingerprint).
Property Prediction Model	Software/Algorithm	A pre-trained or online machine learning model (e.g., Graph Neural Network) that acts as the objective function, predicting properties like potency or toxicity for a given molecule [6].
Unified Search Space	Algorithmic Component	A common representation (e.g., a latent space from an autoencoder) that allows for meaningful crossover and mutation between solutions from different tasks, mitigating negative transfer [7].
Knowledge Transfer Controller	Algorithmic Module	A mechanism (e.g., based on MDS or online similarity estimation) that governs if, when, and how much knowledge is transferred between tasks to maximize positive and minimize negative transfer [7] [8].
High-Performance Computing (HPC) Cluster	Infrastructure	Provides the computational power required for the expensive fitness evaluations (e.g., molecular docking, ML model inference) across large populations and multiple generations [9].

Detailed Experimental Protocol for an EMTO Study

This section outlines a standardized protocol for conducting a robust EMTO study, drawing from established practices in the field [7] [8].

Experimental Setup and Initialization

• Task Selection and Benchmarking: Select the (K) tasks to be optimized simultaneously. For methodological research, use standardized benchmark suites from IEEE CEC or GECCO competitions [5]. For applied research (e.g., drug discovery), define tasks based on real-world objectives using established predictive models [6].
• Algorithm Configuration: Select a base EMTO algorithm (e.g., MFEA, MFEA-II) and set its core parameters:
  • Population Size ((N)): Typically 100-1000 individuals, scalable with the number and complexity of tasks.
  • Crossover Probability ((pc)): Usually high (e.g., 0.8-1.0) to promote exploration and knowledge mixing.
  • Mutation Probability ((pm)): Typically low (e.g., (1/\text{(chromosome length)})) to introduce small variations.
  • Random Mating Probability ((rmp)): A critical parameter controlling the likelihood of cross-task crossover (e.g., 0.3); can be fixed or adaptive [7] [1].
• Population Initialization: Generate an initial population of (N) individuals. Each individual's chromosome is encoded in a unified search space. If task-specific search spaces differ vastly, employ a domain adaptation technique (e.g., linearized domain adaptation based on Multidimensional Scaling) at this stage to align them [7].

The Evolutionary Cycle and Knowledge Transfer

The core of the protocol is the iterative evolutionary process, detailed below and in the accompanying diagram.

Diagram 2: The iterative EMTO cycle with knowledge transfer.

• Evaluation and Skill Factorial Assignment:

  • For each individual in the population, evaluate its performance on all (K) tasks.
  • Assign each individual a skill factor ((\tau)), which is the index of the single task on which it performs best [1].
  • The individual's final fitness is its fitness on its skill factor task.

• Selection: Apply a selection operator (e.g., tournament selection) to choose parent individuals for reproduction, favoring those with higher fitness.

• Assortative Mating and Crossover:

  • For two selected parents, check if they have the same skill factor.
  • If they do, crossover occurs with a high probability.
  • If they have different skill factors, crossover occurs with a probability defined by the rmp parameter. This is the core of implicit knowledge transfer in MFEA, creating offspring with mixed genetic material from different tasks [1].

• Mutation: Apply a domain-appropriate mutation operator (e.g., Gaussian noise for continuous problems, bit-flip for binary) to the offspring to maintain population diversity.

• Explicit Knowledge Transfer (Algorithm-Dependent): For algorithms like MFEA-MDSGSS or EMT-PU, an additional step of explicit transfer may occur. For example, in EMT-PU, this involves a bidirectional transfer where one population helps another refine its search direction through a hybrid update strategy [8].

• Termination Check: The cycle repeats until a termination condition is met (e.g., a maximum number of generations, convergence stagnation, or a target solution quality is reached).

Performance Evaluation and Metrics

Evaluating an EMTO algorithm requires metrics that capture both per-task performance and the synergistic benefits of multitasking.

• Average Best Fitness (ABF): The mean of the best fitness values found for each task over multiple runs. This indicates the baseline optimization capability [7].
• Multitasking Performance Gain (MPG): A metric that quantifies the improvement achieved by multitasking over solving the tasks independently. For a given task, ( \text{MPG}i = (\text{Fitness}{STO} - \text{Fitness}{EMTO}) / \text{Fitness}{STO} ), where STO is Single-Task Optimization. A positive average MPG across tasks indicates successful knowledge transfer [1].
• Convergence Speed: The number of generations or function evaluations required to reach a satisfactory solution. A key claimed advantage of EMTO is faster convergence due to positive knowledge transfer [1].

Evolutionary Multitask Optimization has firmly established itself as a powerful and versatile paradigm within computational optimization. By moving beyond the traditional single-task focus, EMTO offers a framework that more closely mirrors complex real-world challenges, where multiple objectives must be balanced simultaneously. The core concepts of knowledge transfer, embodied in algorithms like the Multifactorial Evolutionary Algorithm, provide a mechanism to exploit synergies between tasks, leading to gains in efficiency and solution quality [1] [5].

For researchers and professionals in fields like drug discovery, the implications are profound. EMTO provides a structured methodology to navigate the high-dimensional, constrained, and often contradictory landscape of molecular optimization [6]. As the field continues to mature, addressing fundamental challenges such as the robust mitigation of negative transfer, the development of fair and comprehensive benchmarks, and the creation of more efficient knowledge transfer mechanisms will be critical [2]. Future progress will likely hinge on a deeper theoretical understanding of task relatedness and the development of more adaptive algorithms that can autonomously learn the best strategies for knowledge exchange, further solidifying EMTO's role as an indispensable tool for tackling the most intricate optimization problems in science and industry.

Evolutionary Multi-Task Optimization (EMTO) represents a paradigm shift in how evolutionary algorithms solve complex problems. Unlike traditional evolutionary approaches that tackle optimization tasks in isolation, EMTO solves multiple self-contained optimization tasks simultaneously within a single run by leveraging knowledge transfer between them [10]. This approach mirrors human cognitive capabilities where knowledge gained from solving one problem constructively informs the solution of another, rather than starting each new problem from scratch [1].

The foundational algorithm for this emerging field is the Multifactorial Evolutionary Algorithm (MFEA), introduced by Gupta et al., which creates a multi-task environment where a single population evolves to solve multiple tasks concurrently [1]. In MFEA, each task is treated as a unique "cultural factor" influencing the population's evolution, with knowledge transfer occurring through specialized algorithmic modules—assortative mating and selective imitation [1]. The mathematical formulation of an MTO problem consisting of K optimization tasks seeks to find optimal solutions {x1, x2, ..., x*K} for all tasks simultaneously [7].

The Mechanism of Implicit Parallelism

Fundamental Principles

Implicit parallelism in EMTO arises from the inherent ability of a single population to simultaneously address multiple optimization tasks while facilitating automatic knowledge exchange between them. This parallelism differs fundamentally from traditional parallel computing; rather than simply distributing computational load, it creates a symbiotic environment where tasks cooperatively evolve toward better solutions [1]. The population-based search nature of evolutionary algorithms provides a natural foundation for this mechanism, as a diverse population can maintain and process genetic material relevant to different tasks within a unified gene pool [10].

This approach mimics a key characteristic of natural evolution, which Professor Yaqing Hou describes as "a massive multi-task engine where each niche forms a task in an otherwise complex multifaceted fitness landscape, and the population of all living organisms is simultaneously evolving to survive in one niche or the other" [11]. In this biological metaphor, genetic material evolved for one ecological niche may prove beneficial for another, facilitating evolutionary leaps through inter-task genetic transfers—a phenomenon that EMTO algorithms systematically exploit.

Implementation Architecture

The architectural implementation of implicit parallelism occurs through several key mechanisms. Skill factor assignment ensures each individual in the population is evaluated against a specific task, with the scalar fitness serving as a unified performance metric across all tasks [10]. Assortative mating allows individuals with similar skill factors to preferentially mate, maintaining task-specific search trajectories, while vertical cultural transmission enables offspring to inherit skill factors from parents, preserving valuable task-specific knowledge across generations [1].

The implicit transfer of knowledge occurs when genetic material from individuals optimized for one task influences the evolutionary path of individuals focused on another task during crossover operations [7]. This creates a continuous, automated exchange of potentially beneficial genetic material without requiring explicit mapping between task domains.

Table 1: Core Components Enabling Implicit Parallelism in EMTO

Component	Function	Implementation in MFEA
Unified Search Space	Creates common representation for multiple tasks	Chromosomal encoding that accommodates all task dimensions
Skill Factor	Identifies which task an individual is optimized for	Task-specific evaluation and tagging of individuals
Scalar Fitness	Provides unified performance measure across tasks	Conversion of task-specific performance to comparable fitness values
Assortative Mating	Controls transfer between task groups	Mating preference based on skill factor similarity

Accelerated Convergence Through Knowledge Transfer

Mechanisms of Convergence Acceleration

The accelerated convergence observed in EMTO systems stems from the constructive exchange of useful genetic material between tasks during optimization. When solving multiple related tasks simultaneously, knowledge transfer allows the algorithm to bypass redundant search efforts by leveraging discovered patterns, promising regions, or structural information from one task to benefit another [1]. This cross-pollination of solutions creates a compounding effect where the collective knowledge of the population grows faster than would be possible when solving tasks in isolation.

Theoretical analyses have confirmed EMTO's superiority over traditional single-task optimization in convergence speed [1]. This acceleration is particularly pronounced when tasks share commonalities in their fitness landscapes, global optima locations, or problem structures [11]. As the algorithm progresses, the continuous knowledge exchange creates a virtuous cycle where improvements in one task catalyze improvements in others, leading to progressively faster refinement across all tasks.

Quantitative Performance Evidence

Recent empirical studies provide compelling quantitative evidence of EMTO's convergence advantages. In comprehensive benchmarking experiments, EMTO algorithms have demonstrated significant performance improvements across various test suites, including the CEC 2017 evolutionary MTO competition suite and more recent CEC 2025 competition benchmarks [7] [11].

Table 2: Convergence Performance Comparison Across Algorithm Types

Algorithm Type	Convergence Speed	Solution Quality	Negative Transfer Risk
Single-Task EA	Baseline	Baseline	Not applicable
Basic MFEA	25-40% improvement	15-30% improvement	High for dissimilar tasks
Advanced EMTO (MFEA-MDSGSS)	50-70% improvement	35-60% improvement	Controlled via specialized mechanisms

The performance gains are especially notable in complex optimization scenarios. For example, the proposed MFEA-MDSGSS algorithm, which incorporates multidimensional scaling and golden section search, demonstrated superior overall performance in experiments conducted on both single-objective and multi-objective MTO problems [7]. These improvements manifest not only in faster initial convergence but also in the ability to escape local optima and discover higher-quality solutions than single-task approaches.

Experimental Protocols and Benchmarking

Standard Evaluation Methodology

Robust experimental evaluation of EMTO algorithms requires standardized protocols to ensure fair comparison and reproducible results. The CEC 2025 Competition on Evolutionary Multi-task Optimization provides comprehensive guidelines for benchmarking both single-objective and multi-objective MTO algorithms [11].

For Multi-task Single-Objective Optimization (MTSOO) evaluation, the test suite contains nine complex MTO problems (each with two tasks) and ten 50-task MTO benchmark problems. Each algorithm should be executed for 30 independent runs with different random seeds, with maximum function evaluations (maxFEs) set to 200,000 for 2-task problems and 5,000,000 for 50-task problems [11]. Critical performance metrics include the Best Function Error Value (BFEV) recorded at predefined evaluation checkpoints throughout the optimization process.

For Multi-task Multi-Objective Optimization (MTMOO), the test suite similarly includes nine complex problems and ten 50-task benchmarks. The Inverted Generational Distance (IGD) metric should be recorded at evaluation checkpoints to assess convergence and diversity performance [11]. The same 30-run protocol applies, with algorithms required to use identical parameter settings across all benchmark problems to prevent over-specialization.

Workflow for EMTO Benchmarking

The following diagram illustrates the standardized experimental workflow for comprehensive EMTO evaluation:

Advanced MFEA-MDSGSS Protocol

Algorithm Specification

The MFEA-MDSGSS algorithm represents a significant advancement in addressing key EMTO limitations, particularly the risk of negative transfer between tasks and premature convergence. This algorithm incorporates two innovative components: a linear domain adaptation method based on multidimensional scaling (MDS) and a linear mapping strategy utilizing golden section search (GSS) [7].

The MDS-based linear domain adaptation method addresses negative transfer by identifying a low-dimensional intrinsic manifold for each task, enabling robust linear mapping in a compact latent space. This facilitates effective knowledge transfer even between tasks with different dimensionalities [7]. Meanwhile, the GSS-based linear mapping strategy promotes exploration of new regions in the search space, helping populations escape local optima and maintain diversity throughout the optimization process [7].

Implementation Protocol

Implementing MFEA-MDSGSS requires careful attention to both components. The MDS-based domain adaptation involves constructing low-dimensional subspaces for each task, learning mapping matrices between subspaces of different tasks, aligning the subspaces to enable cross-task solution mapping, and executing knowledge transfer in the aligned latent space [7].

The GSS-based exploration enhancement involves identifying stagnation points where task improvement plateaus, applying golden section search to explore promising directions from the current population, generating new candidate solutions through linear mapping, and selectively integrating successful candidates into the population [7].

Implementation requires specific parameter configuration. For MDS operations, the intrinsic dimensionality for each task should be determined through eigenvalue analysis of the distance matrix. For GSS operations, the golden ratio (φ ≈ 0.618) guides the sectioning process, with boundary updates following Fibonacci sequences. Population management should maintain 70-80% of individuals through standard evolutionary operations and 20-30% through GSS-enhanced exploration [7].

The following diagram illustrates the integrated workflow of the MFEA-MDSGSS algorithm:

The Scientist's Toolkit: Research Reagent Solutions

Table 3: Essential Research Components for EMTO Implementation

Research Component	Function	Implementation Example
CEC Benchmark Suites	Standardized performance evaluation	CEC 2017 & CEC 2025 MTO test problems [7] [11]
Multidimensional Scaling (MDS)	Enables cross-task knowledge transfer	Dimensionality reduction for task alignment in MFEA-MDSGSS [7]
Golden Section Search (GSS)	Prevents premature convergence	Systematic exploration in promising directions [7]
Skill Factor Mechanism	Tracks task specialization	Assigning individuals to specific optimization tasks [10]
Scalar Fitness Metric	Enables cross-task performance comparison	Unified ranking system for multi-task environments [10]
Assortative Mating Control	Regulates knowledge transfer	Probability-based mating restriction by skill factors [1]
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The dual advantages of implicit parallelism and accelerated convergence position Evolutionary Multi-Task Optimization as a transformative approach for complex problem-solving domains. By harnessing the implicit parallelism inherent in population-based search and structuring knowledge transfer mechanisms between tasks, EMTO algorithms achieve accelerated convergence while maintaining solution diversity. The continued refinement of algorithms like MFEA-MDSGSS, with sophisticated transfer control mechanisms and exploration enhancements, promises further performance gains across increasingly complex single-objective and multi-objective optimization landscapes. For researchers and practitioners, these advances offer powerful tools for tackling computationally challenging problems where traditional single-task approaches reach their limitations.

Evolutionary Multi-Task Optimization (EMTO) is a paradigm that enables the simultaneous solution of multiple optimization problems by exploiting their inherent correlations. The core principle is that valuable knowledge exists across tasks, and its transfer can enhance convergence speed and solution quality for all problems involved. A critical aspect of EMTO algorithm design involves the method of knowledge transfer (KT), which fundamentally divides approaches into two categories: implicit and explicit knowledge transfer. Implicit methods facilitate transfer through algorithmic operations like crossover and cultural imitation, where knowledge is contained within the solutions themselves. In contrast, explicit methods construct direct mappings or models to represent and transfer specific problem-solving knowledge. This taxonomy provides a structured framework for researchers to understand, select, and implement appropriate KT strategies, particularly within computationally intensive fields like drug development where EMTO offers significant potential for accelerating discovery pipelines.

Theoretical Foundation of Knowledge Transfer in EMTO

Implicit Knowledge Transfer

Implicit knowledge transfer leverages the implicit parallelism of population-based searches, where knowledge is encoded directly within the genotypes of individuals and transferred through evolutionary operations. The seminal algorithm in this category is the Multifactorial Evolutionary Algorithm (MFEA), which operates on a unified search space. In MFEA, knowledge transfer occurs primarily through assortative mating and selective imitation. When two parent individuals from different tasks are selected for crossover based on a random mating probability (rmp), they exchange genetic material, thereby transferring knowledge. The offspring then imitate the skill factor (i.e., the assigned task) of one parent. This process does not require an explicit model of the knowledge being transferred; it relies on the belief that beneficial genetic building blocks will propagate through the population. The simplicity of this approach is its strength, but it also introduces a risk of negative transfer—where the exchange of unhelpful or misleading information between unrelated tasks can degrade performance.

Explicit Knowledge Transfer

Explicit knowledge transfer strategies were developed to mitigate the risks of negative transfer and to exert greater control over the transfer process. These methods involve the active extraction and representation of knowledge before it is transferred. Common mechanisms include the use of probabilistic models, mapping matrices, or auto-encoders to capture and translate knowledge. For instance, some algorithms draw compact probabilistic models from elite individuals of a source task, which then guide the search in a target task. Others, like the Domain Adaptation Multitask Optimization (DAMTO) algorithm, use transfer component analysis to map populations from different tasks into a shared feature space, reducing the probability of negative transfer by aligning their distributions. Explicit transfer allows for more informed and selective knowledge exchange, often leading to more robust performance, though it incurs additional computational overhead for model building and management.

Comparative Analysis

The choice between implicit and explicit transfer is a trade-off between simplicity, computational cost, and robustness.

Table 1: Comparison of Implicit and Explicit Knowledge Transfer Strategies

Feature	Implicit Knowledge Transfer	Explicit Knowledge Transfer
Core Mechanism	Evolutionary operations (e.g., crossover, imitation)	Constructed models or mappings (e.g., probabilistic models, auto-encoders)
Knowledge Representation	Encoded within solution chromosomes	External, explicit models or mapped solutions
Computational Overhead	Relatively low	Higher due to model construction and maintenance
Risk of Negative Transfer	Higher, due to uncontrolled exchange	Lower, through similarity measurement and selective transfer
Implementation Complexity	Lower, easier to integrate	Higher, requires design of transfer model
Typical Algorithms	MFEA, MFDE	MOMFEA-SADE, DAMTO, Algorithms with explicit genetic transfer

Advanced EMTO Frameworks and Protocols

Hybrid and Adaptive Transfer Approaches

Recent research focuses on hybridizing implicit and explicit strategies or making them adaptive to dynamically manage inter-task relatedness. The Collaborative Knowledge Transfer-based Multiobjective Multitask PSO (CKT-MMPSO) is a notable example that employs a bi-space knowledge reasoning method. This protocol extracts knowledge from both the search space (distribution of solutions) and the objective space (evolutionary information), thereby breaking the limitations of single-space knowledge. An information entropy-based mechanism then divides the evolutionary process into stages, adaptively activating different knowledge transfer patterns to balance convergence and diversity. Another advanced protocol, the Self-adaptive Multi-Task Differential Evolution (SaMTDE), uses a knowledge source pool for each task. It dynamically adjusts the probability of transferring knowledge from one task to another based on a historical record of successful and unsuccessful transfers, effectively learning the inter-task relatedness online.

Protocol for Implementing a Self-Adaptive Knowledge Transfer Strategy

The following protocol, inspired by SaMTPSO and SaMTDE, provides a detailed methodology for implementing an adaptive KT mechanism.

Objective: To dynamically control knowledge transfer between K optimization tasks in an EMTO environment, maximizing positive transfer and minimizing negative transfer. Materials: A computing environment capable of running evolutionary algorithms; a defined set of K related optimization tasks. Procedure:

• Initialization:
  • For each task ( Tt ) (where ( t = 1, ..., K )), initialize a sub-population ( popt ).
  • For each task, initialize a knowledge source pool containing all K tasks. Set the initial selection probability ( p{t,k} = 1/K ) for all source tasks ( k ) in the pool of ( Tt ).
  • Initialize a success memory and a failure memory for each task, both of length ( LP ) (e.g., ( LP = 50 ) generations).

• Evolutionary Cycle (per generation):

  • For each task ( Tt ):
    • For each individual in ( popt ):
      • Using a roulette wheel selection based on probabilities ( p{t,k} ), select a knowledge source task ( Tk ).
      • If ( Tk = Tt ), the individual evolves using only its own task's information.
      • If ( Tk \neq Tt ), the individual generates an offspring by incorporating knowledge (e.g., through crossover or mutation) from a randomly selected individual from ( popk ).
    • Evaluate all newly generated offspring for task ( Tt ).
  • Population Update: Combine parents and offspring, select the fittest individuals to form the new ( pop_t ).

• Probability Adaptation (every generation):

  • For each task ( Tt ) and each knowledge source ( Tk ), count the number of offspring generated via ( Tk ) that successfully entered the new population (( ns{t,k} )) and those that failed (( nf_{t,k} )).
  • Update the success rate for each source: ( SR{t,k} = \frac{\sum{j=g-LP+1}^{g} ns{t,k}^j}{\sum{j=g-LP+1}^{g} ns{t,k}^j + \sum{j=g-LP+1}^{g} nf_{t,k}^j + \epsilon} + bp ) where ( \epsilon ) is a small constant preventing division by zero, and ( bp ) is a base probability assigned to rarely used sources.
  • Update the selection probability: ( p{t,k} = SR{t,k} / \sum{i=1}^{K} SR{t,i} ).

• Focus Search Trigger:

  • Monitor the success memory for ( Tt ). If for the last ( LP ) generations, ( ns{t,k} = 0 ) for all ( k \neq t ), activate the focus search strategy. This forces all individuals in ( popt ) to only use ( Tt ) as their knowledge source for a predefined period, preventing continued negative transfer.

• Termination: Repeat the Evolutionary Cycle until a termination criterion is met (e.g., maximum generations, convergence).

Diagram 1: Self-adaptive knowledge transfer protocol workflow.

Application Notes for Drug Development

The parallel and knowledge-sharing nature of EMTO is exceptionally suited for the multi-faceted challenges in drug development, where multiple related optimization problems are commonplace.

Virtual Screening and Molecular Docking: A multi-task scenario can be established by considering docking against a series of structurally related target proteins (e.g., a protein family like kinases) or against different conformations of the same target. An EMTO algorithm can simultaneously optimize multiple docking tasks. Implicit KT can help discover molecular scaffolds with broad selectivity across the target family, while explicit KT can be used to build a mapping between the energy landscapes of different proteins, guiding the search towards promising chemical spaces more efficiently than single-task optimization.

Pharmacokinetic and Toxicity Prediction: Developing quantitative structure-property relationship (QSPR) models for properties like absorption, distribution, metabolism, excretion, and toxicity (ADMET) is crucial. These properties, while distinct, are often governed by underlying shared molecular descriptors. EMTO can be applied to simultaneously train multiple QSPR models. Knowledge transfer between tasks can lead to more robust and generalizable models, especially for toxicity endpoints with limited data, by leveraging information from data-rich properties.

Protocol for Multi-Objective Drug Optimization with CKT-MMPSO: Objective: To discover lead compounds that simultaneously optimize potency against a primary target, minimize binding to an off-target, and satisfy drug-likeness rules (Lipinski's Rule of Five). Materials: A compound library; molecular docking software for the primary and off-target; a scoring function for drug-likeness. Procedure:

• Problem Formulation: Define a multi-objective, multi-task problem.
  • Task 1: Minimize docking score to the primary target.
  • Task 2: Maximize docking score (minimize binding) to the off-target.
  • Objective 3: Maximize drug-likeness score. Each candidate solution is a molecular representation (e.g., a fingerprint or a SMILES string translation).

• Algorithm Setup: Configure the CKT-MMPSO algorithm.

  • The bi-space knowledge reasoning (bi-SKR) component will extract information from the molecular descriptor space (search space) and the multi-objective fitness space (objective space).
  • The information entropy-based collaborative KT (IECKT) will manage transfer between the two docking tasks based on the evolutionary stage.

• Execution: Run the CKT-MMPSO. The algorithm will evolve a population of molecules, using KT to transfer promising sub-structures or property profiles between the two tasks (primary and off-target optimization).

• Output Analysis: The result is a Pareto-optimal set of non-dominated solutions, representing the best trade-offs between high on-target activity, low off-target activity, and good drug-likeness. This set provides a shortlist of candidate molecules for synthesis and experimental validation.

Table 2: Key Research Reagents and Computational Tools for EMTO in Drug Development

Tool/Reagent	Type	Function in EMTO Protocol
CEC17/CEC22 MTO Benchmarks	Software Benchmark	Standardized test problems for validating and tuning EMTO algorithms before application to molecular problems.
Molecular Descriptor Libraries	Computational Tool	Provides a unified numerical search space (e.g., ECFP fingerprints) for molecular representation in KT.
AutoDock Vina / GOLD	Docking Software	Acts as the "objective function evaluator" for tasks related to binding affinity.
RDKit	Cheminformatics Library	Used for molecule manipulation, descriptor calculation, and enforcing chemical constraints during evolution.
Knowledge Source Pool	Algorithmic Component	Maintains and manages the different optimization tasks (e.g., different protein targets) for selective knowledge transfer.
Success/Failure Memory	Algorithmic Component	Tracks the history of inter-task transfers to enable adaptive probability adjustment and mitigate negative transfer.

The Scientist's Toolkit

This section provides a concise reference for the core components necessary for designing and executing EMTO experiments.

Table 3: Essential Components for an EMTO Research Framework

Category	Component	Description & Implementation Note
Core Algorithm	Evolutionary Solver	A base EA (e.g., DE, PSO, GA). Choose based on problem nature; DE is often effective for continuous domains.
Knowledge Transfer Mechanism	Implicit Crossover	Implement using a unified representation space and a controlled rmp parameter.
	Explicit Mapping	Requires a separate function to build a model (e.g., a probability distribution or a regression model) from elite solutions and use it to generate new solutions in a target task.
Adaptation Layer	Probability Manager	A software module that dynamically adjusts KT parameters (like rmp or source probabilities) based on online performance metrics.
Task Management	Skill Factor / Population Tagging	A system to assign and track which task a solution is being evaluated for, crucial for implicit MFEA-like algorithms.
Benchmarking & Validation	Single-Task EA Baselines	Essential for comparative analysis. Always run traditional single-task EAs on each problem to quantify the performance gain from multitasking.
Domain-Specific Interfacing	Solution Decoder	A function that translates a general-purpose EA genotype (a real-valued vector) into a domain-specific solution (e.g., a molecular structure).
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Evolutionary Multitask Optimization (EMTO) represents a paradigm shift in computational problem-solving, moving beyond traditional single-task optimization frameworks. This approach leverages the implicit parallelism of evolutionary algorithms and the transfer of knowledge across tasks to solve multiple optimization problems simultaneously [12]. The core premise is that many real-world optimization tasks are related, and the experience gained from solving one problem can be leveraged to accelerate and improve the solution of others [13] [14]. This article explores the expanding research landscape of EMTO, analyzing publication trends, emerging methodologies, and practical applications across scientific domains.

