An interactive opposition matrix and regulatory gene fusion strategy for evolutionary multitasking

Abstract

Evolutionary multitasking algorithms (EMT) study how to solve multiple optimization tasks simultaneously by evolutionary computation, and investigate how knowledge sharing can accelerate the convergence of individual tasks, meaning that useful knowledge gained in solving one task can be used to solve other tasks. However, as the evolutionary search continues, the learnability among tasks may decrease, leading to a decrease in the efficiency of knowledge transfer and affecting the population evolution. To solve this problem, a new multifactorial evolutionary algorithm (MFEA-VOM) is proposed in this paper, which applies to three strategies, namely, implicit conversion strategy, opposition matrix strategy, and regulatory gene fusion strategy. The implicit conversion strategy is applied to minimize the threat of negative knowledge migration and reduce the impact caused by negative knowledge migration. The proposed opposition matrix strategy explores more unknown areas of the population and improves the exploration ability of the population by further exploring and utilizing the unified search space, transforming the parent individuals into an appropriate task through mapping relationships, and reducing the gap between tasks. The proposed regulatory gene fusion strategy is applied to the reproduction of individuals to produce better individuals applicable to the task, submitting the efficiency of knowledge transfer. Through a comprehensive experimental analysis of the EMT optimization problem, the experimental results demonstrate the better performance of MFEA-VOM compared to other EMT algorithms.

Keywords

Evolutionary multitasking knowledge transfer opposition matrix implicit conversion regulatory gene fusion

1 Introduction

Influenced by the Darwinian doctrine of Natural Selection [1], evolutionary algorithms (EAs) were proposed and solved various optimization problems, such as job-shop scheduling [2] and vehicle routing problems [3]. Among optimization algorithms, multi-task evolutionary algorithms are regarded as a new research field. As we all know real problems rarely exist in isolation. Principle of the multi-task evolutionary algorithm is to assume that there is some common useful knowledge between solving related tasks, so useful knowledge can be acquired in the process of solving a task. If there is some connection between different tasks, then between tasks performing knowledge transfer can accelerate the convergence of tasks more effectively than processing tasks individually. Multitask evolution was first proposed by Gupta et al. [4] and became one of the earliest algorithms to solve multiple optimization tasks.

EMT has been successfully applied to solve various optimization problems. For example, Bali et al. [5] proposed the LDA-MFEA algorithm to transform the search space of simple tasks into complex tasks and promote knowledge transfer. Lin et al. [6] proposed that a transfer is positive if the solution of the transfer is non-dominant in its target task. The neighbors of this positive transfer solution are selected as the next generation of transfer solutions. Wu et al. [7] proposed a synchronous optimization strategy for MTGA-based fuzzy system design, estimating the evaluation between two tasks and then clearing in chromosome transfer so that the tasks optimally solve each other. Liang et al. [8] proposed a multi-objective EMT algorithm based on subspace alignment and adaptive differential evolution. The subspace is used to transform the search space of the population to reduce the probability of negative knowledge migration between tasks. Zhong et al. [9] combined a multifactorial evolutionary algorithm (MFEA) and SL-GEP for new population generation. In population selection, a one-to-one strategy is used to compare children with their fathers. An incremental learning EMT algorithm was proposed by Lin et al. [10] the transferred solutions are selected by in incremental classifier to choose the valuable knowledge transfer solutions and explore the search space of the task. Ding et al. [11] proposed a multi-task evolutionary optimization framework to solve computationally intensive problems, G-MFEA is proposed to solve the problem that the optimal solution is not in the same position or the dimension of the decision space is different when the effect is not added. Song et al. [12] proposed a multitask multigroup optimization algorithm (MTMSO) that extends the popular dynamic multigroup optimization (DMS-PSO) algorithm to multitask scenarios. Wen and Ting [13] classified the offspring into three types by the attributes of the parents. Resource allocation is made based on the type of offspring. Zheng et al. [14] based on the fact that the intensity of knowledge transfer in MFEA is algorithmically configured and may hinder knowledge sharing and utilization. A self-adjusting algorithm is proposed to address the common useful knowledge of related tasks. Chen et al. [15] perceive the relationship between tasks in a system and propose an adaptive strategy to select the most appropriate task for knowledge transfer with the current task.

Although evolutionary algorithms have been well developed, evolutionary computation researchers continue to face two major challenges [16]. Computational volume: Compared to other optimization algorithms, evolutionary algorithms usually require a larger computational volume to guarantee convergence performance. Poor generalization: for most of the existing studies optimization experience does not need to be taken into account when performing the task and EAs can only be designed from scratch. however, considering the currently prevailing domain-specific computing infrastructure [17]. EAs should have the ability to handle complementary optimization tasks using optimization experience. To solve the above problem, the usual strategy is to jointly exploit knowledge from different optimization tasks. Knowledge from one optimization task is extracted to another optimization task, and information from different domain scopes is exchanged, thus improving generalization capabilities. By dealing with different problems, the computational effort can be reduced, thus increasing the efficiency of problem solving. Various studies have been generated by the idea of extracting knowledge. These studies mainly include multitask learning and multitask optimization. In multi-task evolutionary algorithm studies it was found that the effect of knowledge transfer is very limited in the early stages of the whole evolutionary process. As the evolutionary process proceeds, the effect of knowledge transfer will become inefficient [18]. Alternatively, as the evolutionary process proceeds, the relevance between tasks gradually decreases and external resources are continuously consumed [19]. In this paper, the inefficient change of knowledge transfer is defined as negative knowledge transfer. To solve the problem of negative knowledge transfer, one approach is to reduce the communication between tasks during the evolutionary process, however, this is not desirable. Another approach is to increase the efficiency of knowledge transfer between tasks. This is the main way to solve the inefficient knowledge transfer. In MFEA, as the population iteration proceeds, the efficiency of inter-population communication decreases and knowledge transfer gradually becomes inefficient, which seriously affects the development of the population and is detrimental to the generation of offspring individuals and the evolution of the population. In the process of population generation of individuals, due to the limitation of MFEA itself, only local area search can be performed, and the solutions that generate offspring often do not survive, resulting in the waste of resources and affecting the selection of individuals in the population.

In [20], Ma et al. proposed a two-level transfer learning method for evolutionary multitasking, with the upper level achieving inter-task knowledge transfer through genetic crossover and the lower level performing intra-task transfer learning based on information transfer of decision variables, which achieved significant results, but failed to adequately address the occurrence of negative knowledge transfer due to the method’s less exploration of the unified search space. And along with the evolutionary process, it is easy to fall into local optimal solutions. To solve this problem, this work combines three strategies with a multi-task evolutionary algorithm to improve the efficiency of knowledge transfer by introducing three strategies and to accelerate the process of population convergence by reducing the negative knowledge transfer. The main contributions of this paper are summarized as follows:

By setting the rmp, the process of population evolution is regulated, which facilitates the achievement of better solutions. The implicit conversion strategy is proposed.

The proposed opposition matrix strategy improves the exploration ability of the population and accelerates the evolutionary process of the population by further exploring and utilizing the unified search space, transforming the parent individuals into suitable tasks through mapping relationships, and reducing the gap between tasks to explore more unknown domains.

The proposed strategy of regulatory gene fusion is applied to the reproduction of individuals, through the transformation of genes so that the parental generation in order to produce better individuals suitable for the task, submitting to the efficiency of knowledge transfer.

In this work, all mathematical symbols are listed in Table 1, and the meaning of each symbol is explained.

Table 1
Symbol description table

Parameters Description

M Number of tasks

$x_{i}^{}$ Solutions for the corresponding tasks

$Ψ_{j}^{i}$ the factorial cost

λ a large penalizing multiplier

$f_{j}^{i}$ the objective value

$δ_{j}^{i}$ the total constraint violation

$r_{j}^{i}$ factorial rank

ψ_i scalar fitness

τ_i the index of the task

D _multitask the unified search space

MM the values of the mapping constants

NN the values of the mapping constants

gen* the number of iterations

mutate _a variance matrices

mutate _b variance matrices

mutate _c variance matrices

mutate _d variance matrices

OM ₁ opposition matrix

OM ₂ opposition matrix

RU ₁ the regulatory vectors

RU ₂ the regulatory vectors

Parameters	Description
M	Number of tasks
$x_{i}^{*}$	Solutions for the corresponding tasks
$Ψ_{j}^{i}$	the factorial cost
λ	a large penalizing multiplier
$f_{j}^{i}$	the objective value
$δ_{j}^{i}$	the total constraint violation
$r_{j}^{i}$	factorial rank
ψ_i	scalar fitness
τ_i	the index of the task
D _multitask	the unified search space
MM	the values of the mapping constants
NN	the values of the mapping constants
gen	the number of iterations
mutate _a	variance matrices
mutate _b	variance matrices
mutate _c	variance matrices
mutate _d	variance matrices
OM ₁	opposition matrix
OM ₂	opposition matrix
RU ₁	the regulatory vectors
RU ₂	the regulatory vectors