The foundational algorithm in this field, the Multifactorial Evolutionary Algorithm (MFEA), introduced the concept of multifactorial optimization, where a single population evolves with skill factors that determine how individuals are evaluated on different tasks [14] [15]. Since this pioneering work, the EMTO research landscape has diversified substantially, with new algorithms addressing key challenges such as negative transfer (where inappropriate knowledge hinders progress), task dissimilarity, and computational efficiency [7] [13] [16]. The growing interest in this field is evidenced by the proliferation of algorithm variants and their application to increasingly complex real-world problems.

Quantitative Analysis of Publication Trends

Evolution of EMTO Algorithm Development

Table 1: Chronological Development of Key EMTO Algorithms

Year	Algorithm Name	Core Innovation	Base Evolutionary Algorithm
2016	MFEA	Introduced multifactorial inheritance & cultural transmission [14] [15]	Genetic Algorithm
2017	Multiobjective MFEA	Extended multitasking to multiobjective optimization [15]	Genetic Algorithm
2019	MFEA with Online Transfer Parameter Estimation (MFEA-II)	Adaptive knowledge transfer via online parameter estimation [7] [16]	Genetic Algorithm
2019	Evolutionary Multitasking via Explicit Autoencoding	Explicit transfer through autoencoding networks [7] [15]	Genetic Algorithm
2020	Self-Regulated EMTO	Self-regulated transfer intensity based on task relatedness [15]	Genetic Algorithm
2021	Multifactorial Differential Evolution	Incorporated differential evolution into MFEA framework [13]	Differential Evolution
2022	MFEA with Adaptive Knowledge Transfer	Adaptive configuration of crossover operators [7] [16]	Genetic Algorithm
2024	MFEA-MDSGSS	Multidimensional scaling & golden section search to reduce negative transfer [7]	Genetic Algorithm
2024	MFDE-AMKT	Adaptive Gaussian-mixture-model-based knowledge transfer [13]	Differential Evolution
2024	Multitask Level-Based Learning Swarm Optimizer	First major PSO-based approach with level-based learning [14]	Particle Swarm Optimization

The development of EMTO algorithms has followed a clear trajectory from foundational frameworks to increasingly sophisticated and specialized methods. Early algorithms like MFEA established the basic mechanisms for implicit knowledge transfer [14]. The subsequent generation of algorithms focused on adapting transfer parameters (e.g., MFEA-II) and enabling explicit transfer mechanisms (e.g., EMEA) to reduce negative transfer [7] [15]. Recent approaches (2022-2024) have incorporated more advanced machine learning techniques and probabilistic models to further enhance the efficiency and robustness of knowledge transfer [7] [13].

A notable trend is the expansion from Genetic Algorithms to other evolutionary paradigms. While GA-based approaches dominated the early landscape, recent years have seen significant growth in Differential Evolution-based approaches (e.g., MFDE, MFDE-AMKT) and the emergence of PSO-based methods (e.g., MTLLSO) [13] [14]. This diversification reflects the field's maturation and the recognition that different base optimizers may be suited to different problem types.

Research Focus Areas and Application Domains

Table 2: Research Focus Areas in EMTO (2020-2025)

Research Focus Area	Proportion of Publications	Key Challenges Addressed	Representative Algorithms
Transfer Mechanism Design	~35%	Negative transfer, transfer efficiency [7] [13]	MFEA-MDSGSS [7], MFDE-AMKT [13]
Many-Task Optimization	~20%	Computational complexity, many-task scalability [16]	MGAD [16], GMFEA [16]
Multiobjective EMTO	~15%	Balancing multiple objectives across tasks [14] [15]	Multiobjective MFEA [15]
Real-World Applications	~15%	Domain adaptation, practical constraints [4] [16]	P-MFEA [16], BACOHBA [16]
Algorithmic Frameworks	~10%	Generalization, benchmarking [12] [15]	SREMTO [15]
Theoretical Foundations	~5%	Convergence analysis, complexity theory [15]	First Complexity Results for EKT [15]

The application domains for EMTO have expanded significantly, demonstrating the versatility of the paradigm. Key application areas highlighted in recent literature include:

• Industrial Automation and Robotics: EMTO methods are being applied to problems such as robot path planning, task scheduling, and predictive maintenance [17]. The ability to simultaneously optimize multiple related control or planning tasks is particularly valuable in these domains.
• Transportation and Logistics: The multi-arbitrary vehicle routing problem has been addressed using P-MFEA, demonstrating the practical utility of EMTO in complex logistics optimization [16].
• Energy Systems: Photovoltaic model parameter optimization has been tackled using SGDE, showcasing EMTO's applicability to sustainable energy challenges [16].
• Healthcare and Drug Development: While still emerging, applications in medical image analysis, treatment optimization, and biomolecular structure prediction represent promising frontiers [14] [17]. The recent development of algorithms for UAV inspection tasks in healthcare contexts (BACOHBA) further demonstrates this potential [16].
• Network Design and Communications: Network robustness optimization has been improved using MFEA-net, illustrating EMTO's relevance to telecommunications and infrastructure planning [16].

Experimental Protocols in EMTO Research

Standardized Benchmarking Methodology

Table 3: Standard EMTO Benchmark Evaluation Framework

Component	Specification	Purpose
Benchmark Suite	CEC2017 Evolutionary MTO Competition Problems [7] [14]	Standardized performance comparison
Performance Metrics	Average Fitness Gain, Convergence Speed, Negative Transfer Frequency [7] [13]	Quantify optimization performance
Statistical Validation	Wilcoxon Signed-Rank Test (p < 0.05) [7] [14]	Ensure statistical significance
Comparison Baseline	Single-Task Evolutionary Algorithms [13]	Establish performance improvement
Computational Environment	Common hardware/platform specifications [7]	Ensure reproducibility

The experimental protocol for evaluating EMTO algorithms follows a standardized approach to ensure fair comparison and reproducibility. The methodology typically includes:

• Problem Selection: Researchers select a diverse set of problems from established benchmark suites, particularly the CEC2017 Evolutionary MTO competition problems [7] [14]. These benchmarks include tasks with varying degrees of similarity and different landscape characteristics to thoroughly evaluate algorithm performance.

• Algorithm Configuration: Each algorithm is configured with its optimal parameter settings as reported in the respective literature. Population sizes are typically set between 100-500 individuals, with evolution continuing for 1000-5000 generations depending on problem complexity [7].

• Performance Assessment: Multiple independent runs (typically 30) are conducted to account for stochastic variations. Performance is evaluated using metrics such as:

  • Average Fitness Gain: Improvement in objective function value compared to baseline single-task optimization [13].
  • Convergence Speed: Number of generations or function evaluations required to reach a target solution quality [7].
  • Negative Transfer Frequency: Proportion of knowledge transfer events that degrade performance [7] [13].

• Statistical Analysis: Results undergo rigorous statistical testing, typically using Wilcoxon signed-rank tests with a significance level of p < 0.05, to confirm that performance differences are statistically significant [7] [14].

Protocol for Knowledge Transfer Mechanism Validation

A specialized experimental protocol is used to validate knowledge transfer mechanisms, which are central to EMTO performance:

The knowledge transfer validation protocol focuses on isolating and quantifying the effects of cross-task knowledge exchange:

• Similarity Assessment: Prior to transfer, task similarity is quantified using measures such as Maximum Mean Discrepancy (MMD) or overlap of subpopulation distributions [13] [16]. This establishes a baseline for expected transfer utility.

• Controlled Transfer Experiments: Knowledge transfer is selectively activated and deactivated in controlled experiments. This involves:

  • Comparing performance with and without transfer mechanisms enabled.
  • Varying transfer frequency and intensity to identify optimal settings [16].
  • Islecting different source-task combinations to evaluate transfer specificity.

• Negative Transfer Quantification: The incidence of negative transfer is carefully monitored by tracking performance degradation following knowledge exchange events. This is quantified as the percentage of transfers that result in fitness deterioration [7] [13].

• Ablation Studies: Comprehensive ablation studies are conducted to isolate the contribution of individual algorithm components. For example, in MFEA-MDSGSS, separate experiments evaluate the individual contributions of the MDS-based linear domain adaptation and the GSS-based linear mapping strategy [7].

Visualization of EMTO Workflows

Generalized EMTO Algorithmic Framework

The generalized EMTO workflow illustrates the core processes common to most algorithms in this domain. The process begins with a unified population initialization where a single population is created to address all tasks simultaneously. Each individual then undergoes factorial cost evaluation, where their fitness is assessed across all tasks [14]. Based on these evaluations, skill factors are assigned to each individual, indicating which task they are most proficient at solving [14].

The key knowledge transfer occurs during assortative mating, where individuals may reproduce with partners from the same or different tasks, enabling implicit knowledge exchange [14]. This is followed by vertical cultural transmission, where offspring inherit traits from parents potentially skilled at different tasks [14]. Finally, environmental selection preserves the fittest individuals for the next generation, and the process repeats until convergence criteria are met [14].

Knowledge Transfer Taxonomy in EMTO

The knowledge transfer methods in EMTO can be categorized into three main approaches, each with distinct characteristics and representative algorithms:

• Implicit Transfer: These methods do not explicitly transform or adapt knowledge between tasks. Instead, they rely on mechanisms like assortative mating (as in MFEA) where chromosomes from different tasks recombine, implicitly sharing genetic material [7] [14]. MFEA-II extends this approach with online transfer parameter estimation to adaptively control the transfer process [7] [16].

• Explicit Transfer: These methods employ dedicated mechanisms to directly transform and transfer knowledge between tasks. Explicit autoencoding (EMEA) uses neural networks to learn mappings between task representations [7] [15], while Linear Domain Adaptation (LDA) methods learn linear transformations to align task subspaces [7] [13].

• Model-Based Transfer: Recent approaches use probabilistic models to capture and transfer knowledge. Gaussian Mixture Models (as in MFDE-AMKT) represent subpopulation distributions and enable comprehensive knowledge transfer [13], while Anomaly Detection Transfer filters out potentially harmful knowledge before transfer [16].

Table 4: Essential Research Reagents and Computational Resources for EMTO

Resource Category	Specific Tools/Platforms	Function in EMTO Research	Application Examples
Benchmark Problems	CEC2017 Evolutionary MTO Suite [7] [14]	Standardized algorithm performance evaluation	Comparative studies of new EMTO algorithms
Evolutionary Algorithm Frameworks	Genetic Algorithms, Differential Evolution, Particle Swarm Optimization [13] [14]	Base optimization engines for multitask extension	MFEA (GA-based), MFDE (DE-based), MTLLSO (PSO-based)
Similarity Measurement Tools	Maximum Mean Discrepancy, Kullback-Leibler Divergence, Grey Relational Analysis [16]	Quantify inter-task relationships for transfer guidance	Source task selection in MGTS strategy [16]
Knowledge Transfer Mechanisms	Linear/Nonlinear Mapping, Model-Based Transfer, Anomaly Detection [7] [13] [16]	Enable cross-task knowledge exchange	MDS-based LDA in MFEA-MDSGSS [7]
Programming Environments	Python, MATLAB, C++ with CUDA for GPU acceleration [17] [15]	Algorithm implementation and experimentation	High-performance EMTO implementation [15]
Performance Metrics	Average Fitness Gain, Convergence Speed, Negative Transfer Frequency [7] [13]	Quantitative algorithm evaluation	Statistical comparison of EMTO variants

The experimental toolkit for EMTO research comprises several essential components that enable comprehensive investigation and development:

• Benchmarking Resources: The CEC2017 Evolutionary MTO competition problems serve as the standard benchmark for comparing algorithm performance [7] [14]. These include diverse problem types with varying degrees of inter-task similarity to thoroughly evaluate algorithm robustness.

• Similarity Assessment Tools: Accurate measurement of task relationships is crucial for effective knowledge transfer. Maximum Mean Discrepancy provides a non-parametric measure of distribution similarity [16], while Grey Relational Analysis captures dynamic evolutionary trends [16]. These tools help predict transfer potential and reduce negative transfer.

• Transfer Mechanisms: The core innovations in EMTO often reside in novel transfer mechanisms. Multidimensional Scaling enables dimensionality reduction and subspace alignment for tasks with different dimensionalities [7]. Gaussian Mixture Models provide probabilistic representations of population distributions for comprehensive knowledge capture [13].

• Computational Infrastructure: As EMTO problems grow in scale and complexity, GPU-accelerated computing frameworks have become essential for handling thousands of tasks efficiently [15]. These platforms significantly reduce search time and enable research on many-task optimization problems.

The research landscape for Evolutionary Multitask Optimization has expanded dramatically since its inception, evolving from a niche concept to a robust paradigm with diverse methodologies and applications. The publication trends reveal a field in rapid maturation, with growing theoretical foundations, increasingly sophisticated algorithms, and expanding real-world applications. The emergence of specialized approaches for handling negative transfer, many-task scenarios, and diverse problem domains indicates the field's responsiveness to fundamental challenges.

Future research directions appear to be focusing on several key areas: scaling EMTO to handle very large numbers of tasks (many-task optimization), improving theoretical understanding of convergence and complexity, enhancing adaptability to dynamic environments, and expanding applications in critical domains such as healthcare, sustainable energy, and complex systems design. As EMTO continues to evolve, its integration with other advanced machine learning techniques and computational frameworks will likely open new frontiers in efficient, knowledge-sharing optimization systems.

Algorithmic Innovations and Real-World Deployment of EMTO

Evolutionary Multitask Optimization (EMTO) represents a paradigm shift in computational optimization, enabling the simultaneous solution of multiple, potentially interrelated, optimization tasks. The foundational Multifactorial Evolutionary Algorithm (MFEA), inspired by biocultural models of multifactorial inheritance, introduced a robust framework for such concurrent problem-solving by evolving a single population of individuals, each associated with a specific task through a skill factor [18]. However, a significant limitation of many early EMTO algorithms, including the canonical MFEA, is their reliance on a single type of evolutionary search operator (ESO), such as a specific genetic algorithm (GA) crossover and mutation scheme, throughout the entire search process [19].

The core challenge is that no single ESO is universally superior for all types of optimization problems. Different problems, characterized by varying fitness landscapes—such as those that are multimodal, deceptive, or high-dimensional—respond differently to the exploration and exploitation properties of various operators [20]. Using a single, fixed ESO for multiple diverse tasks can hinder performance, as the operator may be well-suited to one task but poorly matched to another, ultimately restricting the algorithm's adaptability and convergence precision [19].

This application note explores the frontier of EMTO research that moves beyond this single-operator constraint. We detail the theory, application, and protocol design for leveraging a spectrum of evolutionary search operators—including Genetic Algorithms (GA), Differential Evolution (DE), and Particle Swarm Optimization (PSO)—within a single multitasking environment. By systematically integrating and adaptively controlling these operators, researchers can achieve a more effective balance between global exploration and local exploitation across diverse tasks, leading to enhanced optimization performance and more robust solutions for complex scientific challenges, such as those encountered in drug development.

A Spectrum of Evolutionary Search Operators

The effectiveness of an EMTO algorithm is heavily influenced by the search dynamics of its constituent operators. The integration of a diverse portfolio of ESOs allows the algorithm to leverage the unique strengths of each, adapting to the distinct characteristics of different tasks. The following table summarizes the core mechanisms, strengths, and ideal use cases for three primary operators.

Table 1: Key Evolutionary Search Operators and Their Characteristics in EMTO

Operator	Core Mechanism	Key Strengths	Ideal for Task Types
Genetic Algorithm (GA)	Simulates natural selection using crossover (e.g., Simulated Binary Crossover - SBX) and mutation (e.g., Polynomial Mutation - PM) [21].	Strong exploration capabilities; effective at discovering diverse regions of the search space [19].	Tasks with complex, multi-modal fitness landscapes requiring broad exploration [19].
Differential Evolution (DE)	Utilizes vector differences for mutation (e.g., DE/rand/1: ( vi = x{r1} + F \cdot (x{r2} - x{r3}) )) and a crossover operation to generate trial vectors [19].	Strong exploitation and reliable local search; often demonstrates fast convergence [19].	Tasks with smoother, uni-modal landscapes or when fine-tuning solutions near a promising region is critical [19].
Particle Swarm Optimization (PSO)	Models social behavior; particles update their positions based on their own experience and the experience of neighboring particles [22].	Efficiently handles high-dimensional problems; effective balance between personal and social influence [22].	High-dimensional expensive problems (e.g., gene selection) where surrogate models may be employed [22] [23].

The selection of ESOs for an EMTO platform is not arbitrary. Empirical studies on benchmark problems like CEC17 have quantitatively demonstrated that the performance of ESOs is task-dependent. For instance, the DE/rand/1 operator has been shown to outperform GA-based operators on the CIHS and CIMS problems, whereas GA operators are more effective on the CILS problem [19]. This evidence underscores the necessity of moving beyond a one-size-fits-all approach.

Advanced Multi-Operator EMTO Frameworks and Protocols

Building on the understanding of individual operator strengths, researchers have developed sophisticated EMTO frameworks that manage multiple ESOs. These frameworks can be broadly categorized into fixed, random, and adaptive strategies, with the latter being the most advanced.

The Hybrid Operator-based MFEA (HOMFEA)

Concept: The HOMFEA framework innovates by integrating multiple advanced ESOs directly into the MFEA's evolutionary fabric. It employs a hybrid operator (HO) algorithm that combines the exploitation power of DE Mutation with the exploration capabilities of Simulated Binary Crossover (SBX) and Polynomial Mutation (PM) [21]. This creates a multi-layer search strategy within a single population.

Key Protocol: The core of HOMFEA lies in its memetic algorithm theory, where the hybrid operator is applied to generate offspring. This is coupled with a Vertical Cultural Transmission (VCT) algorithm, which governs the inheritance of skill factors from parents to offspring, thereby managing implicit knowledge transfer between tasks [21].

Table 2: Experimental Protocol for HOMFEA-based Inverse Design of Soft Network Materials

Step	Procedure	Parameters & Notes
1. Problem Formulation	Define the inverse-engineering design task: Reproduce a target J-shaped stress-strain curve of a biological tissue using a soft network material (SNM).	The objective function is the error between the target curve and the curve produced by a candidate SNM structure [21].
2. Population Initialization	Randomly generate an initial population of SNM designs. Use two distinct modeling methods (Mode-1, Mode-2) to increase diversity [21].	Population size (N) is typically set between 100-500 individuals. The two modes prevent premature convergence to a local optimum [21].
3. HOMFEA Execution	For each generation, create offspring using the Hybrid Operator (HO). Assign skill factors via Vertical Cultural Transmission (VCT). Evaluate fitness using finite element analysis (FEA) [21].	HO: Applies DE mutation, SBX, and PM. VCT: Uses assortative mating with a random mating probability (rmp) to control cross-task reproduction [21].
4. Model Validation	Validate the final optimal SNM design by comparing its FEA-simulated stress-strain curve against the target curve.	The HOMFEA-generated design should achieve higher accuracy than designs from semi-rational strategies [21].

The Adaptive Bi-Operator MFEA (BOMTEA)

Concept: BOMTEA represents a significant advancement in adaptive control. It maintains two ESOs—GA and DE—and dynamically adjusts their selection probability based on real-time performance feedback [19]. This allows the algorithm to automatically identify and favor the operator best suited for the current state of each task.

Key Protocol: BOMTEA's innovation is its performance-based adaptive strategy. The algorithm tracks the number of improved offspring generated by each ESO over a window of generations. The selection probability for an operator is then proportional to its recent success rate, creating a competitive environment that drives efficiency [19].

Diagram 1: BOMTEA Adaptive Operator Selection Workflow (Title: Adaptive ESO Control in BOMTEA)

Multi-Swarm and Surrogate-Assisted Approaches

Concept: For high-dimensional expensive problems (e.g., gene selection for tumor identification), a multi-swarm approach using PSO and DE in tandem, assisted by surrogate models, has proven effective [22] [23]. This framework often employs a hierarchical or cooperative structure.

Key Protocol: The Multi-Surrogate Assisted Multi-Tasking (MSAMT) algorithm first uses a space-partitioning strategy to divide the search space. Local surrogate models (e.g., Radial Basis Functions) are built for each region to guide global exploration. Near the current best solution, an ensemble surrogate is used for intensive local exploitation. The Generalized MFEA (G-MFEA) then optimizes these multiple surrogate models as related tasks [22].

The Scientist's Toolkit: Essential Research Reagents for Multi-Operator EMTO

Implementing and testing advanced multi-operator EMTO algorithms requires a suite of computational tools and benchmark problems.

Table 3: Research Reagent Solutions for Multi-Operator EMTO

Reagent / Tool	Function / Description	Application Example
CEC17 & CEC22 MTO Benchmarks	Standardized sets of multitask benchmark functions. Used to quantitatively compare and validate the performance of new EMTO algorithms against state-of-the-art methods [19].	Testing BOMTEA's adaptive operator selection against fixed-operator MFEAs [19].
Python with DEAP, PyGMO, or PlatypUS	Open-source libraries for evolutionary computation. Provide pre-implemented ESOs (GA, DE, PSO) and tools for building custom EMTO frameworks [21].	Prototyping a new hybrid operator combining DE mutation and SBX crossover [21].
ABAQUS with Python API	Finite Element Analysis (FEA) software. Used for high-fidelity fitness evaluation in engineering design problems (e.g., simulating stress-strain curves of soft network materials) [21].	Validating the mechanical performance of an SNM design generated by HOMFEA [21].
Radial Basis Function (RBF) / Kriging Surrogates	Machine learning models used as inexpensive approximations of costly objective functions. Critical for making high-dimensional expensive problems tractable [22].	Building local and global surrogates in the MSAMT algorithm to reduce computational cost [22].
Random Mating Probability (rmp)	A key cultural transmission parameter in MFEA-based frameworks. Controls the frequency of knowledge transfer (crossover) between individuals of different tasks [18].	Tuning the balance between cross-task knowledge transfer and independent task optimization to mitigate negative transfer [18].
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Visualization of Knowledge Transfer and Operator Interaction

A critical aspect of multi-operator EMTO is understanding how knowledge is exchanged both between tasks and between different search strategies. The following diagram illustrates this complex interaction within a single population.

Diagram 2: Multi-Operator Knowledge Transfer Ecosystem (Title: Knowledge Transfer in Multi-Operator EMTO)

The strategic integration of a spectrum of evolutionary search operators—GA, DE, and PSO—marks a significant evolution in the field of Evolutionary Multitask Optimization. Moving beyond the single-operator confinement of the early MFEA model, advanced frameworks like HOMFEA and BOMTEA demonstrate that adaptive and hybridized operator strategies are paramount to achieving superior performance. These methods dynamically leverage the unique exploration-exploitation characteristics of each operator, leading to enhanced convergence precision, robustness, and scalability across a wider range of complex, real-world problems. For researchers in drug development and biomedical sciences, where problems like high-dimensional gene selection and multi-objective molecule optimization are prevalent, adopting these sophisticated multi-operator EMTO platforms offers a powerful pathway to accelerating discovery and optimizing outcomes. The future of EMTO lies in increasingly intelligent, self-adaptive systems that not only select operators but also synthesize new, problem-specific search behaviors.

Evolutionary Multitask Optimization (EMTO) represents a paradigm shift in evolutionary computation. It moves beyond solving single, isolated problems to addressing multiple optimization tasks concurrently. The core premise is that by leveraging implicit parallelism and promoting knowledge transfer between related tasks, EMTO can accelerate convergence and improve solution quality for all tasks involved [24]. However, a significant historical limitation in this field has been the reliance on a single evolutionary search operator (ESO) throughout the evolutionary process. This "one-size-fits-all" approach is inherently limited, as no single operator is optimally suited for all problem types or all stages of the evolutionary search on a given problem [19].

The integration of hybrid and adaptive algorithms addresses this critical shortcoming. These algorithms strategically combine the strengths of different operators, such as the global exploration capabilities of Differential Evolution (DE) and the local exploitation strengths of Genetic Algorithms (GA). Furthermore, they incorporate adaptive mechanisms to dynamically control which operator is applied, based on its real-time performance. This enables the algorithm to automatically discover and exploit the most suitable search operator for various tasks, leading to significant performance improvements, as demonstrated on standard benchmarks like CEC17 and CEC22 [19]. This Application Note provides a detailed guide to the protocols and practical implementation of such hybrid and adaptive strategies within EMTO.

Core Principles and Quantitative Comparisons

The Case for Hybrid Operators

The limitation of single-operator algorithms is empirically demonstrated in studies comparing the Multifactorial Evolutionary Algorithm (MFEA), which uses a GA, and the Multifactorial DE (MFDE), which uses DE/rand/1. Research shows that their performance is task-dependent; for instance, MFDE outperforms MFEA on complete-intersection, high-similarity (CIHS) and medium-similarity (CIMS) problems, while MFEA is more effective on complete-intersection, low-similarity (CILS) problems [19]. This evidence underscores that the suitability of an ESO is contingent on the specific characteristics of the task at hand, making a strong case for a hybrid approach.

Table 1: Performance Comparison of Single-Operator vs. Hybrid Algorithms on MTO Benchmarks

Algorithm	Primary Operator(s)	Reported Performance on CEC17 Benchmarks	Key Characteristics
MFEA [19]	Genetic Algorithm (GA)	Better on CILS problems	Fixed operator; limited adaptability
MFDE [19]	DE/rand/1	Better on CIHS & CIMS problems	Fixed operator; limited adaptability
EMEA [19]	GA & DE (Fixed)	Better than single-operator algorithms	Hybrid but fixed/random operator use
RLMFEA [19]	GA & DE (Random)	Better than single-operator algorithms	Hybrid but lacks adaptive mechanism
BOMTEA [19]	GA & DE (Adaptive)	Outstanding results, significantly outperforms others	Adaptive bi-operator strategy

Foundational Evolutionary Search Operators

A hybrid adaptive system typically builds upon well-understood, complementary operators. Two of the most commonly combined are DE and Simulated Binary Crossover (SBX).

Differential Evolution (DE): DE is a population-based optimizer known for its robust exploration capabilities. A common mutation strategy is DE/rand/1: v_i = x_r1 + F × (x_r2 - x_r3) [19] Here, v_i is the mutant vector, x_r1, x_r2, x_r3 are distinct randomly selected population vectors, and F is a scaling factor. The mutant vector then undergoes a crossover operation (typically binomial) with a target vector to produce a trial vector, which is selected for the next generation if it improves fitness [19].

Simulated Binary Crossover (SBX): SBX is a crossover operator predominantly used in Genetic Algorithms and is designed to simulate the behavior of single-point crossover on binary strings for real-valued representations. It produces two offspring from two parents with a spread factor (β) that is randomly sampled from a distribution, encouraging offspring near the parents while allowing for occasional exploration further away [19]. The offspring are calculated as: c1 = 0.5 * [(1 - β) * p1 + (1 + β) * p2] c2 = 0.5 * [(1 + β) * p1 + (1 - β) * p2] This operator provides strong local exploitation properties.

Protocol for Implementing an Adaptive Bi-Operator Algorithm (BOMTEA)

The following protocol outlines the steps for implementing a Multitasking Evolutionary Algorithm via an Adaptive Bi-Operator Strategy (BOMTEA), which has demonstrated outstanding performance on standard benchmarks [19].

Experimental Setup and Workflow

The overall workflow of the BOMTEA protocol integrates initialization, parallel task evaluation, adaptive operator selection, and knowledge transfer. The process is designed to dynamically leverage the strengths of different evolutionary operators.

Materials and Reagents: The Algorithmic Toolkit

Table 2: Essential Research Reagents for Hybrid Adaptive EMTO

Research Reagent	Function / Role in the Experiment	Key Parameters & Specifications
Population	A set of candidate solutions representing potential answers to the optimization tasks.	Size (N), Encoding (e.g., real-valued vector), Skill Factor (task assignment)
Operator Pool	A collection of distinct evolutionary search operators (ESOs) that provide diverse search dynamics.	DE/rand/1, SBX, Particle Swarm Optimization, etc.
Performance Metric	Quantifies the effectiveness of each operator to guide the adaptive selection mechanism.	Improvement in offspring fitness, success rate of generated offspring
Adaptive Selection Controller	Dynamically adjusts the probability of selecting an operator based on its recent performance.	Probability update rule, sliding window for performance history
Knowledge Transfer Mechanism	Facilitates the exchange of genetic material between populations solving different tasks.	Random Mating Probability (RMP), transfer frequency, anomaly detection [16]
Benchmark Problems	Standardized test functions to validate and compare algorithm performance.	CEC17 MTO, CEC22 MTO benchmarks (e.g., CIHS, CIMS, CILS)
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Step-by-Step Procedural Details

Step 1: Initialization

• Initialize a unified population for K tasks. Each individual is a vector in the combined search space.
• Assign a skill factor (Ï„) to each individual, indicating the task on which it is evaluated.
• Set initial equal selection probabilities for all operators in the pool (e.g., 0.5 for DE and 0.5 for SBX).

Step 2: Main Generational Loop Repeat the following steps until a stopping criterion is met (e.g., maximum number of generations).

Step 2.1: Offspring Generation with Adaptive Operator Selection

• For each individual (or pair of individuals) selected for variation:
  • Choose an ESO from the operator pool probabilistically based on the current selection probabilities.
  • If DE is selected: Generate a trial vector using the DE/rand/1 mutation and binomial crossover strategy [19]. The scaling factor F and crossover rate Cr are key parameters.
  • If SBX is selected: Generate two offspring from two parent individuals using the SBX operation [19]. The distribution index (ηc) controls the spread of the offspring.
• The newly generated offspring inherits the skill factor of its parent(s) or is assigned via a specific strategy.

Step 2.2: Knowledge Transfer Execution

• Implement a knowledge transfer strategy. This can be similar to MFEA's assortative mating, where individuals from different tasks can crossover if a random number is less than the Random Mating Probability (rmp) [19].
• Alternatively, employ more advanced strategies like the Anomaly Detection Transfer (ADKT), which filters out valuable individuals from source tasks before transfer to prevent negative transfer [16].

Step 2.3: Evaluation and Selection

• Evaluate all offspring on their respective tasks (as defined by their skill factor).
• Perform environmental selection to choose survivors for the next generation, typically based on fitness and, in multifactorial environments, on skill factor.

Step 2.4: Operator Probability Update

• Calculate the performance of each operator based on the offspring generated in the current generation. A common metric is the ratio of successful offspring (which survived selection) produced by an operator.
• Update the selection probability for each operator. A simple method is: P_new(Op_i) = (1 - α) * P_old(Op_i) + α * (Success_Rate(Op_i) / Total_Success_Rates) where α is a learning rate. This adaptively boosts the probability of using well-performing operators.

Advanced Adaptive Strategies for Knowledge Transfer

While BOMTEA focuses on adapting operator selection, modern EMTO algorithms also incorporate adaptation in other critical areas.

Adaptive Knowledge Transfer Probability (ARMP)

Instead of using a fixed Random Mating Probability (rmp), advanced algorithms like those discussed in [16] employ an Adaptive RMP (ARMP). This strategy measures the convergence speed of tasks by monitoring changes in the population (e.g., fitness improvement) and dynamically adjusts the knowledge transfer probability. This creates a balance between a task's self-evolution and the influx of knowledge from other tasks, leading to more efficient use of computational resources and accelerated convergence [16].

Transfer Source Selection (MGTS) and Anomaly Detection (ADKT)

Selecting the right source for knowledge transfer is crucial to avoid negative transfer. The MMD and GRA-based Source Task Selection (MGTS) strategy considers both population similarity (via Maximum Mean Discrepancy - MMD) and the similarity of dynamic evolutionary trends (via Grey Relational Analysis - GRA) [16]. This ensures that knowledge is imported from tasks that are not only structurally similar but are also evolving in a compatible direction. Furthermore, the Knowledge Transfer strategy based on Anomaly Detection (ADKT) filters individuals from the source task population, weeding out "anomalous" individuals that are unlikely to be beneficial before transfer occurs [16].

Troubleshooting and Validation

Common Experimental Challenges and Solutions

• Negative Transfer: This occurs when knowledge from a dissimilar or incompatible task degrades performance.
  • Solution: Implement robust transfer source selection (e.g., MGTS) and individual filtering (e.g., ADKT) [16]. Reduce the rmp or transfer frequency adaptively.
• Operator Dominance: One operator may dominate the selection probability early on, preventing the use of other potentially useful operators.
  • Solution: Incorporate a minimal selection probability or use a sliding window for performance calculation to allow recently improved operators to recover.
• Premature Convergence: The population loses diversity and converges to a sub-optimal solution.
  • Solution: Ensure the operator pool contains diverse operators (e.g., both explorative and exploitative). Tune parameters like F in DE and ηc in SBX to balance exploration and exploitation.

Validation and Benchmarking Protocol

To validate the implementation and performance of a hybrid adaptive EMTO algorithm, follow this benchmarking protocol:

• Select Benchmark Suites: Use widely recognized MTO benchmarks, specifically the CEC17 and CEC22 test problems [19]. These contain various task similarities (CIHS, CIMS, CILS).
• Define Performance Metrics: Common metrics include:
  • Accuracy: The best objective function value found for each task at the end of the run.
  • Convergence Speed: The number of generations or function evaluations required to reach a predefined satisfactory value.
• Compare Against Baselines: Run the hybrid adaptive algorithm alongside single-operator MTEAs (e.g., MFEA, MFDE) and other hybrid algorithms (e.g., EMEA, RLMFEA) on the same benchmarks.
• Statistical Testing: Perform statistical significance tests (e.g., Wilcoxon signed-rank test) on the results from multiple independent runs to confirm that performance improvements are significant.

Multi-objective multi-task optimization (MOMTO) represents a significant evolution in computational problem-solving, enabling the simultaneous addressing of multiple, often conflicting, objectives across several related optimization tasks. Within the paradigm of evolutionary multi-task optimization (EMTO), this approach leverages genetic transfer learning, allowing for the exchange of valuable knowledge between tasks. This concurrent resolution enhances overall efficiency and solution quality, proving particularly powerful in complex domains like drug discovery [25] [26]. These frameworks manage the inherent trade-offs between objectives—such as efficacy, safety, and synthesizability in drug candidates—while mitigating the risk of negative transfer, where unhelpful or detrimental information is shared between tasks [25]. Their application is expanding the problem scope that can be tackled computationally, facilitating more integrated and intelligent research and development pipelines.

Foundational Algorithmic Frameworks

The core principle of MOMTO is to exploit synergies between tasks. Several specialized evolutionary algorithms have been developed to implement this principle effectively.

Key Algorithmic Variants

• MOMFEA-STT (Multi-Objective Multi-task Evolutionary Algorithm based on Source Task Transfer): This algorithm introduces a dynamic knowledge transfer strategy. It establishes an online parameter-sharing model between a historical source task and the current target task. By matching the static features of the source task with the dynamic evolutionary trend of the target task, it enables adaptive knowledge transfer. To prevent the algorithm from converging prematurely to local optima, it employs a novel offspring generation method called the random step spiral generation method (SSM) [25].

• MOMTPSO (Multi-Objective Multi-Task Particle Swarm Optimization): This framework utilizes a particle swarm optimization (PSO) backbone, enhanced with three key components for MOMTO: an Adaptive Knowledge Transfer Probability (AKTP) that controls the intensity and timing of transfer based on swarm quality; a Guiding Particle Selection (GPS) mechanism that divides the objective space to select diverse and informative guiding particles; and an Adaptive Acceleration Coefficient (AAC) that configures the influence of transferred knowledge based on inter-task similarity and archive distribution [26].

• Data-Driven MOMTO (DDMTO): This framework integrates machine learning to smooth complex, rugged fitness landscapes, making them easier to navigate. It formulates the original problem and its smoothed counterpart as two separate but related tasks within an EMTO framework. Knowledge is transferred synchronously between these tasks, which avoids error propagation and improves the global search ability without a significant increase in computational cost [27].

Comparative Analysis of Algorithmic Features

Table 1: Comparison of Multi-Objective Multi-Task Optimization Frameworks

Algorithm	Core Optimization Mechanism	Key Transfer Strategy	Primary Innovation	Reported Application Context
MOMFEA-STT [25]	Evolutionary Algorithm	Dynamic source task matching	Online parameter-sharing model & spiral search	General multi-objective benchmarks
MOMTPSO [26]	Particle Swarm Optimization	Adaptive guiding particle selection	Objective space division & adaptive coefficients	CEC competition problems, sensor coverage
DDMTO [27]	Multi-task EA with ML surrogate	Synchronous optimization of original and smoothed tasks	Using ML models as data-driven filters for landscape smoothing	High-dimensional benchmark functions

Figure 1: Generalized Workflow for Implementing a Multi-Objective Multi-Task Optimization Framework.

Application in Drug Discovery and Bioinformatics

The pharmaceutical industry, with its inherently multi-faceted and high-stakes problems, has become a prime area for applying MOMTO frameworks. These frameworks excel at integrating diverse predictive and generative tasks.

Integrated Drug Design Platforms

• DeepDTAGen: This multitask deep learning framework unifies two critical drug discovery tasks: predicting drug-target affinity (DTA) and generating novel, target-aware drug molecules. It uses a shared feature space for both tasks, ensuring that the generated molecules are informed by the structural and interaction knowledge learned during affinity prediction. A key innovation is the FetterGrad algorithm, which mitigates optimization challenges and gradient conflicts between the distinct tasks, leading to more stable and effective learning [28].

• Baishenglai (BSL) Platform: BSL is a comprehensive, deep learning-enhanced platform that integrates seven core drug discovery tasks within a unified framework: molecular generation, molecular optimization, molecular property prediction, drug-target interaction prediction, drug-drug interaction prediction, drug-cell response prediction, and retrosynthesis analysis. Its modularity and emphasis on Out-of-Distribution (OOD) generalization make it particularly robust for real-world applications where novel molecular structures are encountered. The platform has demonstrated practical utility by successfully identifying novel bioactive compounds for the GluN1/GluN3A NMDA receptor [6].

Predictive Modeling with Evolutionary Relatedness

• Evolutionary Relatedness in Multi-Task QSAR: In quantitative structure-activity relationship (QSAR) modeling, the limited bioactivity data for natural products poses a significant challenge. Instance-based Multi-Task Learning (IBMTL) has been optimized using evolutionary relatedness metrics of target proteins. This approach leverages the similarity between protein targets (e.g., based on amino acid sequence similarity) to improve the prediction of natural product bioactivity, especially for targets with scarce data. This method has shown significant performance improvements in protein groups like kinases and cytochrome P450s [29].

Experimental Protocols and Reagent Solutions

Protocol: Implementing a Multi-Task Evolutionary Algorithm for Drug-Target Affinity Prediction

This protocol outlines the steps for employing a framework like DeepDTAGen to predict drug-target binding affinity.

I. Data Preprocessing and Feature Representation 1. Data Curation: Compile a benchmark dataset such as KIBA, Davis, or BindingDB, which provides validated drug-target pairs with binding affinity values [28]. 2. Drug Representation: Represent drug molecules via their Simplified Molecular Input Line Entry System (SMILES) strings. These can be converted into numerical representations using 1D convolutional neural networks (CNNs) or graph neural networks (GNNs) that capture atomic features and molecular structure [28]. 3. Target Representation: Represent target proteins by their amino acid sequences. These sequences are typically converted into embeddings using 1D CNNs to capture sequential and structural motifs [28].

II. Model Architecture and Multi-Task Setup 1. Shared Encoder: Design a shared feature extraction module that processes both drug and target representations, learning a joint latent space that encapsulates interaction information. 2. Task-Specific Heads: - Affinity Prediction Head: Attach a regression head (e.g., fully connected layers) to the shared encoder to predict continuous binding affinity values. The loss function is typically Mean Squared Error (MSE). - Generation Head: For a full MOMTO framework like DeepDTAGen, attach a decoder (e.g., a Transformer) that conditions on the shared latent representation and the target protein to generate novel drug SMILES strings [28].

III. Multi-Task Optimization and Training 1. Loss Balancing: Combine the losses from the prediction and generation tasks. The FetterGrad algorithm can be employed to manage gradient conflicts by minimizing the Euclidean distance between task gradients, ensuring aligned learning [28]. 2. Training Regime: Train the model using a stochastic gradient descent optimizer (e.g., Adam) with mini-batching. Performance is evaluated using metrics like Concordance Index (CI) and r²m for regression, and validity, novelty, and uniqueness for the generated molecules [28].

Protocol: Leveraging Evolutionary Relatedness for Bioactivity Prediction

This protocol describes the use of Instance-Based Multi-Task Learning (IBMTL) for predicting the bioactivity of natural products [29].

I. Dataset Construction and Task Formulation 1. Compound Sourcing: Curate a dataset of predicted natural products and their bioactivities against a group of evolutionarily related target proteins (e.g., a kinase family) from databases like ChEMBL. 2. Task Definition: Define each prediction task for a specific protein target within the group. In IBMTL, the dataset is reformatted by duplicating compounds and adding a task identifier column and evolutionary similarity metrics as additional features.

II. Model Training with Relatedness Metrics 1. Feature Engineering: Calculate evolutionary relatedness metrics between the protein targets, such as global or local amino acid sequence similarity. 2. Algorithm Selection: Employ a traditional machine learning algorithm capable of handling the reformatted IBMTL data frame, such as Random Forest, which has been shown to be effective in this context [29]. 3. Training and Validation: Train the model on the expanded dataset. The model learns to leverage the evolutionary relatedness information to share knowledge across the prediction tasks for different targets, improving performance, especially on data-scarce tasks.

Research Reagent Solutions

Table 2: Key Computational Tools and Platforms for MOMTO in Drug Discovery

Tool/Platform Name	Type	Primary Function in Research	Access Information
Baishenglai (BSL) [6]	Integrated Software Platform	End-to-end virtual screening covering 7 core tasks (e.g., DTI, generation, DDI).	Open Access: https://www.baishenglai.net
DeepDTAGen [28]	Multitask Deep Learning Framework	Simultaneously predicts drug-target affinity and generates target-aware drug molecules.	Methodology described in academic literature.
ChEMBL Database [29]	Bioactivity Database	Provides curated bioactivity data for drug-target pairs; used for model training and validation.	Open Access: https://www.ebi.ac.uk/chembl/
Random Forest with IBMTL [29]	Machine Learning Algorithm	Performs bioactivity prediction using evolutionary relatedness between protein targets.	Implemented in standard ML libraries (e.g., Scikit-learn).

Multi-objective multi-task frameworks represent a paradigm shift in computational optimization, moving from isolated problem-solving to a more holistic, synergistic approach. By formally managing multiple objectives and enabling adaptive knowledge transfer across tasks, algorithms like MOMFEA-STT, MOMTPSO, and integrated platforms like DeepDTAGen and Baishenglai are dramatically expanding the scope and complexity of problems that can be addressed. In drug discovery, this translates to accelerated, more efficient, and more intelligent pipelines for identifying and designing novel therapeutic compounds. As these frameworks continue to evolve, particularly with advancements in handling out-of-distribution data and mitigating negative transfer, their impact is poised to grow, further solidifying their role as indispensable tools in modern scientific research and development.

Evolutionary Multitasking Optimization (EMTO) represents a paradigm shift in how complex optimization problems are approached. It enables the concurrent solution of multiple tasks by strategically exploiting their underlying similarities [19]. In the context of complex network systems, this approach is particularly valuable as it allows knowledge gained while optimizing one network property to accelerate the solution of related problems on the same or different networks [30] [31]. The multifactorial evolutionary algorithm (MFEA) laid the foundation for this field by introducing skill factors to implicitly divide populations and enable knowledge transfer through assortative mating and selective imitation [19] [31].

Combinatorial problems in complex networks—such as community detection, influence maximization, and resource allocation—are often NP-complete, making traditional optimization approaches computationally prohibitive at scale [31]. EMTO frameworks address this challenge by leveraging the implicit parallelism of evolutionary algorithms to exploit synergies between tasks, often leading to significant performance improvements compared to solving problems in isolation [30]. This case study examines the application of advanced EMTO frameworks to combinatorial problems in complex networks, with particular emphasis on the emerging paradigm of Multi-Domain Evolutionary Optimization (MDEO).

Technical Foundation: EMTO Methodologies

Core EMTO Frameworks and Transfer Mechanisms

EMTO implementations generally follow two primary population models: single-population and multi-population approaches [31]. The single-population model, exemplified by MFEA, uses skill factors to implicitly partition the population into subpopulations specialized for different tasks, with knowledge transfer occurring through crossover operations [19]. Multi-population models maintain explicitly separate populations for each task, allowing more controlled inter-task interactions [31].

Knowledge transfer in EMTO occurs through three primary schemes:

• Unified representation: Aligns chromosomes from different tasks in a normalized search space, enabling direct chromosomal crossover [31]
• Probabilistic modeling: Extracts compact probabilistic models from elite solutions to represent task knowledge [31]
• Explicit auto-encoding: Creates direct mappings between search spaces of different tasks using encoding-decoding mechanisms [31]

The adaptive bi-operator evolution for multitasking (BOMTEA) represents a significant advancement by dynamically selecting between evolutionary search operators [19]. Unlike approaches that use fixed operators, BOMTEA adaptively controls the selection probability of genetic algorithms (GA) and differential evolution (DE) based on their performance, determining the most suitable operator for various problem types [19].

Multi-Domain Evolutionary Optimization (MDEO)

MDEO extends the EMTO concept by transferring knowledge across different network domains rather than just between tasks within a single domain [30]. This approach recognizes that real-world networks from different domains (social, biological, technological) often share structural properties like power-law distributions, small-world characteristics, and community structures [30]. The MDEO framework consists of four key components:

• Community-level graph similarity measurement to manage knowledge transfer among domains
• Graph learning-based network alignment to establish node correspondences between different networks
• Self-adaptive mechanism to determine the number of solutions transferred between domains
• Knowledge-guided mutation that redefines mutation candidates to utilize cross-domain knowledge [30]

Table 1: Comparison of EMTO Frameworks for Combinatorial Optimization

Framework	Domain Scope	Problem Type	Space Type	Task Scope	Key Features
MTEO-ConO [31]	Single-domain	Continuous optimization	Continuous	Multiple	Unified representation, implicit transfer
MTEO-ComO [30] [31]	Single-domain	Combinatorial optimization	Discrete	Multiple	Multi-population models, explicit transfer
MDEO [30]	Multi-domain	Combinatorial optimization	Discrete	Single/Multiple	Cross-domain transfer, network alignment

Application Protocol: MDEO for Network Combinatorial Problems

Experimental Setup and Workflow

The following protocol outlines the implementation of MDEO for solving combinatorial problems across multiple network domains, using community deception as a representative problem.

Research Reagent Solutions and Essential Materials:

Table 2: Key Research Reagents and Computational Resources

Item	Function	Specifications
Complex Network Datasets	Provide benchmark problems for evaluation	Multiple real-world networks from different domains (social, biological, technological) with verified structural properties
Graph Autoencoder Framework	Generates graph embeddings for network alignment	Python implementation with customizable layer architecture and loss functions
Evolutionary Algorithm Library	Provides core optimization capabilities	Support for multiple operators (GA, DE) with adaptive selection mechanisms
Network Analysis Toolkit	Quantifies structural properties and similarity	Includes community detection, centrality measures, and graph similarity metrics
High-Performance Computing Cluster	Enables distributed computation for large-scale problems	Multi-node setup with parallel processing capabilities

Step 1: Problem Formulation and Network Preparation

• Select multiple real-world networks from different domains (e.g., social networks, biological networks, infrastructure networks)
• Formulate identical combinatorial optimization tasks across all networks (e.g., community deception, influence maximization, or resource allocation)
• For community deception tasks, define the objective as minimizing the effectiveness of community detection algorithms by strategically modifying network connections [30]

Step 2: Community-Level Similarity Measurement

• Apply community detection algorithms to all networks to identify densely connected groups
• Compute similarity between networks using a novel community-level measurement that quantifies structural closeness
• Create a similarity graph where nodes represent networks and edges represent significant similarity relationships
• Use this similarity measurement to focus knowledge transfer on closely related networks, reducing computational burden [30]

Step 3: Network Alignment via Graph Embedding

• Employ graph autoencoders to generate embeddings that capture node similarity and higher-order interactions
• Implement a community-level anchor node selection method to build training sets for supervised alignment
• Combine supervised and unsupervised learning to create a network alignment model that establishes node correspondences between different networks
• Validate alignment accuracy using known network isomorphic relationships [30]

Step 4: Multi-Domain Evolutionary Optimization

• Initialize separate populations for each network domain
• Implement a self-adaptive mechanism that determines the number of solutions to transfer between networks based on their similarity
• Apply knowledge-guided mutation that redefines mutation candidates using knowledge from other domains
• Transfer solutions between similar networks using the node mappings obtained through network alignment
• Execute evolutionary operations (crossover, mutation, selection) with adaptive operator selection [30]

Step 5: Performance Evaluation and Knowledge Retention

• Evaluate solution quality on each network using task-specific fitness functions
• Compare performance against classical evolutionary optimization approaches
• Analyze cross-domain transfer effectiveness by tracking the contribution of transferred solutions to fitness improvement
• Archive high-quality solutions and transfer patterns for future problem-solving sessions [30]

Workflow Visualization

Results and Performance Analysis

Quantitative Performance Metrics

Experimental validation of the MDEO framework on eight real-world networks demonstrates its effectiveness for combinatorial optimization tasks. The framework was evaluated on both edge-level and node-level tasks, with community deception serving as the primary illustrative problem [30].

Table 3: Performance Comparison of MDEO vs. Classical Evolutionary Optimization

Network Domain	Network Size (Nodes)	Classical EO Fitness	MDEO Fitness	Improvement	Convergence Speed Increase
Social Network	1,892	0.73	0.89	21.9%	2.4x
Biological Network	3,567	0.68	0.85	25.0%	2.8x
Power Grid	4,941	0.71	0.87	22.5%	2.1x
Communication	2,348	0.75	0.91	21.3%	2.6x
Collaboration	5,248	0.69	0.84	21.7%	2.3x
Transportation	3,845	0.72	0.88	22.2%	2.5x
Protein	2,967	0.70	0.86	22.9%	2.7x
Metabolic	4,126	0.74	0.90	21.6%	2.2x

The performance evaluation reveals that MDEO consistently achieves higher fitness values compared to classical evolutionary optimization approaches across all tested network domains [30]. The framework demonstrates particular strength in biological and protein networks, where structural similarities may be more pronounced and thus more amenable to knowledge transfer.

Knowledge Transfer Effectiveness Analysis

The adaptive knowledge transfer mechanism in MDEO successfully identifies beneficial cross-domain transfers while minimizing negative transfers that can hinder optimization progress [30]. The community-level similarity measurement proves effective in predicting which network pairs will benefit most from knowledge exchange.

Table 4: Knowledge Transfer Effectiveness Across Network Domains

Source Domain	Target Domain	Similarity Score	Transfer Success Rate	Average Fitness Improvement
Social	Collaboration	0.87	92.5%	18.7%
Biological	Protein	0.91	96.2%	24.3%
Power Grid	Transportation	0.82	88.9%	16.5%
Communication	Social	0.79	85.3%	15.8%
Collaboration	Biological	0.76	83.1%	14.2%
Transportation	Power Grid	0.84	89.7%	17.1%

The data indicates that higher similarity scores generally correlate with more successful knowledge transfer, validating the community-level similarity measurement approach [30]. The self-adaptive mechanism effectively utilizes these similarity metrics to determine the appropriate number of solutions to transfer between domains.

Advanced Protocol: Adaptive Bi-Operator Evolution

BOMTEA Implementation for Multitasking

The BOMTEA framework addresses the limitation of single-operator evolutionary approaches by combining the strengths of genetic algorithms and differential evolution with adaptive selection [19].

Experimental Protocol:

Step 1: Initial Operator Probability Setup

• Initialize selection probabilities for GA and DE operators at 0.5 each
• Define performance metrics for evaluating operator effectiveness (fitness improvement, diversity maintenance)

Step 2: Population Evaluation and Operator Assessment

• Generate offspring using both GA and DE operators in parallel
• Evaluate fitness of offspring populations from each operator
• Track success rates for each operator based on offspring survival

Step 3: Adaptive Probability Adjustment

• Adjust selection probabilities based on relative performance:
  • Increase probability for operators demonstrating higher success rates
  • Apply smoothing to prevent rapid probability fluctuations
• Maintain minimum probability thresholds (e.g., 0.1) to ensure operator diversity

Step 4: Knowledge Transfer Integration

• Implement implicit transfer through chromosomal crossover between tasks
• Apply explicit transfer through solution mapping for highly similar tasks
• Balance transfer intensity based on measured task relatedness [19]

BOMTEA Performance on Benchmark Problems

Testing on CEC17 and CEC22 multitasking benchmarks demonstrates that BOMTEA significantly outperforms single-operator approaches and fixed operator combinations [19]. The adaptive mechanism successfully identifies the most suitable operator for different problem types: DE/rand/1 performs better on complete-intersection, high-similarity (CIHS) and complete-intersection, medium-similarity (CIMS) problems, while GA shows superiority on complete-intersection, low-similarity (CILS) problems [19].

This case study demonstrates that evolutionary multitasking and multi-domain optimization frameworks offer significant advantages for solving combinatorial problems in complex networks. The MDEO and BOMTEA frameworks outperform classical evolutionary approaches by effectively leveraging structural similarities between networks and adaptively selecting appropriate search operators [19] [30].

Future research directions include extending these frameworks to dynamic network environments where structures evolve over time, developing more sophisticated transferability assessment mechanisms to minimize negative transfer, and scaling the approaches to extremely large networks with millions of nodes [30] [32]. Additionally, integration with emerging AI techniques like hypergraph neural networks shows promise for handling higher-order constraints and relationships [32].

The protocols and methodologies presented provide researchers with practical frameworks for applying advanced evolutionary multitasking approaches to combinatorial problems across diverse network domains, potentially accelerating discoveries in fields ranging from drug development to infrastructure resilience planning [30] [31].

Evolutionary Multitask Optimization (EMTO) has emerged as a powerful paradigm in computational intelligence that enables the simultaneous solving of multiple, related optimization problems. By promoting knowledge transfer between different tasks, EMTO algorithms can leverage shared information to accelerate convergence and improve solution quality beyond what is possible when solving tasks in isolation [16] [24]. This approach mirrors concepts from transfer learning and multitask learning in mainstream artificial intelligence, but applies them within an evolutionary computation framework [24]. The fundamental premise is that similar or related optimization tasks can be solved more efficiently using knowledge gained from other tasks rather than solving each task separately [16].

The practical utility of EMTO has been demonstrated across diverse domains, from engineering design to computational biology, where it has shown powerful scalability and search capabilities [16]. As a growing research frontier, EMTO addresses critical challenges in knowledge transfer probability, migration source selection, and transfer mechanisms to maximize positive transfer while minimizing negative interference between tasks [16] [33]. This article explores the domain applications of EMTO with detailed protocols and analyses of its implementation in both engineering and biological contexts.

EMTO in Engineering Design and Optimization

Engineering design presents numerous complex optimization challenges that naturally align with EMTO capabilities. In this domain, researchers create mathematical models and develop computer methods and tools that advance diverse engineering applications, including geometric and physical modeling, finite-element methods, computational fluid dynamics, multi-physics modeling, and shape and topology optimization [34].

Computational Design Frameworks

Computational design represents an emerging design method that uses a combination of algorithms and parameters to solve design problems with advanced computer processing [35]. This approach includes three primary subsets:

• Parametric Design: An interactive process that uses parameters and rules to create easily modifiable design solutions, ideal for complex architectural geometries [35]
• Generative Design: An iterative process that employs artificial intelligence to produce multiple design options evaluated against success metrics [35]
• Algorithmic Design: A method guided by algorithms to produce architectural models through defined rules rather than individual element definition [35]

Implementation Protocols for Engineering Design Optimization

Protocol 1: Multi-Factorial Evolutionary Algorithm for Structural Design

• Objective: Optimize structural parameters for multiple design scenarios simultaneously
• EMTO Framework: Multi-factorial evolutionary framework with adaptive knowledge transfer
• Procedure:
  • Encode design parameters for all tasks in a unified search space
  • Initialize population with random solutions across task domains
  • Evaluate individuals using multifactorial ranking considering all tasks
  • Apply crossover and mutation with adaptive knowledge transfer probability
  • Select offspring based on skill-factor and scalar fitness
  • Update transfer parameters based on convergence feedback
  • Repeat steps 3-6 until termination criteria met
• Key Parameters: Population size = 100, Generations = 200, Knowledge transfer probability = 0.3-0.7 (adaptive) [16]

Protocol 2: Many-Task Optimization for Complex Systems

• Objective: Solve many-task optimization problems with multiple related engineering tasks
• EMTO Framework: Multi-population framework with explicit collaboration
• Procedure:
  • Maintain separate populations for each task with periodic knowledge exchange
  • Measure population similarity using Maximum Mean Discrepancy (MMD)
  • Assess evolutionary trend similarity via Grey Relational Analysis (GRA)
  • Select transfer sources based on combined MMD and GRA metrics
  • Implement knowledge transfer through elite individual migration
  • Apply anomaly detection to filter negative transfer candidates
  • Adjust transfer frequency based on task convergence rates [16]
• Key Parameters: Migration interval = 10-50 generations, MMD threshold = 0.1-0.3, Population size per task = 50-100 [16]

EMTO in Computational Biology and Drug Discovery

In computational biology, EMTO has demonstrated significant potential for addressing complex biological optimization problems, particularly in drug discovery where it enables more efficient exploration of vast chemical spaces.

Drug Discovery Applications

The RosettaEvolutionaryLigand (REvoLd) protocol represents a cutting-edge application of EMTO in computational biology. This approach uses an evolutionary algorithm to search ultra-large make-on-demand chemical libraries efficiently without enumerating all molecules [36]. REvoLd exploits the combinatorial nature of these libraries, constructed from substrate lists and chemical reactions, to explore vast search spaces for protein-ligand docking with full ligand and receptor flexibility [36].

Table 1: REvoLd Performance Across Drug Targets

Drug Target	Hit Rate Improvement Factor	Unique Molecules Docked	Runs Conducted
Target 1	1622x	49,000-76,000	20
Target 2	1425x	49,000-76,000	20
Target 3	1187x	49,000-76,000	20
Target 4	931x	49,000-76,000	20
Target 5	869x	49,000-76,000	20

Implementation Protocols for Drug Discovery

Protocol 3: REvoLd for Ultra-Large Library Screening

• Objective: Identify promising drug candidates from billion-compound libraries
• EMTO Framework: Evolutionary algorithm with combinatorial space exploration
• Procedure:
  • Generate initial population of 200 random ligands from combinatorial building blocks
  • Evaluate fitness using RosettaLigand flexible docking with full receptor flexibility
  • Select top 50 individuals for reproduction based on docking scores
  • Apply crossover operations between high-fitness molecules
  • Implement mutation steps: fragment switching and reaction changes
  • Conduct secondary reproduction cycle excluding fittest molecules
  • Advance 50 individuals to next generation
  • Continue for 30 generations to balance convergence and exploration [36]
• Key Parameters: Initial population = 200, Generations = 30, Selection rate = 25%, Mutation rate = 0.1-0.3 [36]

Protocol 4: Multitask QSAR Optimization with Evolutionary Relatedness

• Objective: Enhance QSAR-based bioactivity predictions for natural products
• EMTO Framework: Instance-based Multitask Learning (IBMTL) with evolutionary relatedness metrics
• Procedure:
  • Curate dataset of predicted natural products with bioactivity against enzyme targets
  • Implement binary classification filtering using Avalon fingerprint with random forest
  • Apply feature-based MTL across proteins within each protein group
  • Incorporate evolutionary relatedness metrics using amino acid sequence similarity
  • Optimize model with random forest regression across multiple task domains
  • Validate through virtual screening of natural product libraries [29]
• Key Parameters: Avalon fingerprint, Random forest algorithm, Protein classification hierarchy = 6-level [29]

Table 2: Multitask Learning Performance Comparison

Method	Protein Group	Performance Metric	Advantage Over STL
STL	Kinase	RMSE	Baseline
FBMTL	Kinase	RMSE	+5.2%
IBMTL (AA-GSS)	Kinase	RMSE	+12.7%
IBMTL (AA-LSS)	Kinase	RMSE	+15.3%
STL	Cytochrome P450	RMSE	Baseline
FBMTL	Cytochrome P450	RMSE	+3.8%
IBMTL (AA-GSS)	Cytochrome P450	RMSE	+9.1%
IBMTL (AA-LSS)	Cytochrome P450	RMSE	+11.6%

Advanced EMTO Methodologies and Technical Implementations

Knowledge Transfer Strategies

Effective knowledge transfer represents the core of successful EMTO implementation. Recent advances have addressed three critical aspects of transfer mechanisms:

Adaptive Knowledge Transfer Probability (ARMP) This strategy addresses varying knowledge requirements at different evolutionary stages by measuring task convergence speed through population changes and dynamically adjusting transfer probability accordingly [16]. This achieves a dynamic balance between task self-evolution and knowledge transfer, accelerating convergence while allocating computing resources more rationally [16].

MMD and GRA-based Source Task Selection (MGTS) The MGTS strategy considers both population similarity and dynamic evolutionary trend similarity among tasks. It uses Maximum Mean Discrepancy to quantify population distribution similarity and Grey Relational Analysis to assess evolutionary trend similarity, enabling more appropriate transfer source selection [16].

Anomaly Detection-based Knowledge Transfer (ADKT) This approach filters valuable individuals for knowledge transfer by constructing anomaly detection models for current tasks to identify abnormal individuals from source task populations. It then integrates individuals with similar characteristics through K-means clustering to enhance positive transfer [16].

Domain Adaptation Techniques

Progressive Auto-Encoding (PAE) PAE addresses limitations of static pre-trained models in domain adaptation by enabling continuous updates to domain representations throughout the evolutionary process [33]. This technique includes two complementary strategies:

• Segmented PAE: Employs staged training of auto-encoders for structured domain alignment across optimization phases
• Smooth PAE: Utilizes eliminated solutions for gradual and continuous domain refinement [33]

The Scientist's Toolkit: Research Reagent Solutions

Table 3: Essential Research Tools for EMTO Implementation

Tool/Resource	Function	Application Domain
RosettaLigand	Flexible protein-ligand docking with full receptor flexibility	Drug discovery, Computational biology
Enamine REAL Space	Make-on-demand combinatorial chemical library with billions of compounds	Virtual screening, Drug discovery
Dynamo	Visual programming tool for parametric design	Engineering design, Structural optimization
Grasshopper	Algorithmic modeling tool with node-based interface	Engineering design, Architectural geometry
MToP Benchmarking Platform	Standardized evaluation of EMTO algorithms	Algorithm development, Performance comparison
Avalon Fingerprint	Molecular structure representation for QSAR modeling	Chemoinformatics, Drug discovery
Random Forest Algorithm	Machine learning for classification and regression	QSAR modeling, Bioactivity prediction
D-ribo-Phytosphingosine-13C2,d2	D-ribo-Phytosphingosine-13C2,d2|CAS 237757-10-7
2,4-Bis[(trimethylsilyl)oxy]pyridine	2,4-Bis[(trimethylsilyl)oxy]pyridine, CAS:40982-58-9, MF:C11H21NO2Si2, MW:255.46 g/mol	Chemical Reagent

Workflow and System Diagrams

EMTO Knowledge Transfer Framework

EMTO System Workflow - This diagram illustrates the integrated knowledge transfer framework in evolutionary multitask optimization, showing how similarity analysis and adaptive probability control information exchange between task populations.

Drug Discovery Optimization Pipeline

Drug Discovery Pipeline - This workflow visualizes the evolutionary algorithm for drug discovery, showing the iterative process of generating, evaluating, and refining chemical compounds through docking assessments and genetic operations.

Evolutionary Multitask Optimization represents a transformative approach to complex optimization problems across engineering and computational biology domains. By enabling efficient knowledge transfer between related tasks, EMTO algorithms demonstrate significant improvements in convergence speed, solution quality, and computational efficiency compared to traditional single-task optimization methods. The continued development of adaptive transfer mechanisms, sophisticated domain adaptation techniques, and problem-specific implementations promises to further expand the applicability and performance of EMTO across an increasingly diverse range of real-world challenges. As evidenced by the protocols and analyses presented herein, EMTO stands as a powerful methodology at the intersection of evolutionary computation and transfer learning, offering substantial benefits for researchers and practitioners in both engineering design and computational biology.

Overcoming EMTO Challenges: Negative Transfer and Convergence Issues

Identifying the Root Causes of Negative Transfer

Negative transfer (NT) describes the phenomenon where knowledge transfer between tasks in an optimization process inadvertently degrades performance, slowing convergence or leading to premature convergence on suboptimal solutions [37]. Within evolutionary multitasking optimization (EMTO), which solves multiple tasks simultaneously by leveraging inter-task knowledge, NT poses a significant challenge [7] [38]. The effectiveness of EMTO hinges on successfully exploiting synergies between tasks; however, negative transfer can undermine these benefits, reducing the overall efficiency and effectiveness of the optimization process [7] [37]. This application note delineates the root causes of negative transfer and provides detailed protocols for its mitigation, framed within the context of advanced EMTO applications.

Root Cause Analysis: A Systematic Breakdown

The primary causes of negative transfer in EMTO can be categorized into three interrelated areas: issues arising from task dissimilarity, challenges in the transfer process itself, and problems related to solution quality and population diversity.

Table 1: Fundamental Causes of Negative Transfer in EMTO

Root Cause Category	Specific Cause	Impact on Evolutionary Optimization
Task Dissimilarity	Misaligned Fitness Landscapes [7]	Transferred genetic material pulls the search process into the basin of attraction of a local optimum for the target task.
	High-Dimensional Search Spaces [7]	Mappings learned from sparse population data become unstable, leading to ineffective or detrimental transfers.
	Heterogeneous Task Formulations [39]	Differences in decision spaces, objective functions, or constraints make direct knowledge transfer inappropriate.
Transfer Process	Uncontrolled/Random Transfer [40] [41]	Lack of strategic selection for transfer timing and content results in the influx of unhelpful genetic material.
	Single-Operator Reliance [41]	Using a limited set of evolutionary operators fails to adapt to the varied characteristics of different tasks.
Solution Quality & Diversity	Premature Convergence of Source Task [7]	Knowledge transferred from a task trapped in a local optimum misleads other tasks, propagating poor solutions.
	Concept Drift in Streaming Data [38]	Historical knowledge becomes obsolete as the population evolves, reducing the accuracy of transfer selection models.

Illustrative Mechanism of Negative Transfer

The following diagram visualizes a classic negative transfer scenario between two dissimilar tasks, where the global optimum of one task corresponds to a local optimum of the other.

Quantitative Assessment and Benchmarking

Establishing quantitative metrics is crucial for diagnosing and comparing the impact of negative transfer across different EMTO algorithms and problem sets.

Table 2: Key Metrics for Assessing Negative Transfer

Metric Category	Metric Name	Description	Interpretation
Performance Degradation	Single-Task Performance Gap [37]	( \text{Perf}{\text{ST}} - \text{Perf}{\text{MT}} ) where ( \text{Perf}{\text{ST}} ) is performance in single-task mode and ( \text{Perf}{\text{MT}} ) is in multitask mode.	A positive value indicates potential negative transfer.
	Convergence Speed Slowdown [7] [40]	The number of additional generations/FEs required to reach a target fitness in MT vs. ST.	Directly measures efficiency loss due to transfer.
Transfer Quality	Positive Transfer Rate [41]	Proportion of cross-task offspring that outperform the median parent in the target task.	Measures the immediate effectiveness of individual transfers.
	Negative Transfer Incidence [38]	Frequency of generations where the best fitness deteriorates following a transfer operation.	Identifies periods of harmful interference.

Detailed Experimental Protocols for NT Diagnosis

This section provides a step-by-step methodology for empirically identifying and quantifying negative transfer in a controlled EMTO environment.

Protocol: Baseline Performance Establishment

Objective: To establish single-task optimization performance as a baseline for detecting negative transfer.

• Task Selection: Select a set of benchmark functions (e.g., from CEC 2017 MTO suite [7]) with known, diverse landscape properties.
• Algorithm Configuration: Configure a standard EA (e.g., DE, CMA-ES) as the single-task optimizer.
• Experimental Setup:
  • Population Size: 100 per task.
  • Termination Condition: 50,000 Fitness Evaluations (FEs) per task or convergence to a threshold of (10^{-8}).
  • Independent Runs: 30 per task to ensure statistical significance.
• Data Collection: Record the best fitness value at fixed intervals (e.g., every 1,000 FEs) and the final best-found solution.

Protocol: Multitasking with Transfer Intervention

Objective: To evaluate the same tasks under an EMTO framework and measure the impact of knowledge transfer.

• Algorithm Configuration: Implement an EMTO algorithm (e.g., MFEA [7] [40], or a custom one).
• Incorporating Transfer:
  • Use a probabilistic transfer mechanism (e.g., ( rmp = 0.5 ) in MFEA).
  • Alternatively, implement an explicit transfer strategy (e.g., elite individual transfer [42] or autoencoder-based mapping [7]).
• Experimental Setup:
  • Total Population Size: ( 100 \times K ) for ( K ) tasks.
  • Termination Condition: ( 50,000 \times K ) FEs in total.
  • Independent Runs: 30.
• Data Collection: Record the same performance data as in Protocol 4.1. Additionally, log all cross-task transfers and the fitness of the offspring generated.

Protocol: Controlled Negative Transfer Induction

Objective: To deliberately induce negative transfer by optimizing dissimilar tasks, demonstrating its effects clearly.

• Task Pairing: Deliberately pair optimization tasks known to have misaligned optima (e.g., a task where the global optimum of Task A resides in a local optimum basin of Task B, as conceptualized in [7]).
• Procedure: Follow Protocol 4.2 with this specific task pairing.
• Analysis: Compare the convergence curves and final results with the baselines from Protocol 4.1. A clear performance degradation in the multitasking scenario confirms negative transfer.

The Researcher's Toolkit: Reagents & Computational Solutions

Table 3: Essential Research Reagents and Computational Tools for NT Analysis

Item Name	Type	Function in NT Research	Example/Reference
Benchmark Suites	Dataset	Provides standardized MTO problems for fair comparison and NT induction.	CEC 2017 MTO [7], WCCI 2020 MO-MTO [38]
Domain Adaptation Modules	Algorithm	Aligns search spaces of different tasks to enable more robust transfer.	Linear Domain Adaptation (LDA) [7], PCA-based Subspace Alignment [39]
Similarity Metrics	Metric	Quantifies inter-task relationship to preemptively gauge NT risk.	MMD [43], KLD [43], Task Transfer Distance (theoretical) [37]
Online Classifiers	Model	Dynamically identifies valuable knowledge for transfer from streaming population data.	Budget Online Learning Naive Bayes [38], Support Vector Classifier (SVC) [39]
Reinforcement Learning Agent	Framework	Automates the decision of "when" and "how" to transfer based on evolutionary state.	Learning to Transfer (L2T) Actor-Critic Network [41]
2,2-Diethoxy-N-ethyl-1-ethanamine	2,2-Diethoxy-N-ethyl-1-ethanamine, CAS:69148-92-1, MF:C8H19NO2, MW:161.24 g/mol	Chemical Reagent	Bench Chemicals
(2R,3R)-2-methylbutane-1,2,3,4-tetrol	(2R,3R)-2-methylbutane-1,2,3,4-tetrol | High Purity	(2R,3R)-2-methylbutane-1,2,3,4-tetrol for research. A chiral polyol building block. For Research Use Only. Not for human or veterinary use.	Bench Chemicals

Integrated Workflow for NT Diagnosis and Mitigation

The following diagram outlines a comprehensive experimental and analytical workflow, integrating the protocols and tools described to diagnose and counteract negative transfer.

Effectively identifying the root causes of negative transfer—ranging from fundamental task dissimilarity to suboptimal algorithmic policies—is paramount for advancing EMTO research. The structured metrics, detailed experimental protocols, and toolkit of computational reagents provided herein offer a concrete foundation for researchers to diagnose, understand, and ultimately mitigate this pervasive challenge. By systematically applying these guidelines, the development of more robust and efficient evolutionary multitasking systems for complex applications, including those in drug development, becomes a more attainable goal.

In Evolutionary Multitask Optimization (EMTO), the simultaneous solving of multiple optimization problems is achieved by leveraging the implicit parallelism of population-based search and transferring knowledge between tasks [4] [1]. This paradigm significantly improves convergence speed and solution quality compared to traditional single-task evolutionary algorithms [1]. However, two major challenges persistently hinder its performance: the risk of negative transfer between tasks, particularly those with different dimensionalities or dissimilar fitness landscapes, and the tendency for tasks to fall into premature convergence [7] [33]. To combat these issues, Domain Adaptation (DA) and Linear Mapping techniques have emerged as powerful strategies. Domain adaptation focuses on aligning the search spaces of different tasks to create a more conducive environment for knowledge exchange, thereby mitigating negative transfer [33] [44]. Concurrently, linear mapping strategies help to explore promising regions of the search space, preventing stagnation in local optima [7]. Within the context of a broader thesis on EMTO applications, this document provides detailed application notes and experimental protocols for implementing these mitigation strategies, serving as a practical guide for researchers and drug development professionals aiming to harness EMTO for complex optimization problems.

Domain Adaptation Strategies in EMTO

Domain adaptation in EMTO aims to bridge the distribution gap between the search spaces of different tasks, known as the source domain and target domain [44]. This alignment facilitates more effective and robust knowledge transfer, reducing the negative impact of domain shift [45] [46]. The following sections detail the primary domain adaptation strategies.

Linearized Domain Adaptation (LDA)

Linearized Domain Adaptation employs linear transformations to project solutions from different tasks into a shared, low-dimensional subspace where knowledge transfer becomes more effective [7] [47].

• MDS-based LDA: This method uses Multidimensional Scaling (MDS) to first establish a low-dimensional subspace for each task's decision space. A linear mapping is then learned to align these subspaces, enabling stable knowledge transfer even between tasks of different dimensionalities [7].
• Transfer Component Analysis (TCA): TCA maps populations from different tasks, which may have different marginal probability distributions, into a shared reproducing kernel Hilbert space (RKHS). This mapping minimizes the distribution mismatch between tasks, making the transferred knowledge more beneficial [19].

Table 1: Linearized Domain Adaptation (LDA) Techniques in EMTO

Technique	Core Mechanism	Key Advantage	Primary Reference
MDS-based LDA	Learns a linear mapping between low-dimensional MDS subspaces of different tasks.	Enables robust knowledge transfer between tasks with different dimensionalities.	[7]
Transfer Component Analysis (TCA)	Maps task populations to a shared RKHS to minimize distribution distance.	Effective for homogeneous DA where feature spaces are identical but distributions differ.	[19]
Subspace Alignment	Finds a linear transformation that makes the source domain distribution match the target domain.	A direct approach to solving the covariate shift problem in domain adaptation.	[46]

Progressive Auto-Encoding (PAE)

Progressive Auto-Encoding is an advanced, non-linear domain adaptation technique that addresses the limitations of static pre-trained models. It dynamically updates domain representations throughout the evolutionary process, allowing the algorithm to adapt to changing populations [33].

• Segmented PAE: This strategy involves the staged training of auto-encoders. Different auto-encoders are trained and utilized during distinct phases of the optimization process, providing structured domain alignment tailored to the evolutionary stage [33].
• Smooth PAE: This approach performs gradual and continuous domain refinement. It leverages eliminated solutions—individuals discarded during selection—to update the auto-encoders, ensuring that valuable knowledge from past populations is not lost [33].

The following workflow diagram illustrates the integration of PAE within a standard Multi-Task Evolutionary Algorithm (MTEA) framework:

Classification of Domain Adaptation Problems

The nature of the domain adaptation problem varies based on the relationship between the source and target domains. Two primary categorizations are essential for selecting the appropriate strategy [45].

• Homogeneous vs. Heterogeneous DA: Homogeneous DA deals with scenarios where the source and target domains share the same feature space and dimensionality, but the data distributions differ. Heterogeneous DA is a more challenging setting where the domains may have different feature spaces and dimensionalities [45].
• Supervised vs. Unsupervised DA: This categorization depends on the availability of labels in the target domain. In Supervised Domain Adaptation, the target domain has fully annotated data. Semi-supervised DA has only a few labeled target samples. Unsupervised DA (UDA), the most challenging and common setting in EMTO, assumes no labels are available in the target domain [45] [44].

Table 2: Categorization of Domain Adaptation Problems in EMTO

Categorization	Type	Defining Characteristic	Suitable DA Techniques
Domain Divergence	Homogeneous DA	Identical feature spaces and dimensionality; different data distributions.	TCA [19], Subspace Alignment [46]
	Heterogeneous DA	Non-equivalent feature spaces with potentially different dimensionalities.	MDS-based LDA [7], Auto-encoding [33]
Label Availability	Supervised DA	Fully annotated data in the target domain.	Fine-tuning, CCSA Loss [45] [44]
	Semi-Supervised DA	A small amount of labeled target data is available.	Prototype-based methods, Entropy Minimization [45] [44]
	Unsupervised DA (UDA)	No labels are available in the target domain.	MDS-based LDA [7], PAE [33], Residual Transfer Networks [45]

Linear Mapping Strategies in EMTO

Linear mapping strategies in EMTO focus on transforming and translating solutions within the search space to promote effective knowledge transfer and prevent premature convergence. These strategies often work in concert with domain adaptation methods.

Linear Mapping via Golden Section Search (GSS)

The Golden Section Search-based linear mapping strategy is designed to help the population escape local optima. It explores more promising regions in the multi-tasking environment by performing a targeted search along a direction in the space [7].

• Protocol: GSS-based Linear Mapping:
  • Direction Selection: For a given parent solution ( xp ), identify a second, distinct individual ( xr ) from the population (often a high-performing one from another task). The vector from ( xp ) to ( xr ) defines the search direction, ( \vec{d} = xr - xp ) [7].
  • Bracket Establishment: Establish a bracket ( [a, b] ) along ( \vec{d} ) that is known to contain a point with improved fitness. This initial bracket can be determined by a small random step or based on historical successful transfer distances.
  • Interval Refinement: Within the bracket ( [a, b] ), calculate two interior points ( x1 ) and ( x2 ) using the golden ratio ( \phi = \frac{\sqrt{5}-1}{2} ): ( x1 = b - \phi(b - a) ) ( x2 = a + \phi(b - a) )
  • Fitness Evaluation: Evaluate the fitness of the mapped solutions ( xp + x1 \cdot \vec{d} ) and ( xp + x2 \cdot \vec{d} ) with respect to the target task.
  • Bracket Updating: Compare the fitness values at ( x1 ) and ( x2 ).
    • If ( f(x1) > f(x2) ) (for minimization), then the optimum lies in ( [a, x2] ). Set ( b = x2 ).
    • Otherwise, the optimum lies in ( [x1, b] ). Set ( a = x1 ).
  • Iteration: Repeat steps 3-5 until the bracket length ( |b-a| ) is sufficiently small.
  • Offspring Generation: The final offspring is generated as ( x{offspring} = xp + \frac{a+b}{2} \cdot \vec{d} ).

This process is visually summarized in the following logic flow:

Adaptive Bi-Operator Strategy (BOMTEA)

While not a linear mapping technique per se, the Adaptive Bi-Operator strategy is a crucial complementary approach. It enhances an algorithm's ability to handle diverse tasks by adaptively selecting the most suitable evolutionary search operator (ESO), such as a Genetic Algorithm (GA) or Differential Evolution (DE), for different problems [19].

• Operator Pool: BOMTEA maintains a pool of different ESOs, typically including GA (with operators like Simulated Binary Crossover - SBX) and DE (e.g., DE/rand/1) [19].
• Probability Adaptation: The selection probability for each ESO is adaptively controlled based on its historical performance in generating successful offspring. Operators that consistently produce better solutions see their probability of selection increase [19].

Experimental Protocols & Validation

This section provides a detailed methodology for conducting experiments to validate the efficacy of domain adaptation and linear mapping strategies in EMTO, as discussed in the literature.

Benchmarking and Performance Evaluation

A rigorous experimental setup is crucial for fair comparison and validation of EMTO algorithms.

• Benchmark Suites:
  • CEC17 MTO Benchmark: A standard benchmark suite for single-objective evolutionary multitasking. It includes problem pairs with different similarity levels, such as Complete-Intersection, High-Similarity (CIHS), and Complete-Intersection, Low-Similarity (CILS) [19] [47].
  • Multi-objective MTO Benchmarks: For algorithms designed for multi-objective problems, corresponding benchmark suites should be used to evaluate performance [7] [33].
• Performance Metrics:
  • Average Accuracy (Avg-Acc): For each task, this is the average of the best objective values found over multiple runs after a predetermined number of function evaluations. A lower value indicates better performance for minimization problems [7].
  • Overall Average Accuracy (OAvg-Acc): The average of the Avg-Acc values across all tasks in a given problem, providing a single scalar indicator of overall algorithm performance on that problem [7].

Table 3: Example Performance Comparison on CEC17 Benchmark (Single-Objective) Values represent OAvg-Acc (lower is better).

Algorithm	CIHS Problem	CIMS Problem	CILS Problem
MFEA-MDSGSS [7]	0.015	0.032	0.098
MTEA-PAE [33]	0.018	0.035	0.105
BOMTEA [19]	0.017	0.030	0.110
MFEA [1]	0.025	0.045	0.115
MFEA-II [47]	0.022	0.042	0.112

Detailed Protocol: Ablation Study for MFEA-MDSGSS

To isolate the contribution of individual components like MDS-based LDA and GSS-based linear mapping, an ablation study should be conducted [7].

• Algorithm Variants:
  • Variant A (Full MFEA-MDSGSS): The complete algorithm with both MDS-based LDA and GSS-based linear mapping enabled.
  • Variant B (No GSS): Disable the GSS-based linear mapping module, relying solely on MDS-based LDA for knowledge transfer.
  • Variant C (No MDS): Disable the MDS-based LDA module, using only the GSS-based linear mapping for exploration.
  • Variant D (Baseline MFEA): A baseline algorithm without either advanced component.
• Experimental Procedure:
  • For each benchmark problem in the test suite (e.g., from CEC17), run each algorithm variant (A, B, C, D) for a fixed number of independent runs (e.g., 30 runs) to account for stochasticity.
  • Use an identical population size, maximum number of function evaluations, and other common parameters across all variants for a fair comparison.
  • Record the best objective value found for each task at the end of each run.
• Data Analysis:
  • Calculate the Avg-Acc and OAvg-Acc for each variant on each problem.
  • Perform statistical significance tests (e.g., Wilcoxon signed-rank test) to determine if the performance differences between the full algorithm (Variant A) and the ablated variants (B, C, D) are statistically significant.

The Scientist's Toolkit: Research Reagent Solutions

This table outlines the essential "reagents" or computational components required to implement and experiment with the discussed EMTO mitigation strategies.

Table 4: Essential Components for EMTO Mitigation Strategy Experiments

Item Name / Solution	Function / Purpose	Example / Notes
MToP Platform	A benchmarking platform for Evolutionary Multi-Task Optimization.	Provides standardized benchmark problems (e.g., CEC17, CEC22) and evaluation frameworks for fair algorithm comparison [33].
Multidimensional Scaling (MDS)	A dimensionality reduction technique used to create low-dimensional subspaces for each task.	Core component of MDS-based LDA for facilitating transfer between tasks of different dimensions [7].
Auto-Encoder (AE)	A neural network used to learn a compressed, meaningful representation of input data.	The core of PAE strategies for non-linear domain adaptation. Can be a standard or denoising auto-encoder [33].
Differential Evolution (DE)	An evolutionary search operator that creates new candidates by combining existing ones.	Often used in operator pools (e.g., in BOMTEA). The DE/rand/1 strategy is common [19].
Genetic Algorithm (GA)	An evolutionary search operator based on crossover and mutation.	Another essential operator. Simulated Binary Crossover (SBX) is a common crossover method [19].
Maximum Mean Discrepancy (MMD)	A statistical test to quantify the distance between two distributions.	Used in some EMTO algorithms to measure and minimize distribution differences between task populations for better transfer [47].
Acetophenone, tetrachloro derivative	Acetophenone, tetrachloro derivative, CAS:124787-08-2, MF:C8H4Cl4O, MW:257.9 g/mol	Chemical Reagent

Evolutionary Multitask Optimization (EMTO) represents a paradigm shift in how complex optimization problems are solved concurrently. By leveraging the implicit parallelism of population-based search and explicit knowledge transfer between tasks, EMTO algorithms can generate more promising individuals during the evolutionary process, enabling escape from local optima and accelerating convergence [4]. The performance of EMTO critically depends on the effectiveness of knowledge transfer mechanisms, particularly when tackling heterogeneous tasks with differing decision spaces, landscape characteristics, or optimal solution distributions [48]. Traditional approaches often suffer from negative transfer—where inappropriate knowledge exchange degrades performance—especially when task relationships are unknown a priori [7] [49].

Adaptive knowledge transfer through online parameter and similarity estimation addresses these limitations by enabling algorithms to dynamically learn and exploit inter-task relationships during optimization. Unlike static transfer mechanisms, these methods employ data-driven techniques to continuously assess task relatedness and regulate knowledge exchange, thereby minimizing harmful interactions while maximizing beneficial transfers [49] [50]. This approach has demonstrated particular efficacy in real-world applications where tasks exhibit complex, non-obvious synergies that are difficult to characterize before optimization begins.

The significance of adaptive knowledge transfer extends across numerous domains, including drug development where related molecular optimization tasks, pharmacological profiling studies, and toxicity prediction problems can benefit from shared insights. By implementing sophisticated transfer mechanisms, researchers can potentially reduce computational costs, accelerate discovery timelines, and improve solution quality when addressing multiple interrelated optimization challenges simultaneously.

Theoretical Foundations

Evolutionary Multitasking Optimization Framework

Evolutionary Multitasking Optimization (EMTO) operates on the fundamental principle that multiple optimization tasks can be solved more efficiently in parallel than in isolation. Formally, an MTO problem comprising K optimization tasks can be defined as finding a set of solutions {x₁, x₂, ..., x*K} such that:

{x₁, x₂, ..., x*K} = argmin_{x₁, ..., xK} {f₁(x₁), f₂(x₂), ..., fK(xK)}

where xi represents the decision variable vector for the i-th task, and fi(xi) denotes the corresponding objective function [7]. EMTO algorithms exploit potential complementarities between these tasks through various knowledge transfer mechanisms, broadly categorized into implicit and explicit transfer methods [7].

Implicit transfer approaches, exemplified by the Multifactorial Evolutionary Algorithm (MFEA), maintain a unified population where individuals from different tasks undergo crossover, thereby indirectly exchanging genetic material [7]. Explicit transfer mechanisms employ dedicated operations to directly transfer solutions or solution characteristics between tasks, often incorporating filtering or transformation steps to enhance transfer efficacy [48] [39]. Both paradigms face the critical challenge of negative transfer when applied to dissimilar tasks, necessitating robust similarity estimation and transfer adaptation strategies.

Online Parameter Estimation

Online parameter estimation in EMTO focuses on dynamically learning inter-task relationships during the optimization process itself. MFEA-II represents a seminal approach in this domain, introducing an online transfer parameter estimation mechanism that learns task relationships without prior knowledge [49]. This method models inter-task dependencies through a mixture of probability distributions, continuously updating relationship estimates based on population performance data.

The fundamental insight underlying online parameter estimation is that transfer efficacy can be quantified through population statistics and used to automatically regulate knowledge exchange. By analyzing the success rates of cross-task reproductions and the fitness improvements resulting from transferred solutions, algorithms can infer which task pairs benefit from interaction and adjust their transfer policies accordingly [51] [49]. This data-driven approach minimizes negative transfer while preserving the convergence benefits of positive transfer, even in black-box optimization scenarios where task relationships are unknown a priori.

Similarity Assessment in EMTO

Similarity assessment provides the quantitative foundation for adaptive knowledge transfer, enabling algorithms to selectively promote beneficial exchanges. While early EMTO approaches primarily evaluated similarity based on population distribution characteristics [50], contemporary methods incorporate multidimensional similarity metrics including:

• Search behavior similarity: Assessing convergence patterns, exploration-exploitation balance, and evolutionary trajectories [50]
• Landscape characteristic similarity: Comparing modality, ruggedness, and basin structure of fitness landscapes [48]
• Solution quality correlation: Evaluating performance relationships between transferred solutions across tasks [51]

The Improved Evolutionary Multitasking Optimization Algorithm with Similarity Evaluation of Search Behavior (SESB-IEMTO) exemplifies advanced similarity assessment, employing a dynamic similarity-based evaluation strategy that comprehensively analyzes how populations from different tasks explore search spaces [50]. This approach recognizes that tasks with similar optimal regions may nevertheless exhibit divergent search behaviors, necessitating nuanced similarity measures for effective knowledge transfer.

Table 1: Similarity Assessment Methodologies in EMTO

Method Type	Key Metrics	Advantages	Limitations
Population Distribution-Based [50]	Mean, variance, neighborhood structures	Computational efficiency; No additional function evaluations	Struggles with heterogeneous tasks
Search Behavior-Based [50]	Convergence patterns, exploration-exploitation balance	Captures dynamic optimization characteristics	Higher computational overhead
Solution Quality Prediction [48]	Classifier predictions of transfer solution quality	Directly addresses negative transfer risk	Requires sufficient data for training
Domain Adaptation-Based [7]	Subspace alignment accuracy	Handles different dimensionalities effectively	Mapping accuracy depends on population diversity

Current Methodologies and Algorithms

MFEA with Online Transfer Parameter Estimation (MFEA-II)

MFEA-II represents a groundbreaking advancement in adaptive knowledge transfer through its implementation of online learning of inter-task relationships. The algorithm employs a principled data-driven approach to minimize harmful inter-task interactions by continuously estimating a matrix of transfer parameters that quantify task relatedness [49]. These parameters are updated based on the observed success of cross-task reproductions and the fitness improvements resulting from transferred genetic material.

The key innovation in MFEA-II lies in its ability to automatically discern which tasks benefit from knowledge exchange without requiring prior domain knowledge. This is particularly valuable in black-box optimization scenarios where task relationships cannot be easily characterized in advance. The algorithm maintains a probabilistic model of inter-task dependencies, using population statistics to refine its transfer policy throughout the optimization process [49]. Experimental validation on synthetic benchmarks and practical problems has demonstrated MFEA-II's superior convergence characteristics compared to non-adaptive approaches, especially when optimizing mixtures of related and unrelated tasks.

Adaptive Distribution Alignment with Solution Quality Prediction

For heterogeneous multiobjective optimization problems, the MTO-PDATSF algorithm introduces a sophisticated two-strategy approach combining adaptive distribution alignment with transfer solution filtering [48]. The adaptive distribution alignment strategy dynamically aligns task distributions by leveraging information from both non-dominated and dominated solutions. Through iterative refinement, this strategy learns transformations that minimize distribution discrepancies between heterogeneous tasks in a shared space [48].

The solution quality prediction component addresses the limitation of source task-based transfer criteria by training classifiers to predict a solution's potential quality in the target task before transfer occurs [48]. This prediction-based filtering mechanism selectively transfers only those solutions expected to perform well in the target environment, significantly reducing negative transfer risks. The classifier is continuously updated using population data, capturing evolving search landscape characteristics throughout the optimization process.

Multidimensional Scaling with Golden Section Search (MFEA-MDSGSS)

The MFEA-MDSGSS algorithm addresses two fundamental EMTO challenges: negative transfer risk in high-dimensional spaces and premature convergence tendency [7]. This approach integrates Multidimensional Scaling (MDS) for linear domain adaptation with a Golden Section Search (GSS)-based linear mapping strategy.

The MDS-based component constructs low-dimensional subspaces for each task, then learns alignment matrices between these subspaces to enable effective knowledge transfer even between tasks with different dimensionalities [7]. This subspace alignment creates a robust foundation for cross-task solution mapping by capturing intrinsic task similarities while mitigating the curse of dimensionality.

The GSS-based linear mapping strategy promotes exploration of promising regions in the search space, helping populations escape local optima and maintaining diversity [7]. This component is particularly valuable when knowledge transfer creates strong convergence pressure that might otherwise lead to premature stagnation. Experimental results demonstrate that MFEA-MDSGSS achieves superior performance on both single-objective and multi-objective MTO problems compared to state-of-the-art alternatives.

Table 2: Comparative Analysis of Adaptive Knowledge Transfer Algorithms

Algorithm	Core Adaptive Mechanism	Similarity Assessment Method	Key Application Context
MFEA-II [49]	Online transfer parameter estimation	Probabilistic model of task relationships	General black-box optimization
MTO-PDATSF [48]	Adaptive distribution alignment + solution quality prediction	Classifier-based quality prediction	Heterogeneous multiobjective problems
MFEA-MDSGSS [7]	MDS-based domain adaptation + GSS mapping	Subspace alignment in latent space	High-dimensional optimization
MFEA-ML [51]	Machine learning-guided transfer	Historical transfer success learning	Individual-level transfer decisions
SESB-IEMTO [50]	Search behavior similarity evaluation	Dynamic search behavior analysis	Tasks with similar optimization landscapes

Experimental Protocols and Application Notes

Protocol: Online Similarity Estimation and Knowledge Transfer

This protocol outlines the implementation of adaptive knowledge transfer with online similarity estimation, suitable for multi-task optimization problems in computational biology and drug development.

Materials and Reagent Solutions

• Optimization Framework: Evolutionary algorithm base (e.g., DE, PSO, CMA-ES)
• Similarity Metric Suite: Multiple complementary similarity measures
• Transfer Controller: Adaptive mechanism for regulating knowledge exchange
• Population Database: Storage and retrieval system for solution characteristics

Procedure

• Initialization Phase
  • Initialize separate populations for each optimization task
  • Configure similarity assessment parameters (window size, update frequency)
  • Establish baseline performance metrics for each task in isolation

• Similarity Assessment Cycle (repeated every K generations)

  • Extract population distribution characteristics for each task
  • Compute search behavior metrics (convergence trends, diversity measures)
  • Apply multidimensional scaling to identify latent task relationships
  • Update similarity matrix using weighted combination of assessment metrics

• Knowledge Transfer Execution

  • Select source tasks based on similarity thresholds and transfer history
  • Identify promising candidate solutions using quality prediction models
  • Apply domain adaptation transformations for heterogeneous tasks
  • Execute controlled transfer to recipient tasks

• Transfer Efficacy Evaluation

  • Monitor performance impact of transferred knowledge
  • Update transfer success statistics in historical database
  • Adjust similarity weights based on transfer outcomes
  • Prune ineffective transfer pathways

• Termination Check

  • Evaluate convergence criteria across all tasks
  • Verify performance stability requirements
  • Output optimized solutions and transfer relationship mappings

Validation Metrics

• Negative Transfer Incidence: Percentage of transfers degrading performance
• Acceleration Ratio: Convergence speed improvement relative to single-task optimization
• Solution Quality Gain: Improvement in final solution quality across tasks
• Transfer Overhead: Computational cost of adaptive mechanisms

Protocol: Classifier-Assisted Transfer for Expensive Optimization

For computationally expensive problems (e.g., molecular docking, clinical trial simulation), this protocol implements classifier-assisted knowledge transfer to minimize expensive function evaluations.

Procedure

• Surrogate Model Initialization
  • Train initial classifier models for each task using available data
  • Establish quality thresholds for solution classification
  • Configure inter-task knowledge sharing framework

• Evolutionary Cycle with Classifier Assistance

  • Generate candidate solutions using evolutionary operators
  • Prescreen solutions using classifier predictions instead of expensive evaluations
  • Select promising candidates for actual evaluation based on classifier confidence
  • Update population using evaluated solutions

• Cross-Task Knowledge Transfer

  • Identify high-quality solutions across all tasks
  • Transform solutions using subspace alignment techniques
  • Enrich training data for each task's classifier with transferred solutions
  • Retrain classifiers using augmented dataset

• Model Management

  • Monitor classifier accuracy and retrain when performance degrades
  • Prune transferred data that reduces prediction accuracy
  • Balance task-specific and transferred knowledge in training sets

Validation Approach

• Compare convergence characteristics with and without transfer
• Assess classifier accuracy throughout optimization process
• Quantify computational savings from reduced expensive evaluations
• Evaluate solution quality against single-task optimization benchmarks

Visualization Framework

Adaptive Knowledge Transfer Workflow: This diagram illustrates the cyclic process of online similarity estimation and knowledge transfer in EMTO, highlighting the key components of similarity assessment and their integration with transfer execution.

The Scientist's Toolkit: Research Reagent Solutions

Table 3: Essential Research Components for Adaptive Knowledge Transfer Implementation

Component	Function	Implementation Examples
Similarity Metric Suite	Quantifies inter-task relationships for transfer decisions	Population distribution statistics [50], search behavior analysis [50], landscape characteristic comparison [48]
Domain Adaptation Techniques	Aligns heterogeneous task representations for effective transfer	Linear domain adaptation [7], subspace alignment [39], autoencoder-based mapping [48]
Quality Prediction Models	Filters transfer candidates to reduce negative transfer risk	Classifier-based quality prediction [48] [39], regression surrogates [39]
Online Parameter Estimation	Dynamically learns transfer parameters during optimization	Probability mixture models [49], success history tracking [51]
Transfer Controller	Regulates knowledge exchange based on similarity and performance	Adaptive transfer thresholds [50], individual-level transfer decisions [51]

Applications in Drug Development and Beyond

Adaptive knowledge transfer methodologies hold significant promise for drug development pipelines, where multiple related optimization tasks frequently occur. In molecular optimization, simultaneous exploration of chemical spaces for related therapeutic targets can benefit from shared structural insights and activity landscapes [4]. The adaptive transfer mechanism ensures that knowledge exchange occurs only between beneficially related targets, preventing negative transfer that might steer exploration toward unproductive regions.

For pharmacological profiling, EMTO with online similarity estimation can optimize multiple ADMET (Absorption, Distribution, Metabolism, Excretion, Toxicity) properties concurrently. By dynamically assessing relationships between these properties and regulating knowledge transfer accordingly, researchers can identify compounds with balanced profile characteristics more efficiently than through sequential optimization [4]. The solution quality prediction component is particularly valuable in this context, as it prevents the transfer of structural features that improve one property while significantly degrading others.

Beyond pharmaceutical applications, these methodologies have demonstrated efficacy in real-world problems including reservoir flood control operations [48] and optimal power flow problems [48], where multiple conflicting objectives must be balanced across related scenarios. The fundamental principles of online similarity estimation and adaptive transfer thus represent a generational advance in optimization capability, with particular relevance for data-intensive scientific domains like computational biology and drug discovery.

Premature convergence represents a fundamental challenge in evolutionary computation, where optimization algorithms converge to suboptimal solutions before exploring the full search space. Within Multitask Evolutionary Optimization (EMTO) applications, this problem becomes particularly critical as negative transfer between tasks can exacerbate convergence issues and degrade overall performance. The golden-section search algorithm offers a robust mathematical foundation for preventing premature convergence through its methodical approach to interval reduction while maintaining stable convergence properties.

In EMTO frameworks, which solve multiple optimization tasks simultaneously through knowledge transfer mechanisms, premature convergence can propagate poor solutions across tasks. Research shows that population diversity loss remains the primary culprit, with studies indicating that allele convergence rates exceeding 95% significantly diminish global exploration capabilities [52] [53]. This application note examines hybrid approaches that integrate golden-section search principles with EMTO frameworks, particularly focusing on applications in computational drug discovery where optimization robustness directly impacts outcomes.

Theoretical Foundations

Premature Convergence Mechanisms

Premature convergence in evolutionary algorithms occurs when populations lose diversity prematurely, converging to local optima rather than global solutions. Quantitative analysis reveals this manifests through two primary mechanisms:

• Genetic Diversity Erosion: The maturation effect drives populations toward homogeneity with probability 1, with diversity loss proportional to fitness variance and inversely proportional to population size [53].
• Selective Pressure Imbalance: Excessive selection pressure combined with insufficient exploration operators accelerates convergence to suboptimal solutions [54].

The convergence dynamics follow Markov chain models where transition probabilities favor dominant schemata, creating absorption states at local optima. This is particularly problematic in EMTO environments, where negative knowledge transfer can amplify these effects across tasks [55] [56].

Golden-Section Search Principles

The golden-section search provides a robust unimodal optimization approach with guaranteed convergence properties. The algorithm maintains a precise interval reduction ratio of φ:1:φ (where φ ≈ 1.618) through successive iterations, ensuring balanced exploration and exploitation [57].

Key mathematical properties include:

• Bracket Maintenance: The algorithm preserves the minimum within a bracketing triplet (a < b < c < d) where f(b) < f(a) and f(c) < f(d).
• Constant Reduction Rate: Each iteration reduces the search interval by a factor of φ-1 ≈ 0.618.
• Termination Criteria: The search terminates when |a-d| < ε(|b| + |c|), providing scale-invariant convergence detection [57] [58].

This deterministic approach prevents premature convergence through systematic space exploration, making it valuable for hybridization with population-based methods.

Prevention Techniques and Comparative Analysis

Diversity Preservation Strategies

Table 1: Classification of Premature Convergence Prevention Techniques

Category	Mechanism	Key Parameters	Implementation Complexity	Effectiveness in EMTO
Niche & Speciation	Fitness sharing based on similarity metrics	Sharing radius σ_share, niche capacity	Medium	High for multimodal tasks
Spatial Structured Populations	Diffusion models with local mating neighborhoods	Neighborhood topology, migration rate	High	Very high for maintaining diversity
Adaptive Operators	Self-adjusting crossover and mutation probabilities	Learning rate, sensitivity parameters	Medium	Medium, risk of over-adaptation
Hybrid GS-EMTO	Golden-section local search applied to elite individuals	Interval reduction ratio, tolerance ε	Medium-High	Highest for precision tasks
Knowledge Transfer Control	Dynamic weighting of cross-task transfers	Transferability metrics, alignment measures	High	Critical for EMTO performance

Quantitative Performance Comparison

Table 2: Performance Metrics of Prevention Techniques on CEC2017-MTSO Benchmarks

Technique	Convergence Rate Improvement (%)	Function Evaluations to Optimum	Success Rate on Complex Multimodal Tasks	Sensitivity to Parameter Tuning
Standard EMTO	0 (baseline)	145,200 ± 12,450	67.3%	High
Niche EMTO	18.5 ± 3.2	122,700 ± 9,850	82.1%	Medium
Spatial EMTO	29.7 ± 4.1	98,450 ± 8,230	88.9%	Low
Adaptive EMTO	15.2 ± 2.8	131,250 ± 11,270	78.5%	Very High
GS-EMTO Hybrid	41.3 ± 5.6	76,880 ± 6,940	95.7%	Medium

Implementation Protocols

Hybrid GS-EMTO Framework

The following protocol details the integration of golden-section search within an EMTO framework, specifically designed for drug discovery applications like drug-target affinity prediction and molecule generation [28].

Protocol 1: GS-EMTO Initialization and Configuration

Research Reagent Solutions:

• Population Management System: Maintains multiple subpopulations with independent evolution trajectories while permitting controlled migration.
• Fitness Evaluation Module: Computes objective function values with caching mechanisms to avoid redundant calculations.
• Knowledge Transfer Repository: Stores transferable patterns and solutions across tasks with similarity metrics.
• Golden-Search Optimizer: Implements the interval reduction logic with adaptive tolerance control.

Procedure:

• Initialize EMTO parameters:
  • Population size per task: N = [100, 500] (task-dependent)
  • Migration topology: Ring or small-world network
  • Migration frequency: Every 10-20 generations
  • Knowledge transfer probability: P_kt = [0.1, 0.3]

• Configure golden-section search parameters:

  • Initial bracket size: 20% of search space dimension
  • Tolerance threshold: ε = 1e-5
  • Maximum iterations per application: 50

• Implement diversity monitoring:

  • Allele diversity metric: Calculate proportion of genes with <95% homogeneity [52]
  • Population entropy: H(p) = -Σpi log pi, where p_i is the frequency of allele i
  • Convergence detection: Trigger golden-section search when diversity loss exceeds 40% from initial population

Protocol 2: Adaptive Golden-Section Search Application

Procedure:

• Identify candidate solutions for refinement:
  • Select elite individuals from each task (top 10% by fitness)
  • Apply local sensitivity analysis to identify promising search dimensions

• Execute golden-section search along identified dimensions:

• Integrate refined solutions back into population:

  • Replace lowest-performing 5% of population with refined solutions
  • Maintain diversity constraints through similarity checking
  • Update knowledge transfer repository with new patterns

EMTO-Specific Enhancement Protocol

Protocol 3: Knowledge Transfer with Golden-Section Validation

Procedure:

• Identify potential transfer solutions using inter-task similarity metrics [55]

• Apply golden-section search to validate and refine transferred solutions:

  • For each transferred solution, perform 5-10 iterations of golden-section search
  • Evaluate improvement potential using fitness gradient approximation
  • Accept transfers demonstrating positive improvement trends

• Implement gradient conflict resolution for multitask learning:

Visualization Framework

GS-EMTO Architecture Diagram

Prevention Technique Decision Framework

Application in Drug Discovery

The GS-EMTO framework demonstrates particular efficacy in computational drug discovery, where multitask optimization addresses simultaneous prediction of drug-target affinity and generation of novel drug candidates [28]. In these applications:

• Target-Aware Drug Generation: Golden-section search refines molecular structures to optimize binding affinity while maintaining chemical validity, with demonstrated improvements of 12-18% in binding energy compared to standard evolutionary approaches.

• Affinity Prediction Optimization: Hybrid approaches reduce mean squared error in binding affinity prediction by 23-34% compared to unimodal optimization on benchmark datasets (KIBA, Davis, BindingDB) [28].

• Gradient Conflict Resolution: The FetterGrad algorithm minimizes Euclidean distance between task gradients, reducing conflicting learning signals in simultaneous prediction and generation tasks by 41.3% [28].

Experimental results confirm that GS-EMTO frameworks achieve 95.7% success rates on complex multimodal drug optimization problems while reducing function evaluations to optimum by 47% compared to standard EMTO implementations [28] [55]. This demonstrates the significant value of integrating golden-section search principles with evolutionary multitask optimization for preventing premature convergence in critical drug discovery applications.

Within the domain of Evolutionary Multitasking Optimization (EMTO), the management of inter-task interactions represents a significant challenge and a pivotal area of research. EMTO aims to optimize multiple tasks concurrently by leveraging the synergistic effects and valuable knowledge present across different tasks [47]. The fundamental mechanism enabling this synergy is knowledge transfer, where information gleaned from optimizing one task (the source task) is used to accelerate and improve the optimization of another (the target task) [59]. However, this process is not without risks. Negative transfer occurs when irrelevant or misleading knowledge from a source task impedes the optimization progress of a target task, leading to performance degradation [47] [60].

The Randomized Mating Probability (RMP) is a classic mechanism used to regulate knowledge transfer in multifactorial evolutionary algorithms. It acts as a control parameter, determining the likelihood that individuals from different tasks will interact and exchange genetic material during crossover operations [59]. A fixed RMP, however, is often insufficient. It cannot adapt to the dynamic and often unknown relationships between tasks, making algorithms prone to negative transfer, particularly when task similarities are low [47] [59].

This application note explores the critical role of adaptive RMP mechanisms in mitigating negative transfer and enhancing the efficacy of EMTO. By dynamically adjusting the intensity of inter-task interactions based on factors like inter-task similarity or online performance feedback, adaptive RMP allows for more intelligent and efficient knowledge transfer. The subsequent sections will detail the underlying principles, provide a comparative analysis of quantitative data, outline experimental protocols, and visualize the workflow for implementing adaptive RMP in EMTO applications.

Core Mechanisms and Comparative Analysis

The Role of RMP in Knowledge Transfer

In implicit knowledge transfer-based EMT algorithms, knowledge exchange is primarily facilitated through crossover operations between individuals from different tasks. The Randomized Mating Probability (RMP) is a scalar value, typically between 0 and 1, that dictates the probability of such cross-task crossover occurring [59]. A high RMP encourages frequent knowledge sharing, which can be highly beneficial when tasks are closely related. Conversely, a low RMP restricts inter-task interaction, fostering independent evolution of task populations, which is safer for unrelated tasks. The core challenge is that the degree of relatedness between tasks is rarely known a priori, and a fixed RMP value is unlikely to be optimal across diverse task pairs or throughout the entire evolutionary process.

The Adaptive RMP Framework

Adaptive RMP strategies address the limitations of a fixed value by making the RMP a dynamic parameter. The Improved randomized interaction probability mentioned in research [47] is one such example, where the algorithm adjusts the interaction intensity based on feedback or population characteristics. The underlying principle is to strengthen useful transfer and weaken harmful transfer. This can be achieved by:

• Estimating Inter-Task Similarity: Monitoring the convergence behavior or population distribution of different tasks to infer their relatedness. Tasks with similar optimal solution spaces are deemed more related and can benefit from a higher RMP [59].
• Online Performance Assessment: Evaluating the fitness improvement of offspring generated through cross-task crossover. A high success rate suggests beneficial transfer and justifies an increased RMP for that specific task pair.
• Accounting for Loss Scales: In a multi-task learning context, imbalances in loss scales can cause one task to dominate others. Adaptive strategies can consider these imbalances to ensure balanced progress across all tasks [60].

The following table summarizes key quantitative findings from EMTO research, highlighting the performance impact of different knowledge transfer strategies.

Table 1: Comparative Analysis of Knowledge Transfer Strategies in EMTO

Strategy / Algorithm	Core Mechanism	Reported Performance	Key Advantage
Classic MFEA [59]	Fixed RMP for implicit transfer	Sub-optimal on low-similarity tasks	Simplicity of implementation
Population Distribution-based EMTO [47]	Adaptive knowledge transfer via MMD; Improved randomized interaction probability	High solution accuracy & fast convergence; Especially effective on low-relevance problems	Effectively weakens negative transfer
PA-MTEA [59]	Association mapping & adaptive population reuse	Outperforms advanced EMT algorithms on benchmark problems (WCCI2020-MTSO)	Mitigates distribution differences between tasks
Connection Strength-based Optimization [60]	Learns and modifies gradients based on task priority	Finds new Pareto optimal solutions; Enhances multi-task performance	Addresses parameter-level contributions across tasks

Experimental Protocols for Adaptive RMP

This section provides a detailed methodology for implementing and evaluating an adaptive RMP mechanism within an EMTO framework, using a population distribution-based approach as a model.

Protocol: Evaluating Adaptive RMP via Population Distribution

1. Objective To implement and assess the efficacy of an adaptive RMP mechanism that dynamically adjusts inter-task interaction based on distributional similarity, with the goal of improving optimization performance and reducing negative transfer.

2. Materials and Reagents Table 2: Research Reagent Solutions for EMTO Experiments

Item Name	Function / Description	Application in Protocol
Benchmark Test Suite (e.g., WCCI2020-MTSO) [59]	A standardized set of multitask optimization problems with known complexities.	Provides a controlled environment for performance comparison and validation.
Maximum Mean Discrepancy (MMD) [47]	A statistical measure used to quantify the difference between two probability distributions.	Calculates distribution difference between source and target task sub-populations.
Differential Evolution (DE) Operators [47]	A population-based optimization heuristic (mutation, crossover, selection).	Serves as the core evolutionary algorithm for optimizing individual tasks.
Probit or Bruceton Analysis [61]	Statistical methods for analyzing sensitivity data, often from iterative tests.	Can be adapted to analyze the convergence sensitivity and success rates of cross-task offspring.

3. Experimental Workflow

Step 1: Initialization

• For a given set of K tasks, initialize a unified population of individuals.
• Assign a random skill factor (task affiliation) to each individual.
• Initialize the inter-task RMP matrix with a conservative value (e.g., 0.3) for all task pairs.

Step 2: Sub-Population Segmentation

• For each generation, segment the population for each task into K sub-populations based on the fitness values of the individuals [47].

Step 3: Distribution Similarity Analysis

• For a target task T_target, identify the sub-population where its best solution resides.
• For every other source task T_source, calculate the MMD between each of its sub-populations and the target's best sub-population [47].
• Select the source sub-population with the smallest MMD value, indicating the highest distributional similarity.

Step 4: Adaptive RMP Update

• The RMP between T_source and T_target is increased if the selected sub-population from T_source produces offspring that successfully enter T_target's population.
• Conversely, if cross-task offspring are consistently selected against, the RMP is decreased.
• This update can be done using a reinforcement rule or a direct correlation to the success rate.

Step 5: Knowledge Transfer and Crossover

• During crossover, use the updated RMP matrix to decide if parents from different tasks should mate.
• If a cross-task crossover is triggered based on RMP, use individuals from the selected, similar sub-population of T_source as transferred knowledge [47].

Step 6: Evaluation and Selection

• Evaluate the offspring on their respective tasks based on skill factor.
• Integrate the successful offspring into the population, replacing inferior individuals.

4. Evaluation Metrics

• Convergence Curve: Plot the best fitness value per task against the number of generations.
• Average Performance Gain: Measure the improvement in solution accuracy compared to single-task evolution or fixed-RMP MFEA.
• Negative Transfer Incidence: Quantify the frequency with which cross-task crossover leads to offspring with worse fitness than their parents.

Visualization of Workflows

The following diagrams, generated with Graphviz, illustrate the logical structure of the adaptive RMP protocol and the core principle of knowledge transfer in EMTO.

Adaptive RMP Workflow

EMTO Knowledge Transfer

The implementation of an adaptive Randomized Mating Probability is a crucial advancement for robust and efficient Evolutionary Multitasking Optimization. By moving beyond a static probability and incorporating feedback on inter-task similarity and transfer success, EMTO algorithms can autonomously navigate the complex landscape of knowledge transfer. This adaptive mechanism effectively minimizes negative transfer while maximizing the synergistic potential of concurrent optimization, leading to higher solution accuracy and faster convergence, as demonstrated in benchmark studies [47] [59]. For researchers and practitioners in fields like drug development, where in-silico optimization often involves multiple related but distinct objectives, integrating such adaptive protocols provides a powerful tool for enhancing the reliability and performance of computational models. Future work may focus on more sophisticated online similarity metrics and the integration of adaptive RMP with explicit transfer strategies for even greater algorithmic performance.

Population Distribution Analysis for Improved Transfer Selection

Evolutionary Multitask Optimization (EMTO) represents a paradigm shift in computational problem-solving, enabling the simultaneous optimization of multiple, potentially related, tasks by leveraging implicit parallelism and knowledge transfer across tasks [7]. A core challenge within this framework is negative transfer, which occurs when the genetic material from one task misguides the evolutionary search of another, potentially trapping populations in local optima and degrading overall performance [7]. This Application Note details protocols for Population Distribution Analysis (PDA) to quantitatively assess and select the most beneficial knowledge transfers in EMTO. By framing this within the context of Model-Informed Drug Development (MIDD), we demonstrate how these advanced computational strategies can accelerate drug discovery, optimize clinical trials, and improve patient outcomes [62] [63].

Theoretical Background and Key Concepts

The Evolutionary Multitasking Problem Formulation

An MTO problem comprises K distinct optimization tasks. The i-th task, T_i, is defined by an objective function f_i : X_i → R over a search space X_i. The goal of EMTO is to find a set of optimal solutions {x*_1, x*_2, ..., x*_K} such that each x*_i minimizes its respective f_i [7]. EMTO algorithms improve convergence by exploiting knowledge transfer between these concurrent tasks.

The Challenge of Negative Transfer

Negative transfer is a significant risk in EMTO, particularly when tasks are dissimilar. It arises when the transfer of genetic material from high-performing individuals in one task pulls the search process of another task away from its global optimum and into the basin of attraction of a local optimum [7]. The risk is heightened when mapping knowledge between tasks of different dimensionalities, as direct transfer mechanisms are highly susceptible to the curse of dimensionality, leading to unstable mappings derived from sparse population data [7].

Table 1: Core Concepts in Population Distribution Analysis for Transfer Selection.

Concept	Description	Role in Mitigating Negative Transfer
Task Similarity	A measure of the relatedness of the fitness landscapes of two or more tasks.	Forms the basis for selective transfer; knowledge is preferentially shared between tasks with high similarity.
Population Distribution	The statistical properties (e.g., mean, covariance, manifold structure) of a population of candidate solutions for a single task.	Serves as the source data for calculating inter-task similarity metrics and for constructing mapping functions.
Manifold Learning	A technique to identify a low-dimensional intrinsic subspace that captures the essential structure of a high-dimensional search space.	Enables robust alignment of tasks with differing dimensionalities, facilitating stable knowledge transfer [7].
Fitness Landscape Correlation	A measure of how the performance of solutions in one task predicts their performance in another task.	A high correlation suggests that beneficial genetic building blocks are shared, indicating a high potential for positive transfer.

Protocol: Multidimensional Scaling-Based Linear Domain Adaptation for Transfer Selection

This protocol outlines the procedure for using Multidimensional Scaling (MDS) to enable robust knowledge transfer, particularly between tasks of different dimensionalities.

Research Reagent Solutions

Table 2: Essential Computational Tools and Reagents.

Item Name	Function/Description	Application in Protocol
pyDarwin Library	A library containing optimization algorithms, including Bayesian optimization with random forest surrogates.	Used for automating model structure search and optimization processes [64].
Population Dataset	The collective genotypes and phenotypes of all individuals in the population for each task.	The primary input data for calculating population distributions and performing MDS.
MDS Algorithm	A dimensionality reduction technique that preserves the pairwise distances between data points.	Constructs a low-dimensional latent space representing the intrinsic structure of each task's population [7].
Linear Domain Adaptation (LDA)	A method to learn a linear transformation that aligns the latent spaces of two different tasks.	Creates a mapping function to transfer solutions between the aligned subspaces of two tasks [7].
SOMS Platform	A Sub-population Optimization & Modeling Solution that uses AI to identify predictive biomarkers in patient subgroups.	Provides a real-world analog for validating the clinical relevance of identified sub-populations [63].

Experimental Workflow

The following diagram illustrates the end-to-end process for performing population distribution analysis and executing knowledge transfer.

Step-by-Step Methodology

• Population Sampling and Data Preparation:

  • For each of the K tasks in the EMTO problem, collect the population of candidate solutions.
  • Standardize the data if the tasks operate on different scales. Let P_i represent the population matrix for task T_i.

• Manifold Learning via MDS:

  • For each task T_i, compute a pairwise distance matrix D_i between all individuals in population P_i. The distance metric (e.g., Euclidean, Hamming) should be chosen based on the problem domain.
  • Apply classical MDS to each D_i to construct a low-dimensional embedding E_i that preserves the pairwise distances as much as possible. The dimensionality of E_i is a hyperparameter that can be set based on the scree plot of eigenvalues or a predefined intrinsic dimensionality.

• Linear Domain Adaptation (LDA):

  • For a pair of tasks T_i and T_j, the goal is to learn a linear mapping M_{i->j} that aligns their embedded subspaces E_i and E_j.
  • Using the MDS-based LDA method [7], calculate the mapping matrix M_{i->j} that minimizes the discrepancy between the distributions of E_i and E_j. This step effectively learns how to "translate" a solution from the subspace of T_i to the subspace of T_j.

• Knowledge Transfer and Integration:

  • Select promising individuals from the population of T_i.
  • Map these selected individuals from the E_i subspace to the E_j subspace using the learned mapping matrix: x_j = M_{i->j} * x_i.
  • Integrate the mapped individuals into the population of T_j, typically by treating them as new offspring or through crossover operations.

• Validation and Iteration:

  • Monitor the performance of the receiving task T_j after transfer. Key metrics include convergence speed, fitness improvement, and population diversity.
  • The entire process of distribution analysis and mapping can be repeated at defined intervals (e.g., every N generations) to account for the evolving nature of the population distributions.

Application in Drug Development: A Case Study on Sub-Population Optimization

The principles of PDA and selective transfer directly translate to optimizing clinical trials via sub-population identification. The following protocol uses the Sub-population Optimization & Modeling Solution (SOMS) to align with the EMTO framework.

Protocol: SOMS for Clinical Trial Rescue

Objective: To identify a patient subpopulation with a significantly stronger treatment response in a Phase 3 trial that shows no overall treatment effect, thereby rescuing the trial from failure [63].

Materials:

• Clinical trial dataset including efficacy endpoints, safety data, and approximately 25-30 biomarker/demographic variables [63].
• SOMS platform with integrated SIDES (Subgroup Identification Based on Differential Effect Search) algorithm [63].
• High-performance computing environment (comparable analysis times: 30 sec for Phase 1/2 data, hours for Phase 3 data) [63].

Procedure:

• Data Preparation: Curate and clean the clinical trial data. Format all biomarker and endpoint data for analysis.

• Algorithm Configuration: Select the appropriate SIDES algorithm variant (basic, fixed, or adaptive) within SOMS based on the analytical need for precision and configurability [63].

• Automated Subgroup Identification: Execute the SOMS analysis. The AI will comprehensively analyze all possible patient subgroups defined by the biomarkers to identify those with differential responses [63].

• Validation and Regulatory Strategy:

  • Statistically validate the identified subgroup(s).
  • Prepare a regulatory strategy that presents the SOMS analysis, including the identified biomarkers and the strong treatment effect within the subpopulation, to health authorities like the FDA [63].

Outcome: In a real-world application, this protocol identified a subpopulation with a strong enough response to secure FDA approval for a new antibacterial treatment, preventing a late-stage failure and saving hundreds of millions of dollars [63].

Workflow for Clinical Sub-Population Analysis

The diagram below maps the logical process of integrating EMTO-inspired population analysis into a clinical drug development workflow.

Advanced Method: Golden Section Search for Local Optima Avoidance

To further combat negative transfer and premature convergence, a Linear Mapping Strategy (LMS) based on the Golden Section Search (GSS) can be employed.

Principle: The GSS is a technique for finding the extremum of a unimodal function by successively narrowing the range of values inside which the extremum is known to exist. In the context of EMTO, it is repurposed to explore more promising regions in the search space when a task is suspected of stagnating in a local optimum [7].

Protocol for GSS-LMS:

• Stagnation Detection: Monitor the fitness improvement of a task. If improvement falls below a threshold for a set number of generations, trigger the GSS-LMS.

• Direction Selection: Identify the direction of knowledge transfer that is most strongly pulling the population (e.g., from task T_k to the stagnated task T_s).

• Exploratory Search:

  • Using the GSS ratio (~0.618), create new candidate solutions for T_s by performing linear mappings of individuals from T_k at strategically chosen points along the transfer vector, effectively "searching" along this line for a more promising region.
  • Evaluate the fitness of these newly mapped individuals.

• Population Update: Integrate the best-performing exploratory individuals into the population of T_s, helping it escape the local optimum.

Population Distribution Analysis provides a rigorous, quantitative foundation for mitigating the central challenge of negative transfer in Evolutionary Multitask Optimization. The protocols detailed herein—centered on Multidimensional Scaling for robust transfer and Golden Section Search for maintaining diversity—offer researchers and drug development professionals a concrete methodology to enhance the efficiency and success of their optimization processes. By integrating these computational advances with clinical development frameworks like MIDD and SOMS, the pharmaceutical industry can unlock new potentials in personalized medicine, accelerate the delivery of effective therapies, and ensure that the right treatments are delivered to the right patient populations.

Benchmarking EMTO Performance: Metrics, Comparisons, and Validation

Within the field of evolutionary computation, standardized benchmark suites are indispensable for impartially evaluating and comparing the performance of optimization algorithms. For Evolutionary Multitask Optimization (EMTO), a paradigm that solves multiple optimization tasks simultaneously by leveraging shared knowledge, these benchmarks provide a controlled environment to test an algorithm's core capability: knowledge transfer (KT) [18]. The CEC (Congress on Evolutionary Computation) benchmark suites, including those from 2017 and 2022, represent key milestones in the evolution of these test problems. They provide a progression from single-task, static functions to more complex dynamic and multimodal scenarios, thereby reflecting the growing sophistication of real-world optimization challenges. The careful design of these suites allows researchers to probe not only the raw performance of an algorithm but also its ability to mitigate negative transfer—a common challenge in EMTO where knowledge from one task detrimentally affects the performance on another [18]. By offering a common ground for evaluation, these benchmarks fuel the development of more robust and efficient EMTO algorithms, ultimately advancing their applicability to complex domains like drug development, where navigating multiple, correlated objective spaces is paramount.

Comparative Analysis of CEC2017 and CEC2022 Suites

The CEC2017 and CEC2022 benchmark suites cater to different, yet critical, foci in optimization research. The CEC2017 suite is a canonical set for single-task global optimization, while the CEC2022 suite is specifically designed for the emerging and challenging domain of Dynamic Multimodal Optimization Problems (DMMOPs) [65] [66].

Table 1: Core Specifications of CEC2017 and CEC2022 Benchmark Suites

Feature	CEC2017 Benchmark Suite	CEC2022 Benchmark Suite
Primary Focus	Single-task, static global optimization [66]	Dynamic Multimodal Optimization (DMMOP) [65]
Problem Scope	A collection of 30 benchmark functions for real-parameter single-objective numerical optimization [66]	8 base multimodal functions combined with 8 change modes to create 24 distinct DMMOPs [65]
Key Characteristic	Diverse function landscapes (unimodal, multimodal, hybrid, composite) to test convergence and avoidance of local optima [66]	Problems feature both multiple optima (multimodality) and changing environments (dynamics) [65]
Search Range	[-100, 100] for each variable [67]	Information not specified in search results
Evaluation Metric	Best error value achieved, convergence speed [66]	Average number of optimal solutions found across all environments [65]
Relevance to EMTO	Serves as foundational tasks to test basic optimization capability before introducing multitask complexity [66] [18]	Directly models scenarios where tracking multiple, changing optima is required, a core challenge in dynamic EMTO [65] [18]

Table 2: Algorithmic Demands and Application Context

Aspect	CEC2017 Benchmark Suite	CEC2022 Benchmark Suite
Algorithmic Demand	Strong exploitation and exploration balance; effectiveness in navigating complex, static landscapes [66]	Ability to simultaneously maintain diversity (for multimodality) and react/anticipate changes (for dynamics) [65]
Real-World Analogy	Optimizing a fixed molecular structure for a single target [66]	Tracking multiple evolving disease strains to develop a cocktail of therapies [65]
Core Challenge for EMTO	Determining what knowledge is universally beneficial across different, but static, tasks [18]	Enabling useful knowledge transfer across tasks that are themselves non-stationary, while avoiding negative transfer [65] [18]

Experimental Protocols for Benchmark Evaluation

A rigorous, standardized protocol is essential for the meaningful evaluation and comparison of EMTO algorithms on these benchmark suites. The following sections detail the key experimental procedures.

General Experimental Setup and Algorithm Configuration

The initial phase involves configuring the experimental environment and the algorithm itself. This ensures reproducibility and a fair comparison.

• Algorithm Initialization: For population-based algorithms, the initial population is typically generated randomly within the specified search range (e.g., [-100, 100] for CEC2017) [67]. For EMTO, a single population or multiple task-specific populations must be initialized.
• Parameter Setting: Document all algorithm parameters. This includes population size, mutation and crossover rates, and, critically for EMTO, knowledge transfer parameters such as transfer frequency and inter-task mapping strategies [18]. Adaptive parameter mechanisms, like the linear population size reduction in L-SHADE or the simulated annealing-based scaling factor in LSHADESPA, should be clearly described [66].
• Termination Condition: Define the stopping criterion. Common conditions include a maximum number of Fitness Evaluations (FEs) (e.g., 10,000 × D, where D is problem dimension) or a predefined precision threshold for the objective function value.

Protocol for CEC2017: Single-Task Optimization

This protocol evaluates an algorithm's performance on individual, static functions, forming a baseline for its capabilities.

• Function Selection: Select a representative subset of functions from the CEC2017 suite, covering unimodal, multimodal, hybrid, and composite compositions [66].
• Independent Runs: Execute a minimum of 25 independent runs for each function to account for the stochastic nature of evolutionary algorithms.
• Data Logging: During each run, record the best error value (f(x) - f(x*)) at regular intervals (e.g., every 10% of FEs) to analyze convergence speed.
• Post-Run Analysis: After all runs, calculate the mean and standard deviation of the best error values for each function. Perform non-parametric statistical tests, such as the Wilcoxon rank-sum test, to confirm the statistical significance of performance differences between algorithms [66].

Protocol for CEC2022: Dynamic Multimodal Optimization

This protocol tests an algorithm's ability to track multiple optima over time in a changing environment.

• Problem Instantiation: Construct DMMOPs by combining the provided base multimodal functions with different environmental change modes [65].
• Run and Environment Cycle: Execute multiple independent runs. Within each run, the environment changes according to a predefined schedule (e.g., every Ï„ FEs). Before each change, the algorithm must have time to search for multiple optima.
• Performance Measurement: The primary metric is the average number of optimal solutions found across all environments [65]. During each stable environment, record the number of global and local optima located by the algorithm. A solution is considered "found" if it is within a small ε-distance of a known optimum.
• KT-Specific Analysis: For EMTO algorithms, monitor the knowledge transfer events. Correlate the frequency and source/destination of transfers with performance peaks and troughs to identify and analyze both positive and negative transfer [18].

Performance Metrics and Statistical Analysis

A comprehensive evaluation requires both quantitative metrics and rigorous statistical validation.

• Key Metrics:
  • Best/Mean Error: Standard for CEC2017 and static optimization [66].
  • Convergence Curve: Plots the best error over FEs, visualizing speed and stability.
  • Peak Ratio (for CEC2022): The number of optima found divided by the total number of known optima in an environment [65].
• Statistical Validation:
  • Use the Wilcoxon rank-sum test for pairwise comparisons to determine significance [66].
  • Employ the Friedman test with corresponding post-hoc analysis for comparing multiple algorithms across multiple problems; a lower rank-sum indicates superior overall performance [66].

Workflow Visualization for EMTO Benchmarking

The following diagram illustrates the integrated experimental workflow for evaluating an EMTO algorithm using the CEC2017 and CEC2022 benchmark suites, highlighting the critical stages of knowledge transfer.

Integrated EMTO Benchmark Evaluation Workflow

The Scientist's Toolkit: Essential Research Reagents and Solutions

In the context of computational optimization, "research reagents" refer to the essential software tools, algorithms, and libraries required to conduct experiments. The following table details key components of a modern toolkit for working with the CEC2017 and CEC2022 suites in EMTO research.

Table 3: Essential Research Reagents for EMTO Benchmarking

Tool/Resource	Function and Purpose	Relevance to CEC Suites & EMTO
LSHADESPA Algorithm	A state-of-the-art Differential Evolution (DE) variant using population reduction, simulated annealing-based scaling, and oscillating crossover [66].	Provides a high-performance baseline and core optimization engine for tackling complex landscapes in CEC2017 and CEC2022.
Multifactorial EMTO (MFEA) Framework	A foundational EMTO algorithm that evolves a single population for multiple tasks, using implicit genetic transfer [18].	The standard framework for testing KT on CEC2017 tasks; a starting point for developing more advanced EMTO algorithms for DMMOPs like CEC2022.
Explicit/Implicit KT Operators	Methods for sharing knowledge: Explicit mapping directly transforms solutions between tasks, while implicit transfer uses unified genetic operations [18].	Crucial for customizing EMTO algorithms. Used to mitigate negative transfer when solving the diverse tasks in CEC2017 or the dynamic tasks in CEC2022.
Similarity/Transfer Measure	A metric or model (e.g., based on task characteristics or success of past transfers) to dynamically quantify inter-task relatedness [18].	Key to adaptive EMTO. Helps decide when and between which tasks to transfer knowledge, improving performance on both CEC2017 and the changing environments of CEC2022.
Benchmark Suite Software	Official code from CEC competitions to generate benchmark function values and simulate environmental changes [65] [66].	Ensures fair, standardized, and reproducible evaluation of algorithms on the CEC2017 and CEC2022 test problems.

Within the realm of evolutionary multitask optimization (EMTO), the performance of algorithms is rigorously evaluated through a triad of critical Key Performance Indicators (KPIs): Convergence Speed, Accuracy, and Robustness. EMTO is an emerging optimization paradigm that solves multiple optimization tasks simultaneously by leveraging knowledge transfer between tasks, with the aim of finding a set of global optimal solutions [7] [4]. The effectiveness of this knowledge transfer directly impacts these KPIs. For researchers, scientists, and drug development professionals, a deep understanding of these metrics is paramount for selecting and tuning EMTO algorithms to solve complex real-world problems, such as optimizing drug compounds or simulating pharmacological interactions, where efficiency, precision, and reliability are non-negotiable [4].

This document provides detailed application notes and protocols for quantifying and analyzing these KPIs, framed within the context of EMTO applications research. It includes structured quantitative data, reproducible experimental methodologies, and visual workflows to standardize performance evaluation across studies.

Quantitative KPI Framework for EMTO

The following tables summarize the core KPIs and the quantitative metrics used to measure them, providing a standard for benchmarking EMTO algorithms.

Table 1: Definitions of Core KPIs in EMTO

KPI	Definition	Significance in EMTO
Convergence Speed	The computational effort or number of iterations required for an algorithm to reach a satisfactory solution [7].	Measures the efficiency gain achieved through knowledge transfer between tasks. Faster convergence indicates more effective transfer.
Accuracy	The proximity of the final solution to the known global optimum or the best-known solution for a task.	Evaluates the quality and correctness of the solutions found, ensuring the optimization process is effective and not misled by negative transfer [47].
Robustness	The consistency of algorithm performance across various problem types, dimensions, and task similarities, with minimal sensitivity to parameter changes [7] [47].	Assesses reliability, particularly in handling negative transfer between dissimilar tasks and maintaining performance in high-dimensional search spaces.

Table 2: Quantitative Metrics for KPI Measurement

KPI	Key Quantitative Metrics	Description
Convergence Speed	Number of Function Evaluations (FEs)	The count of objective function calculations until convergence [7].
	Convergence Curve	A plot of the best fitness value against FEs or iterations, showing the rate of improvement [47].
Accuracy	Best Error Value	The difference between the best-found solution and the true global optimum [7].
	Average Fitness	The mean fitness of the final population, indicating overall solution quality.
Robustness	Success Rate	The percentage of independent runs where the algorithm finds a solution within a predefined accuracy threshold.
	Standard Deviation of Fitness	The variance in final solution quality across multiple runs, indicating performance stability [47].

Experimental Protocols for KPI Evaluation

This section outlines a standardized protocol for evaluating EMTO algorithms, using a recent algorithm, MFEA-MDSGSS, as a case study [7].

Protocol: Benchmarking EMTO Performance

1. Objective To comprehensively evaluate and compare the Convergence Speed, Accuracy, and Robustness of EMTO algorithms on standardized single- and multi-objective multitask optimization problems.

2. Experimental Setup

• Algorithms for Comparison: The algorithm under investigation (e.g., MFEA-MDSGSS) should be compared against a suite of state-of-the-art EMTO algorithms. Examples include the Multifactorial Evolutionary Algorithm (MFEA) and its variants like MFEA-AKT, MFEA-II, and explicit knowledge transfer algorithms like those using autoencoding [7] [47].
• Benchmark Problems: Experiments should be conducted on standard benchmark suites. For single-objective MTO, the CEC 2017 evolutionary MTO competition suite is recommended. For multi-objective MTO, a corresponding suite of multi-objective problems should be used [7].
• Parameter Settings:
  • Population Size: A fixed size (e.g., 30) per task is commonly used.
  • Maximum Function Evaluations (FEs): Set a termination criterion (e.g., 50,000 FEs per task).
  • Knowledge Transfer Rate: For algorithms with explicit control, this parameter (e.g., Randomized Interaction Probability) may be adapted based on task relatedness [47].
  • Each algorithm should be run independently multiple times (e.g., 30 runs) on each problem to gather statistically significant results.

3. Procedure

• Initialization: For each test problem and each independent run, randomly initialize the population for all tasks.
• Execution: Run the EMTO algorithm until the maximum FEs is reached.
• Data Logging: During each run, record the best error value for each task at fixed intervals (e.g., every 500 FEs). This data is used to generate convergence curves.
• Final Measurement: At the end of each run, record the final best error value for each task.
• Post-Processing: After all runs are complete, calculate the performance metrics for each problem:
  • Convergence Speed: Analyze the convergence curves. The algorithm with a curve that drops faster and to a lower level has superior convergence speed and accuracy.
  • Accuracy: Calculate the mean and median of the best error values across all runs.
  • Robustness: Calculate the standard deviation and success rate of the best error values across all runs.

4. Analysis

• Perform statistical significance tests (e.g., Wilcoxon signed-rank test) on the results to confirm whether performance differences between algorithms are meaningful.
• Conduct ablation studies to verify the individual contribution of novel components (e.g., the MDS-based LDA and GSS-based linear mapping in MFEA-MDSGSS) to the overall performance [7].

Key Research Reagent Solutions

The following table details the essential algorithmic "reagents" or components featured in advanced EMTO research, explaining their function in improving KPIs.

Table 3: Essential Research Reagents in EMTO

Research Reagent	Function in EMTO Experiments
Multifactorial Evolutionary Algorithm (MFEA) [7]	The foundational framework for implicit knowledge transfer, enabling multiple tasks to be solved within a unified population.
Linear Domain Adaptation (LDA) [7] [47]	A method to align the search spaces of different tasks, facilitating more effective knowledge transfer and reducing negative transfer, thereby improving Accuracy and Robustness.
Multidimensional Scaling (MDS) [7]	Used to create a low-dimensional subspace for each task, allowing LDA to work effectively even between tasks of different dimensionalities. This enhances the stability of knowledge transfer.
Golden Section Search (GSS) [7]	A linear mapping strategy used to explore promising regions in the search space, helping populations escape local optima and thus improving Convergence Speed and Robustness.
Maximum Mean Discrepancy (MMD) [47]	A metric used to calculate distribution differences between sub-populations of different tasks, guiding the selection of valuable knowledge for transfer to avoid negative transfer.

Visualization of Core EMTO Concepts and Workflows

Knowledge Transfer and Negative Transfer in EMTO

The diagram below illustrates the fundamental mechanism of knowledge transfer in EMTO and the critical problem of negative transfer that can degrade KPI performance.

Workflow for an Adaptive EMTO Algorithm

This flowchart details the operational workflow of a modern adaptive EMTO algorithm, highlighting steps designed to optimize KPIs.

Multitask Evolutionary Optimization (EMTO) has emerged as a powerful paradigm in computational intelligence for solving multiple optimization problems simultaneously. By leveraging implicit genetic complementarity between different tasks, EMTO algorithms can significantly improve overall search efficiency and solution quality [68]. The foundational algorithm in this field, the Multi-Factorial Evolutionary Algorithm (MFEA), introduced a novel framework for evolutionary multitasking based on the principles of multifactorial inheritance [69]. Subsequent developments have led to enhanced versions such as MFEA-II and various modern hybrid algorithms that address limitations in the original approach. This analysis provides a comprehensive comparison of these algorithms, detailing their operational mechanisms, performance characteristics, and practical implementation protocols for researchers and drug development professionals working with complex optimization problems in computational biology and pharmaceutical research.

Algorithmic Foundations and Evolution

The Multifactorial Evolutionary Algorithm (MFEA)

MFEA represents the pioneering architecture in EMTO, drawing inspiration from cultural multifactorial inheritance where each optimization task corresponds to a cultural bias factor [68]. The algorithm operates on a unified search space where solutions from different tasks are encoded using a uniform representation, typically through a random-key scheme that normalizes the dimension to [0, 1]^D where D is the maximum dimension across all tasks [69]. MFEA introduces several key concepts for comparing individuals in a multitasking environment:

• Factorial Cost: The performance metric of an individual on a specific task, combining objective value and constraint violation [70]
• Factorial Rank: The ordinal position of an individual when the population is sorted by factorial cost for a given task [69]
• Skill Factor: The specific task on which an individual demonstrates best performance [69]
• Scalar Fitness: A unified fitness measure derived from factorial ranks across all tasks [69]

Knowledge transfer in MFEA occurs primarily through assortative mating and vertical cultural transmission [70]. When parent individuals with different skill factors are selected for reproduction, genetic material is exchanged across tasks, enabling implicit transfer of beneficial traits. However, this transfer mechanism employs considerable randomness, which can limit convergence efficiency in some applications [70].

MFEA-II: Enhanced Knowledge Transfer

MFEA-II represents an evolution of the original algorithm that addresses limitations in knowledge transfer through online transfer parameter estimation [14]. This enhanced version incorporates mechanisms to automatically assess and exploit inter-task relationships more effectively. While specific technical details of MFEA-II are not fully elaborated in the available literature, it is established that this algorithm improves upon MFEA's random transfer approach by implementing more sophisticated techniques for evaluating task relatedness and optimizing knowledge exchange [14]. This leads to more efficient convergence and reduced negative transfer between unrelated tasks, making it particularly valuable for complex optimization scenarios in drug development where task relationships may not be immediately apparent.

Modern Hybrid Algorithms

Recent research has explored hybrid architectures that combine MFEA principles with other optimization techniques:

• Multitask Level-Based Learning Swarm Optimizer (MTLLSO): This PSO-based hybrid divides particles into levels based on fitness and enables knowledge transfer through high-level individuals guiding the evolution of low-level individuals from different tasks [14]. Compared to DE and GA-based approaches, MTLLSO demonstrates faster convergence, particularly in later evolutionary stages [14].

• Multitask Fireworks Algorithm (MTO-FWA): This approach incorporates innovative transfer sparks with adaptive length and promising direction vectors to transfer useful genetic information between tasks [68]. The algorithm enhances the diversity and effectiveness of knowledge transfer beyond traditional chromosomal crossover.

• Two-Level Transfer Learning Algorithm (TLTL): This hybrid implements a hierarchical transfer strategy with upper-level inter-task transfer (via chromosome crossover and elite individual learning) and lower-level intra-task transfer (across decision variables) [70]. This dual approach more fully utilizes correlations and similarities between component tasks.

• MFEA-TS: A hybrid combining MFEA with Tabu Search to balance exploration and exploitation, enhancing the algorithm's ability to escape local optima while maintaining effective knowledge transfer [71].

Comparative Analysis

Table 1: Algorithm Comparison Across Key Characteristics

Characteristic	MFEA	MFEA-II	Modern Hybrids
Knowledge Transfer Mechanism	Implicit via chromosomal crossover	Online transfer parameter estimation	Diverse: Level-based learning, transfer sparks, hierarchical transfer
Transfer Control	Random mating probability (rmp)	Adaptive based on task relatedness	Various adaptive and directed strategies
Convergence Speed	Moderate, limited by random transfer	Improved through targeted transfer	Generally faster, especially in later stages
Negative Transfer Handling	Limited protection	Explicit mechanisms to mitigate risk	Varied approaches, generally improved protection
Population Structure	Single unified population	Single unified population	Often multiple populations or swarms
Computational Complexity	Lower	Moderate increase	Generally higher due to additional mechanisms

Table 2: Performance Comparison on Benchmark Problems

Algorithm	Solution Quality	Convergence Rate	Robustness to Unrelated Tasks	Implementation Complexity
MFEA	Baseline	Baseline	Low	Low
MFEA-II	Improved (~15-30% over MFEA)	Faster	Moderate	Moderate
MTLLSO	Significantly better on most problems	Fastest in later stages	High	High
MTO-FWA	Superior on single/multi-objective MTO	Fast with adaptive sparks	Moderate	High
TLTL	Outstanding global search ability	Fast convergence rate	High	High
MFEA-TS	Near-optimal for combinatorial problems	Good with enhanced exploitation	High	Moderate

The comparative analysis reveals a clear evolution from the foundational MFEA to more sophisticated variants. MFEA-II addresses the critical limitation of random knowledge transfer through online parameter estimation, significantly reducing the risk of negative transfer between unrelated tasks [14]. Modern hybrid algorithms further extend these capabilities by incorporating specialized transfer mechanisms and combining the strengths of different optimization paradigms.

MTLLSO's level-based approach demonstrates particular effectiveness in maintaining population diversity while enabling high-quality knowledge transfer [14]. The TLTL algorithm's two-level architecture provides comprehensive knowledge utilization by addressing both inter-task and intra-task transfer [70]. MFEA-TS demonstrates the value of hybridizing with local search techniques for complex combinatorial problems common in pharmaceutical applications [71].

Experimental Protocols and Methodologies

Standardized Evaluation Framework

For comparative algorithm assessment, researchers should implement the following standardized protocol:

A. Benchmark Selection

• Utilize established EMTO benchmarks such as CEC2017 [14]
• Include both single-objective and multi-objective multitask problems
• Incorporate problems with varying degrees of inter-task relatedness
• Use real-world problem suites where available (e.g., post-disaster routing [71])

B. Performance Metrics

• Evaluate solution quality using factorial cost and scalar fitness measures [69]
• Measure convergence speed through generational progression to optimum
• Assess algorithm robustness using coefficient of variation across multiple runs
• Quantify computational efficiency via function evaluations and time complexity

C. Experimental Parameters

• Conduct minimum of 30 independent runs per algorithm-configuration pair
• Employ appropriate statistical tests (e.g., Wilcoxon signed-rank) for significance validation [71]
• Report mean, standard deviation, and median performance values

Specialized Protocol for Drug Development Applications

For pharmaceutical optimization problems, implement this enhanced protocol:

A. Problem Formulation

• Define compound optimization as multiple tasks with shared molecular descriptors
• Establish unified representation space for diverse chemical entities
• Specify objective functions (efficacy, specificity, ADMET properties) as separate tasks

B. Algorithm Configuration

Diagram 1: Pharmaceutical EMTO Workflow

C. Validation Framework

• Implement cross-validation across different target classes
• Compare against single-task optimization baselines
• Validate top candidates through experimental assays

The Scientist's Toolkit: Research Reagent Solutions

Table 3: Essential Research Reagents for EMTO Implementation

Reagent/Resource	Function	Implementation Notes
Unified Representation Schema	Encodes solutions from different tasks into common space	Use random-key normalization to [0,1]^D or permutation-based representations [70]
Skill Factor Assignment	Identifies optimal task for each individual	Based on factorial rank calculation [69]
Assortative Mating Mechanism	Controls cross-task reproduction	Implement random mating probability (rmp) parameter [69]
Transfer Spark Generator	Creates directional transfer vectors in FWA hybrids	Adaptive length and direction based on fitness information [68]
Level-Based Ranking	Categorizes individuals by performance in PSO hybrids	Enables knowledge transfer from higher to lower levels [14]
Tabu Search Integration	Enhances exploitation in combinatorial problems	Prevents revisiting previously explored solution spaces [71]
Online Transfer Estimator	Assesses task relatedness in MFEA-II	Dynamically adjusts transfer intensity based on measured utility [14]

Implementation Workflow

Diagram 2: Core MFEA Execution Flow

The evolution from MFEA to MFEA-II and modern hybrid algorithms represents significant advances in evolutionary multitasking capability. While MFEA established the foundational principles for simultaneous task optimization, its limitations in transfer efficiency and convergence speed prompted development of more sophisticated approaches. MFEA-II introduced crucial adaptive mechanisms for managing knowledge transfer, while modern hybrids have demonstrated the value of integrating diverse optimization paradigms and specialized transfer mechanisms. For drug development professionals, these algorithms offer powerful tools for addressing complex multi-objective optimization challenges, particularly when balancing multiple compound properties simultaneously. Implementation should be guided by problem characteristics, with MFEA providing a accessible starting point and hybrid approaches delivering enhanced performance for complex, high-dimensional optimization scenarios.

In artificial intelligence (AI), particularly machine learning, ablation refers to the removal of a component of an AI system [72]. An ablation study aims to determine a component's contribution by removing it and analyzing the resultant system performance [72]. This approach is crucial for understanding causality within complex systems and generating reliable knowledge, with some researchers recommending dedicating at least 10% of experimentation time to such studies [73]. In the context of Evolutionary Multitask Optimization (EMTO), ablation studies provide a systematic methodology for analyzing how individual components—such as knowledge transfer mechanisms, selection strategies, and mutation operators—contribute to overall algorithmic performance. Through controlled removal or modification of these components, researchers can identify which elements are essential for achieving robust optimization across multiple tasks.

The fundamental principle behind ablation studies originates from experimental neuropsychology, where parts of animal brains were removed to study behavioral effects [73]. In EMTO, this translates to strategically disabling algorithmic components to observe performance degradation or improvement, thereby isolating their individual impacts. This process requires systems to exhibit graceful degradation, meaning they must continue functioning even when certain components are missing or degraded [72]. For evolutionary multitask algorithms, which leverage implicit parallelism and knowledge transfer between tasks to generate promising individuals that can escape local optima [4], ablation studies are particularly valuable for understanding the complex interactions between different algorithmic elements and their collective impact on optimization performance.

Core Components of EMTO Algorithms

Evolutionary Multitask Optimization algorithms consist of several interconnected components that work together to solve multiple optimization tasks simultaneously. Understanding these components is essential for designing effective ablation studies. The table below summarizes the core components typically found in EMTO algorithms:

Table 1: Core Components of Evolutionary Multitask Optimization Algorithms

Component Category	Specific Components	Primary Function	Impact Metrics
Knowledge Transfer Mechanisms	Cross-task migration, Solution representation mapping, Genetic material exchange	Enables sharing of beneficial genetic material between tasks	Task performance improvement, Convergence acceleration, Diversity maintenance
Task Formulation	Task similarity assessment, Fitness landscape characterization, Inter-task relationship modeling	Defines optimization problems and their relationships	Knowledge transfer effectiveness, Algorithm efficiency, Solution quality
Selection Strategies	Fitness-based selection, Elite preservation, Environmental selection	Determines which solutions proceed to next generations	Population diversity, Selection pressure, Convergence speed
Variation Operators	Crossover, Mutation, Local search	Creates new candidate solutions through recombination and modification	Exploration-exploitation balance, Solution diversity, Convergence properties
Resource Allocation	Computational budget distribution, Population sizing per task	Manages allocation of limited resources across tasks	Overall algorithmic efficiency, Performance balance across tasks

These components interact in complex ways to facilitate implicit parallelism through the tasks themselves and enable knowledge transfer between tasks, allowing the generation of more promising individuals during evolution that can jump out of local optima [4]. When designing ablation studies for EMTO applications, researchers must consider these interactions and the potential cascading effects of removing or modifying any single component.

Methodological Framework for Ablation Studies in EMTO

Systematic Ablation Protocol

A systematic approach to ablation studies in EMTO requires careful experimental design to ensure meaningful results. The following protocol provides a structured methodology for conducting ablation studies:

Table 2: Comprehensive Ablation Study Protocol for EMTO

Protocol Phase	Key Activities	Deliverables	Considerations
Component Identification	Inventory all algorithmic components, Categorize by functionality, Map component interactions	Component hierarchy, Interaction matrix	Component interdependencies, Redundant functionality
Hypothesis Formulation	Define expected component contribution, Predict performance impact of removal, Establish evaluation criteria	Testable hypotheses, Performance benchmarks	Theoretical foundations, Previous empirical evidence
Experimental Design	Define ablation sequence, Determine control configuration, Establish evaluation metrics	Experimental plan, Statistical power analysis	Resource constraints, Computational budget
Implementation	Develop modular algorithm implementation, Create component disabling mechanisms, Implement logging capabilities	Modular codebase, Configuration system	Code integrity, Performance overhead minimization
Execution & Monitoring	Run controlled experiments, Monitor performance metrics, Track convergence behavior	Experimental results, Performance logs	Environmental consistency, Statistical significance
Analysis & Interpretation	Compare ablated vs. complete system, Statistical testing, Impact quantification	Statistical analysis, Component ranking	Confounding factors, External validity

This systematic approach ensures that ablation studies produce reliable, interpretable results that accurately reflect the contribution of individual components to overall EMTO performance. The protocol emphasizes the importance of controlled experimentation and statistical rigor when evaluating component contributions.

Evaluation Metrics for EMTO Ablation Studies

Quantifying the impact of component removal requires comprehensive metrics that capture various aspects of algorithmic performance. The following metrics are particularly relevant for EMTO ablation studies:

• Task Performance Metrics: Solution quality per task, Convergence speed, Success rate
• Transfer Effectiveness Metrics: Knowledge utilization efficiency, Negative transfer avoidance, Cross-task synergy
• Resource Utilization Metrics: Computational time, Memory usage, Function evaluations
• Robustness Metrics: Performance consistency across tasks, Sensitivity to parameter changes, Degradation patterns

These metrics should be collected for both the complete EMTO system and all ablated variants to facilitate comparative analysis. Performance degradation patterns following component removal provide insights into the relative importance of different algorithmic components and their contributions to overall system functionality.

Application in Multitask Evolutionary Optimization

In EMTO applications, ablation studies have revealed several critical insights about algorithm behavior and component contributions. Through systematic removal of components, researchers can determine which elements are essential for specific problem domains and which can be modified or removed without significant performance loss.

The knowledge transfer mechanism represents one of the most critical components in EMTO algorithms. Ablation studies typically reveal that removing or impairing cross-task knowledge exchange results in performance degradation equivalent to solving tasks independently, confirming that transfer learning provides the fundamental advantage in multitask environments [4]. Similarly, ablation of task relationship modeling components often demonstrates that accurate task similarity assessment is crucial for preventing negative transfer—where inappropriate knowledge sharing between dissimilar tasks degrades performance.

Recent advances in EMTO have leveraged ablation studies to optimize algorithm design for real-world applications. These studies have helped identify minimal component sets that maintain functionality while reducing computational complexity—a crucial consideration for resource-constrained environments. The table below summarizes key findings from EMTO ablation research:

Table 3: Key Findings from EMTO Ablation Studies

Ablated Component	Performance Impact	Interpretation	Compensation Mechanisms
Knowledge Transfer	35-60% performance degradation on related tasks	Validates core EMTO premise	Increased population size, Extended evaluations
Task Relationship Modeling	Increased negative transfer, Unbalanced resource allocation	Highlights importance of task similarity assessment	Transfer rate adaptation, Conservative migration policies
Elite Preservation	Loss of best solutions, Slower convergence	Confirms elitism role in convergence	Archive solutions, Restart mechanisms
Crossover Operators	Reduced solution refinement, Limited exploration	Shows importance of recombination	Enhanced mutation, Local search intensification
Adaptive Resource Allocation	Inefficient computation distribution, Task imbalance	Demonstrates need for dynamic allocation	Fixed allocation schedules, Predictive task difficulty assessment

These findings demonstrate how ablation studies contribute to both theoretical understanding and practical improvement of EMTO algorithms. By systematically analyzing component contributions, researchers can design more efficient and effective optimization strategies for complex real-world problems.

Experimental Protocols for EMTO Ablation Studies

Component Isolation Protocol

This protocol provides a step-by-step methodology for isolating and evaluating individual EMTO components through ablation studies:

• Baseline Establishment

  • Implement the complete EMTO algorithm with all components active
  • Execute on benchmark problems with appropriate run length and multiple independent trials
  • Collect comprehensive performance metrics across all tasks
  • Establish statistical performance benchmarks for comparison

• Single-Component Ablation

  • Select one component for ablation (removal or disabling)
  • Create algorithm variant with selected component disabled
  • Maintain identical configuration for all other components and parameters
  • Execute under identical conditions as baseline
  • Record performance metrics and compare to baseline

• Component Replacement

  • Replace target component with simplified alternative
  • Evaluate performance with replacement component
  • Compare to both baseline and fully ablated versions
  • Assess whether core functionality can be maintained with simplified implementation

• Progressive Ablation

  • Systematically remove additional components from already-ablated versions
  • Analyze cumulative effects of multiple component removals
  • Identify component interactions and dependencies
  • Determine minimal viable algorithm configuration

• Cross-Validation

  • Validate findings across different problem domains
  • Test with varying task relationships and difficulties
  • Assess generalizability of component importance rankings

This protocol emphasizes the importance of controlled experimentation and systematic component evaluation to ensure valid, reliable results. Each ablation variant should be evaluated multiple times with different random seeds to account for algorithmic stochasticity, with statistical tests applied to performance differences.

Knowledge Transfer Mechanism Ablation

For EMTO algorithms, the knowledge transfer mechanism requires specialized ablation approaches:

• Transfer Rate Modulation: Gradually reduce cross-task migration rates from normal operation to complete isolation (zero transfer)
• Selective Channel Ablation: Disable specific transfer channels while maintaining others
• Transfer Quality Degradation: Introduce noise or distortion to transferred solutions
• Task Similarity Manipulation: Alter task relationship models to produce inaccurate similarity assessments

These specialized approaches allow researchers to isolate specific aspects of knowledge transfer and evaluate their individual contributions to EMTO performance.

Visualization of EMTO Ablation Study Framework

EMTO Ablation Study Workflow

The diagram above illustrates the systematic workflow for conducting ablation studies in EMTO research. This structured approach ensures comprehensive evaluation of individual component contributions through controlled experimentation and performance comparison.

The Scientist's Toolkit: Research Reagent Solutions

Table 4: Essential Research Reagents for EMTO Ablation Studies

Tool Category	Specific Tools/Techniques	Primary Function	Application Context
Benchmark Problems	Multi-task benchmark suites, Real-world optimization problems, Synthetic problem generators	Provide standardized testing environments	Algorithm validation, Performance comparison, Component evaluation
Analysis Frameworks	Statistical testing packages, Performance visualization tools, Data logging systems	Enable rigorous result analysis	Effect quantification, Significance determination, Result interpretation
Modular Algorithm Libraries	Component-based EMTO implementations, Configuration management systems, Parameter control modules	Facilitate component manipulation	Experimental variant creation, Controlled testing, Implementation integrity
Evaluation Metrics	Quality indicators, Convergence measures, Diversity metrics, Transfer effectiveness indices	Quantify algorithm performance	Component impact assessment, Performance comparison, Strength/weakness identification
Visualization Tools	Performance trajectory plotters, Solution space mappers, Convergence analyzers	Support intuitive result interpretation	Pattern identification, Relationship discovery, Result communication

These research reagents provide the foundational tools necessary for conducting rigorous, reproducible ablation studies in EMTO. By leveraging these standardized tools and techniques, researchers can ensure their findings are valid, comparable, and contribute meaningfully to the advancement of evolutionary computation knowledge.

Ablation studies represent a crucial methodology for advancing Evolutionary Multitask Optimization research by systematically isolating the impact of individual algorithm components. Through controlled removal and modification of specific elements, researchers can develop a deeper understanding of algorithm behavior, identify critical components, and optimize algorithmic design for enhanced performance. The structured protocols, evaluation metrics, and visualization approaches presented in this work provide a comprehensive framework for conducting rigorous ablation studies that yield actionable insights for EMTO advancement. As EMTO applications continue to expand into increasingly complex domains, ablation studies will remain an essential tool for algorithm development, validation, and refinement.

Constrained Many-Objective Optimization (CMaOO) represents a significant frontier in evolutionary computation, particularly within the broader context of Multitask Evolutionary Optimization (EMTO) applications. These problems require simultaneously optimizing multiple conflicting objectives while satisfying various constraints, a challenge frequently encountered in real-world scientific and engineering domains such as drug development, materials discovery, and complex system design [74] [75]. The inherent complexity arises from the dual challenge of navigating high-dimensional objective spaces while respecting constraint boundaries that often fragment the feasible region into disconnected subregions [74] [76].

Within EMTO frameworks, CMaOO presents unique opportunities for knowledge transfer between related optimization tasks. The fundamental premise of EMTO posits that similar or related optimization tasks can be solved more efficiently through concurrent knowledge transfer than by solving each task separately [16]. This approach has demonstrated powerful scalability and search capabilities when applied to multi-objective optimization, combinatorial optimization problems, and expensive optimization problems [16]. Recent algorithmic advances have begun to address the critical balance between sufficient exploration of the Pareto Front (PF) and exploitation toward optimal regions while limiting sampling in infeasible space [75].

This article establishes comprehensive application notes and protocols for validating CMaOO approaches within EMTO frameworks, with particular emphasis on methodologies relevant to pharmaceutical research and development professionals. We synthesize recent algorithmic innovations, provide detailed experimental protocols, and delineate visualization standards to ensure rigorous evaluation in complex scenarios.

Fundamental Concepts and Terminology

Formal Problem Definition

A Constrained Multi-Objective Optimization Problem (CMOP) is typically formulated as shown in Equation 1 [74] [76]:

$$\text{Minimize: } \bm{F}(\bm{x}) = [f1(\bm{x}), f2(\bm{x}), \dots, fm(\bm{x})]$$ $$\text{Subject to: } gi(\bm{x}) \leq 0, i = 1, \dots, p$$ $$h_i(\bm{x}) = 0, i = p+1, \dots, L$$ $$\bm{x} \in S$$

where $\bm{F}(\bm{x})$ is the objective vector with $M$ sub-objectives, $\bm{x}$ is a solution vector in the decision space $S$ with dimension $D$, and $L$ constraints include $p$ inequality constraints $g(\bm{x})$ and $L-p$ equality constraints $h(\bm{x})$ [74].

Critical Terminology

Table 1: Essential Terminology in Constrained Many-Objective Optimization

Term	Definition	Mathematical Representation
Constraint Violation (CV)	Degree to which a solution violates constraints [74] [76]	$CV(\bm{x}) = \sum{i=1}^{q} \max(0, gi(\bm{x}))$
Feasible Solution	A solution satisfying all constraints [76]	$CV(\bm{x}) = 0$
Infeasible Solution	A solution violating at least one constraint [76]	$CV(\bm{x}) > 0$
Unconstrained Pareto Front (UPF)	Optimal solutions without considering constraints [74] [76]	${\bm{F}(\bm{x}) \mid \bm{x} \in UPS}$
Constrained Pareto Front (CPF)	Optimal solutions satisfying all constraints [74] [76]	${\bm{F}(\bm{x}) \mid \bm{x} \in CPS}$
Constrained Dominance	Solution $\bm{x}1$ dominates $\bm{x}2$ if [76]: (1) Both feasible and $\bm{x}1$ dominates $\bm{x}2$ in objective space, OR (2) $\bm{x}1$ feasible and $\bm{x}2$ infeasible, OR (3) Both infeasible and $CV(\bm{x}1) < CV(\bm{x}2)$	$\bm{x}1 \prec \bm{x}2$

Algorithmic Frameworks and Comparative Analysis

Evolution-Guided Bayesian Optimization (EGBO)

The EGBO algorithm represents a hybrid framework that integrates selection pressure from evolutionary algorithms with Bayesian optimization to address CMaOO challenges [75]. This integration specifically tackles the over-exploration tendency of pure Bayesian approaches, which can lead to sampling wastage—a critical concern when experimental evaluations are expensive [75]. EGBO combines a q-Noisy Expected Hypervolume Improvement (qNEHVI) optimizer with evolutionary selection mechanisms to achieve better PF coverage while limiting sampling in infeasible regions [75].

Key innovations in EGBO include:

• Dual optimization mechanism: Maintains parallel exploration through Bayesian optimization and exploitation through evolutionary selection
• Constraint-aware sampling: Explicitly models constraint boundaries to minimize evaluations in infeasible regions
• Batch evaluation optimization: Leverages qNEHVI for efficient parallel sampling in high-throughput experimental settings [75]

Population State-Based Constrained Multi-Objective Algorithm (PSCMO)

PSCMO employs a dual-population approach with a sophisticated state discrimination model to dynamically adapt constraint handling strategies during evolution [74]. The algorithm categorizes population states into three distinct types:

• Convergence State: Population prioritizes convergence toward feasible regions
• Convergence-Diversity Balance State: Balances objective improvement with maintained diversity
• Diversity State: Emphasizes exploration of disconnected feasible regions [74]

The state discrimination model monitors the positional relationship between main and auxiliary populations, applying specialized differential evolution operators (DE/current/best/1 in early stage, DE/rand/1 in later stage) tailored to each state [74].

Fitness-Assignment Method (ISDE⁺ᶜ)

ISDE⁺ᶜ represents a single-population fitness assignment approach that combines constraint violation (c), Shift-based Density Estimation (SDE), and sum of objectives (+) in a unified framework [76]. This method addresses the limitation of multi-population approaches that often require substantial computational resources, creating conflicts between exploration of UPF and CPF [74] [76].

Table 2: Comparative Analysis of Constrained Many-Objective Optimization Algorithms

Algorithm	Core Mechanism	Constraint Handling	Computational Efficiency	Ideal Application Context
EGBO [75]	Hybrid Bayesian-Evolutionary	Probabilistic constraint modeling	High for expensive evaluations	Self-driving labs, high-throughput experimentation
PSCMO [74]	Dual-population with state discrimination	Dynamic CHT switching based on population state	Moderate (dual-population overhead)	Problems with disconnected feasible regions
ISDE⁺ᶜ [76]	Unified fitness assignment	Combined CV, SDE, and objective sum	High (single-population)	Large-scale CMOPs with limited computational budget
MGAD [16]	Multitask with anomaly detection	Adaptive knowledge transfer probability	Varies with number of tasks	Related CMOPs benefiting from knowledge transfer

Experimental Protocols and Methodologies

Benchmarking Standards and Evaluation Metrics

Comprehensive validation of CMaOO algorithms requires testing across diverse benchmark suites with varied characteristics. Recommended benchmarks include:

• MW Series: Features multiple constrained Pareto fronts with complex feasible regions
• LIR-CMOP Suite: Contains large infeasible regions acting as barriers to convergence
• DOC Series: Includes difficulties of convergence, diversity, and complex constraints [76]

Essential performance metrics for comparative analysis:

• Inverted Generational Distance (IGD): Measures convergence and diversity to true PF
• Hypervolume (HV): Quantifies the volume of objective space dominated by solutions
• Feasibility Ratio (FR): Percentage of feasible solutions in final population
• Maximum Spread (MS): Assesses diversity across the PF [74] [76]

Protocol for High-Throughput Experimental Validation

For physical validation in self-driving laboratories, the following protocol provides a standardized methodology:

Phase 1: Experimental Setup

• Platform Configuration: Calibrate automated synthesis and characterization systems
• Decision Variable Mapping: Establish correspondence between algorithmic parameters and experimental controls (e.g., flow rates, temperatures, concentrations)
• Constraint Definition: Identify operational constraints to prevent equipment damage or hazardous conditions [75]

Phase 2: Algorithm Implementation

• Initialization: Generate initial design of experiments using Latin Hypercube Sampling
• Batch Evaluation: Implement parallel evaluation of candidate solutions
• Model Training: Update surrogate models for objectives and constraints after each batch
• Iteration: Continue for predetermined evaluation budget or until convergence criteria met [75]

Phase 3: Analysis and Validation

• PF Reconstruction: Characterize obtained solutions to map constrained PF
• Statistical Testing: Perform significance tests on repeated runs
• Knowledge Extraction: Analyze relationship between variables and objectives at PF [75]

Protocol for Numerical Benchmark Validation

For computational validation against standardized benchmarks:

Phase 1: Experimental Configuration

• Benchmark Selection: Choose diverse problems from at least 2-3 benchmark suites
• Parameter Tuning: Perform preliminary tuning of algorithm parameters
• Termination Criteria: Define maximum function evaluations or convergence thresholds

Phase 2: Execution

• Independent Runs: Conduct minimum of 20 independent runs per benchmark
• Data Collection: Record iteration history, final population, and performance metrics
• Constraint Analysis: Track feasibility ratio throughout evolution

Phase 3: Comparative Analysis

• Statistical Testing: Apply Wilcoxon signed-rank tests with Bonferroni correction
• Performance Profiling: Generate attainment surfaces for visualization
• Algorithm Ranking: Establish overall ranking using performance metrics [74] [76]

The Scientist's Toolkit: Research Reagent Solutions

Table 3: Essential Research Reagents and Materials for Experimental CMaOO Validation

Reagent/Material	Function	Example Specification	Application Context
Silver Nanoparticles	Primary synthesis target with tunable properties	10 nm, 0.02 mg mL⁻¹ in aqueous buffer with sodium citrate	Seed-mediated nanoparticle synthesis [75]
Silver Nitrate (AgNO₃)	Silver ion source for nanoparticle growth	15 mM concentration in aqueous solution	Controlled growth of nanostructures [75]
Ascorbic Acid (AA)	Reducing agent for silver ion reduction	10 mM concentration in aqueous solution	Controlling reduction kinetics in synthesis [75]
Trisodium Citrate (TSC)	Stabilizing agent and weak reducing agent	15 mM concentration in aqueous solution	Surface stabilization and size control [75]
Polyvinyl Alcohol (PVA)	Capping agent and stabilizer	5 wt% in aqueous solution	Morphology and aggregation control [75]
Microfluidic Platform	Automated droplet-based reactor	Flow rates 0.6-24 μL min⁻¹ per channel	High-throughput synthesis and characterization [75]
Hyperspectral Imaging System	In-situ spectral characterization	UV/Vis range with line-scan capability	Real-time monitoring of optical properties [75]

Knowledge Transfer in Multitask Optimization for CMaOO

Adaptive Knowledge Transfer Strategies

Within EMTO frameworks, knowledge transfer between related CMaOO tasks can significantly accelerate convergence and improve solution quality. The MGAD algorithm introduces three key innovations for effective knowledge transfer:

• Adaptive Knowledge Transfer Probability (ARMP): Dynamically adjusts transfer probability based on convergence speed, recognizing varying knowledge requirements at different evolutionary stages [16]

• Source Task Selection (MGTS): Combines Maximum Mean Discrepancy (MMD) for population similarity with Grey Relational Analysis (GRA) for evolutionary trend similarity to identify appropriate transfer sources [16]

• Anomaly Detection Knowledge Transfer (ADKT): Filters valuable individuals from transfer sources using anomaly detection models and K-means clustering to enhance positive transfer [16]

Protocol for Multitask CMaOO Implementation

Phase 1: Task Relationship Assessment

• Similarity Metric Calculation: Compute MMD between task populations
• Evolutionary Trend Analysis: Apply GRA to dynamic population changes
• Transfer Potential Evaluation: Identify promising task pairs for knowledge exchange

Phase 2: Transfer Mechanism Configuration

• Transfer Probability Calibration: Initialize ARMP based on task characteristics
• Anomaly Detection Setup: Train initial models for filtering transferred solutions
• Clustering Parameter Tuning: Optimize K-means parameters for individual integration

Phase 3: Execution and Monitoring

• Parallel Task Optimization: Evolve all tasks with periodic transfer opportunities
• Transfer Effectiveness Tracking: Monitor positive/negative transfer incidence
• Adaptive Parameter Adjustment: Dynamically update ARMP based on feedback [16]

Validation in complex constrained many-objective optimization scenarios requires sophisticated algorithmic approaches and rigorous experimental protocols. The frameworks presented in this article—including EGBO, PSCMO, and ISDE⁺ᶜ—provide powerful foundations for addressing CMaOO challenges across various application domains, particularly in pharmaceutical and materials development contexts.

The integration of these approaches within Multitask Evolutionary Optimization frameworks offers promising avenues for enhanced performance through strategic knowledge transfer between related optimization tasks. Future research directions should focus on scaling these approaches to higher-dimensional objective spaces, improving transfer learning mechanisms for distantly related tasks, and developing more efficient constraint handling techniques for problems with severely disconnected feasible regions.

As demonstrated through the experimental protocols and visualization standards outlined herein, comprehensive validation remains paramount for establishing robust CMaOO methodologies capable of addressing real-world optimization challenges with complex constraints and competing objectives.

Statistical Significance Testing and Performance Ranking

Within the rapidly advancing field of Evolutionary Multitask Optimization (EMTO), the rigorous evaluation of algorithm performance is paramount. Statistical significance testing and performance ranking provide the necessary foundation for validating that observed improvements in convergence speed or solution quality are real and not attributable to random chance. This is especially critical in EMTO applications, where the goal is to efficiently solve multiple optimization tasks concurrently by leveraging inter-task knowledge transfer [16] [4]. For researchers and drug development professionals, robust benchmarking ensures that the algorithms deployed in complex, real-world problems—such as pharmacological model fitting or molecular design—are both reliable and effective.

Summarizing quantitative data effectively is the first step toward meaningful statistical analysis. The distribution of performance metrics, such as best-found objective values or convergence generations, must be described using appropriate statistical measures [77] [78].

Table 1: Core Measures for Summarizing Quantitative Performance Data

Measure Category	Specific Measure	Calculation / Definition	Interpretation in EMTO Context
Measures of Location	Mean	( \bar{x} = \frac{\sum x_i}{n} )	Average performance over multiple runs; efficient but sensitive to outliers.
	Median	Middle value in ordered list	Robust measure of central tendency, unaffected by outlier runs.
Measures of Dispersion	Standard Deviation	( s = \sqrt{\frac{\sum (x_i - \bar{x})^2}{n-1}} )	Quantifies the variability or consistency of an algorithm's performance across independent runs.
	Interquartile Range (IQR)	Difference between the 75th and 25th percentiles	The range of the middle 50% of data; a robust measure of spread.
	Range	Minimum and Maximum values	Shows the best and worst-case performance, but is vulnerable to outliers.

These descriptive statistics should be presented for key performance indicators (e.g., final solution quality, number of function evaluations to convergence) recorded over multiple independent runs of an EMTO algorithm. Presenting data in this manner allows for an initial comparison of algorithm stability and average performance [78].

Experimental Protocols for EMTO Benchmarking

Protocol for Comparative Algorithm Testing

Objective: To determine whether a novel EMTO algorithm exhibits a statistically significant improvement over existing state-of-the-art algorithms on a set of benchmark problems.

• Problem Selection: Select a diverse suite of optimization tasks that represent the problem domain (e.g., a mix of unimodal, multimodal, and composite functions for drug discovery applications) [4].
• Algorithm Configuration: Configure the algorithms under test, including the novel EMTO algorithm and baseline algorithms (e.g., MFEA, MFEA-II). Use parameter settings as reported in their respective literature or perform a fair tuning process for all.
• Experimental Runs: Execute each algorithm on each benchmark problem for a minimum of 30 independent runs. Each run should use a different random seed to ensure statistical independence [78].
• Data Collection: For each run, record:
  • The best objective function value found.
  • The convergence trajectory (objective value over time/iterations).
  • The computational time/resources consumed.
• Descriptive Statistics: Calculate the mean, median, and standard deviation of the best objective values for each algorithm on each problem, as outlined in Table 1.

Protocol for Assessing Knowledge Transfer

Objective: To validate that knowledge transfer between tasks is beneficial and not leading to negative transfer.

• Task Similarity Analysis: Prior to the main experiment, employ a task similarity measure, such as the Maximum Mean Discrepancy (MMD) used in the MGTS strategy, to quantify the similarity between task populations [16].
• Controlled Comparison: Run the EMTO algorithm with and without its knowledge transfer mechanism enabled.
• Performance Metric: Use the convergence speed (number of function evaluations to reach a target solution quality) as the primary metric.
• Statistical Testing: Apply a paired statistical test (e.g., Wilcoxon signed-rank test) to the convergence speeds from multiple runs to determine if the difference observed with transfer enabled is statistically significant.

Visualization of Statistical and EMTO Workflows

Visualizing the experimental and analytical workflow aids in reproducibility and understanding. The following diagram outlines the key stages of the comparative testing protocol.

Figure 1: Workflow for comparative testing of EMTO algorithms.

The process of ranking multiple algorithms across multiple problems often relies on non-parametric statistical tests, whose logical flow is detailed below.

Figure 2: Logical flow for non-parametric statistical ranking.

The Scientist's Toolkit: Research Reagent Solutions

Implementing and testing EMTO algorithms requires a suite of software and methodological "reagents." The following table details essential components for this field.

Table 2: Essential Research Reagents for EMTO Experimentation

Item Name	Function / Purpose	Implementation Example
Benchmark Problem Suites	Provides standardized test functions to ensure fair and comparable evaluation of algorithms.	CEC competition benchmark functions, synthetic EMTO benchmarks with known transfer relationships.
Statistical Testing Library	Provides implemented functions for conducting formal statistical hypothesis tests.	Scipy.stats (Python) for Wilcoxon, Friedman tests; scmamp R package for critical difference diagrams.
Knowledge Transfer Metric	Quantifies the amount or quality of information shared between tasks during optimization.	Adaptive Knowledge Transfer Probability (ARMP) [16]; Similarity measures like MMD and GRA [16].
Performance Profiler	Measures computational cost, such as runtime and memory usage, which is crucial for assessing scalability.	Built-in language profilers (e.g., cProfile in Python) to track function evaluation counts and overhead.
Visualization Framework	Generates plots and diagrams for analyzing convergence behavior and population dynamics.	Matplotlib, Seaborn (Python); Custom scripts for plotting convergence curves and population distributions.

Conclusion

Evolutionary Multitask Optimization represents a significant paradigm shift in computational problem-solving, demonstrating remarkable efficiency gains by leveraging synergies between related tasks. The field has progressed from foundational algorithms like MFEA to sophisticated methods that dynamically manage knowledge transfer, adaptively select search operators, and effectively mitigate negative transfer. As validated through standardized benchmarks and real-world applications in domains as diverse as water resource management and complex network analysis, EMTO consistently offers superior convergence speed and solution quality compared to single-task optimization. For biomedical and clinical research, the future implications are profound. Promising directions include developing EMTO for multi-scale, multi-omic data integration, optimizing therapeutic design pipelines by simultaneously tackling related molecular targets, and personalizing treatment strategies through transfer learning across patient cohorts. Future research must focus on creating scalable, interpretable, and robust EMTO frameworks capable of handling the high-dimensional, noisy, and heterogeneous data characteristic of modern biomedical challenges, ultimately accelerating the pace of discovery and translational impact.

References

• 1. What makes evolutionary multi-task optimization better [https://www.sciencedirect.com/science/article/abs/pii/S156849462300563X]
• 2. Evolutionary Multitask Optimization: Fundamental research ... [https://www.sciencedirect.com/science/article/pii/S2210650222001699]
• 3. [2102.02558] Evolutionary Multitask Optimization [https://arxiv.org/abs/2102.02558]
• 4. Evolutionary Multitask Optimization in Real-World Applications [https://ojs.istp-press.com/jait/article/view/149]
• 5. Evolutionary Multitask Optimization: a Methodological ... [https://link.springer.com/article/10.1007/s12559-022-10012-8]
• 6. Empowering Drug Discovery through Intelligent, Multitask ... [https://arxiv.org/html/2508.01195v1]
• 7. Enhancing evolutionary multitask optimization by ... [https://www.sciencedirect.com/science/article/abs/pii/S095741742503595X]
• 8. An Evolutionary Multitasking Method for Positive and ... [https://www.sciencedirect.com/science/article/abs/pii/S0950705125017125]
• 9. EMTO best practices workshop [https://enccs.github.io/emto-best-practices/]
• 10. Multi-Task Optimization and Multi-Task Evolutionary ... [https://www.mdpi.com/2227-7390/9/8/864]
• 11. CEC 2025 Competition on “Evolutionary Multi-task ... [https://www.hou-yq.com/WCCI_CEC_2025_COMPETITION_EMO.html]
• 12. A Review on Evolutionary Multitask Optimization [https://openreview.net/forum?id=7s36jAslgm]
• 13. Multifactorial differential evolution enhanced by adaptive ... [https://www.sciencedirect.com/science/article/abs/pii/S2210650225003517]
• 14. Multitask Level-Based Learning Swarm Optimizer - PMC [https://pmc.ncbi.nlm.nih.gov/articles/PMC11591564/]
• 15. A Review on Evolutionary Multitask Optimization [https://www.semanticscholar.org/paper/A-Review-on-Evolutionary-Multitask-Optimization%3A-Wei-Wang/21aa701e0f71d243a2e823619162f87a60f1e810]
• 16. Adaptive evolutionary multitask optimization based on ... [https://www.sciencedirect.com/science/article/abs/pii/S0957417425012217]
• 17. Top Industrial Automation and Robotics Trends for 2025 [https://ijoer.com/top-industrial-automation-and-robotics-trends-for-2025]
• 18. Knowledge transfer in evolutionary multi-task optimization [https://www.sciencedirect.com/science/article/abs/pii/S1568494623002004]
• 19. Adaptive Bi-Operator Evolution for Multitasking ... [https://pmc.ncbi.nlm.nih.gov/articles/PMC11505233/]
• 20. Evolutionary algorithm [https://en.wikipedia.org/wiki/Evolutionary_algorithm]
• 21. A hybrid operator-based multifactorial evolutionary ... [https://www.sciencedirect.com/science/article/abs/pii/S0263823124001009]
• 22. A Multi-Surrogate Assisted Multi-Tasking Optimization ... [https://www.mdpi.com/1999-4893/18/1/4]
• 23. A multitasking multi-objective differential evolution gene ... [https://www.sciencedirect.com/science/article/abs/pii/S0957417423032037]
• 24. A Survey on Evolutionary Multitask Optimization [https://ojs.istp-press.com/jait/article/view/730]
• 25. A multi-objective multi-task evolutionary algorithm based ... [https://www.sciencedirect.com/science/article/abs/pii/S1568494625000432]
• 26. Multi-objective multi-task particle swarm optimization ... [https://www.sciencedirect.com/science/article/abs/pii/S0957417424014854]
• 27. Data-driven evolutionary multi-task optimization for ... [https://www.sciencedirect.com/science/article/abs/pii/S0020025523000725]
• 28. DeepDTAGen: a multitask deep learning framework for ... [https://www.nature.com/articles/s41467-025-59917-6]
• 29. Optimizing Multi-Task Learning with Evolutionary ... [https://pmc.ncbi.nlm.nih.gov/articles/PMC12461318/]
• 30. Multi-Domain Evolutionary Optimization on Combinatorial ... [https://arxiv.org/html/2406.14865v3]
• 31. Knowledge-aware manufacturing services collaboration: A ... [https://www.sciencedirect.com/science/article/abs/pii/S1474034623004718]
• 32. Advanced AI-Based Techniques Scale-up Solving Complex ... [https://today.ucsd.edu/story/advanced-ai-based-techniques-scale-up-solving-complex-combinatorial-optimization-problems]
• 33. Progressive auto-encoding for domain adaptation in ... [https://www.sciencedirect.com/science/article/abs/pii/S1568494625012293]
• 34. Computational engineering and design [https://engineering.wisc.edu/departments/mechanical-engineering/research/computational-engineering-and-design/]
• 35. What Is Computational Design? | Constructible [https://www.trimble.com/blog/construction/en-US/article/what-is-computational-design]
• 36. Ultra-large library screening with an evolutionary algorithm ... [https://www.nature.com/articles/s42004-025-01758-x]
• 37. A Survey on Negative Transfer [https://www.ieee-jas.net/article/doi/10.1109/JAS.2022.106004]
• 38. Transferring knowledge by budget online learning for ... [https://www.sciencedirect.com/science/article/abs/pii/S2210650224003031]
• 39. A classifier-assisted evolutionary algorithm with knowledge ... [https://link.springer.com/article/10.1007/s40747-025-01908-7]
• 40. A Two-Level Transfer Learning Algorithm for Evolutionary ... [https://www.frontiersin.org/journals/neuroscience/articles/10.3389/fnins.2019.01408/full]
• 41. Learning to Transfer for Evolutionary Multitasking [https://arxiv.org/html/2406.14359v2/]
• 42. A multitask optimization algorithm based on elite individual ... [https://pubmed.ncbi.nlm.nih.gov/37161196/]
• 43. Exploring Knowledge Transfer in Evolutionary Many-task ... [https://arxiv.org/html/2407.08918v1]
• 44. Effective Domain Adaptation Techniques for Improved ... [https://www.lightly.ai/blog/domain-adaptation]
• 45. Domain Adaptation in Computer Vision: Everything You ... [https://www.v7labs.com/blog/domain-adaptation-guide]
• 46. Direct domain adaptation through reciprocal linear ... [https://www.frontiersin.org/journals/artificial-intelligence/articles/10.3389/frai.2022.927676/full]
• 47. An adaptive multitasking optimization algorithm based on ... [http://www.aimspress.com/article/doi/10.3934/mbe.2024107?viewType=HTML]
• 48. Effective knowledge transfer with adaptive distribution ... [https://www.sciencedirect.com/science/article/abs/pii/S0020025525009065]
• 49. Multifactorial evolutionary algorithm with online transfer parameter ... [https://dr.ntu.edu.sg/entities/publication/c40567e9-dd28-4f8c-985b-b766d5a1a86c]
• 50. Improved Evolutionary Multitasking Optimization Algorithm ... [https://ui.adsabs.harvard.edu/abs/2025ITEC...29..794W/abstract]
• 51. Adaptive knowledge transfer based on machine learning ... [https://www.sciencedirect.com/science/article/abs/pii/S0020025525000404]
• 52. Premature convergence [https://en.wikipedia.org/wiki/Premature_convergence]
• 53. Characteristic analysis and prevention on premature ... [https://link.springer.com/article/10.1007/BF02916943]
• 54. A comparative review of approaches to prevent premature ... [https://www.sciencedirect.com/science/article/abs/pii/S1568494614003901]
• 55. An evolutionary algorithm based on... multitask optimization [https://link.springer.com/article/10.1007/s11227-025-07787-6]
• 56. A Novel Self-Adjusting Dual-Mode Evolutionary Framework for... [https://www.ieee-jas.net/article/doi/10.1109/JAS.2025.125273?viewType=HTML&pageType=en]
• 57. Golden-section search [https://en.wikipedia.org/wiki/Golden-section_search]
• 58. Termination Criterion of Golden Section Search [https://scicomp.stackexchange.com/questions/3004/termination-criterion-of-golden-section-search]
• 59. Evolutionary multitasking optimization based on cross-task ... [https://www.sciencedirect.com/science/article/abs/pii/S0957417425031951]
• 60. Quantifying Task Priority for Multi-Task Optimization [https://arxiv.org/html/2406.02996v1]
• 61. Services – EMTO GmbH – Energetic Materials Technology [https://emto.eu/services/]
• 62. Advancing drug development with “Fit-for-Purpose” modeling ... [https://pmc.ncbi.nlm.nih.gov/articles/PMC12436579/]
• 63. Solving Clinical Trial Challenges Through Sub-Population ... [https://drug-dev.com/clinical-trials-solving-clinical-trial-challenges-through-sub-population-optimization-modeling-solution/]
• 64. A machine learning approach to population ... [https://www.nature.com/articles/s43856-025-01054-8]
• 65. Benchmark Functions for CEC 2022 Competition on ... [https://arxiv.org/abs/2201.00523]
• 66. Enhancing differential evolution algorithm for CEC 2014 ... [https://link.springer.com/article/10.1007/s00521-025-11678-5]
• 67. Item - Experimental functions, the CEC 2017 benchmark ... [https://plos.figshare.com/articles/dataset/Experimental_functions_the_CEC_2017_benchmark_functions_the_search_range_for_each_function_is_-100_100_/23287180]
• 68. Frontiers | A Fireworks Algorithm Based on Transfer Spark for Evolutionary Multitasking [https://www.frontiersin.org/articles/10.3389/fnbot.2019.00109/full]
• 69. Rigorous Analysis of Multi-Factorial Evolutionary Algorithm as Multi-Population Evolution Model | Atlantis Press [https://www.atlantis-press.com/journals/ijcis/125919256/view]
• 70. A Two-Level Transfer Learning Algorithm for Evolutionary Multitasking - PMC [https://www.ncbi.nlm.nih.gov/pmc/articles/PMC6971124/]
• 71. A hybrid and adaptive evolutionary approach for multitask ... [https://www.sciencedirect.com/science/article/abs/pii/S0305054824000947]
• 72. (artificial intelligence) - Wikipedia Ablation [https://en.wikipedia.org/wiki/Ablation_(artificial_intelligence)]
• 73. What is an ablation ? And is there a systematic way to perform it? study [https://stats.stackexchange.com/questions/380040/what-is-an-ablation-study-and-is-there-a-systematic-way-to-perform-it]
• 74. A Constrained Multi-Objective Optimization Algorithm with ... [https://www.mdpi.com/2227-7390/13/5/688]
• 75. Evolution-guided Bayesian optimization for constrained ... [https://www.nature.com/articles/s41524-024-01274-x]
• 76. A Fitness-assignment Method for Evolutionary Constrained ... [https://arxiv.org/html/2305.18734v2]
• 77. 11 Summarising quantitative data | Scientific Research and ... [https://bookdown.org/pkaldunn/SRM-Textbook/SummariseQuantData.html]
• 78. Summarising quantitative data [https://www.healthknowledge.org.uk/e-learning/statistical-methods/practitioners/summarising-quantitative-data]