The rest of this paper will be presented below. Section 2 introduces multifactor optimization (MFO), discusses the similarities and differences between multi-task optimization algorithms and traditional optimization algorithms, and introduces the MFEA. Section 3 describes MFEA-VOM in detail, including the general framework, the recessive transformation strategy, the opposition matrix strategy, and the regulatory gene fusion strategy. Section 4 presents a comprehensive experiment for single- and multi-objective MFO problems as a way to evaluate the effectiveness of MFEA-VOM. Section 5 is a discussion of MFEA-VOM. Section 6 concludes the paper and provides a discussion of the future.

2 Preliminary

This section mainly briefly introduces the multifactor optimization paradigm, the similarities and differences multifactor optimization algorithms and traditional optimization algorithms and MFEA.

2.1 Multifactorial optimization

In [4], MFO is defined as an evolutionary multitasking paradigm built on the implicit parallelism of the search for groups, with the aim of searching for the best solution for multiple tasks.

Fig. 1

Basic framework of evolutionary multitasking algorithms.

To solve multiple tasks, traditional SOO requires different solvers to provide a unique problem representation for each task. Unlike traditional SOO, MFO uses a unified problem representation and applies one solver to solve multiple tasks [21]. During the search process, MFO can implicitly pass useful genetic material through the tasks to other tasks, thus enabling an efficient evolutionary process.

Suppose there are M tasks, each of which is a minimization problem. The kth task if denoted as task K_i. The corresponding objective function of K_i. The objective of MFO is to find a set of solutions ${x_{1}^{*}, x_{2}^{*}, \dots, x_{m}^{*}}$ in one single run, where $x_{i}^{*}$ minimizes T_i, i.e., $x_{i}^{*} = argmin f_{i} (x), i = 1, 2, \dots, k$ (1)

To evaluate populations in a multitasking environment each individual was defined to have the following properties in [4]. Individuals are encoded in a uniform search space that contains the search space of all tasks and that is decoded as a representation of the solution specific to the task with respect to each of the k tasks.

Factorial Cost: In a task T_j the factorial cost $ψ_{j}^{i} = λ • δ_{j}^{i} + f_{j}^{i}$ ; where λ is a large penalizing multiplier, $f_{j}^{i}$ and $δ_{j}^{i}$ are the objective value and the total constraint violation, respectively, of p_i with respect to T_j.

Factorial Rank: The factorial rank $r_{j}^{i}$ of individual p_i on task T_j is the index of p_i in the list of population members. Factorial rank is sorted in ascending order by factorial cost. When multiple individuals have the same factorial costs, random tie-breaking is applied to resolve the parity.

Scalar Fitness: The list of factorial rank ${r_{1}^{i}, r_{2}^{i}, \dots, r_{k}^{i}}$ of an individual p_i is reduced to a scalar fitness ψ_i based on its best rank over all tasks; $ψ_{i} = 1 / mi n_{j \in {1, 2 \dots k}} {r_{j}^{i}}$ .

Skill Factor: The skill factor τ_i is the index of the task at which the individual demonstrates the highest competence amongst all tasks; i.e., $τ_{i} = argmi n_{j} {r_{j}^{i}}, j \in {1, 2, \dots, k}$ .

2.2 Differences between MFO and existing optimization algorithms

Genetic algorithms have been widely used in applications, and the paradigm still suffers from problems such as high computational effort and poor generalization ability. To solve these problems, the representative branch proposed is the evolutionary multitasking algorithm. The purpose of multitask optimization is to solve multiple tasks that are independent of each other but similar at the same time, and to accelerate the convergence of tasks by enabling potential similarities or complementarities to be shared among multiple cross-domain tasks through knowledge migration. Existing optimization algorithms, such as: classical simulated annealing methods, ant colony algorithms, and particle swarm algorithms, are mainly used to solve a large number of practical application problems. Although traditional optimization algorithms have been well developed for some practical applications, better solutions are often achieved with the help of multi-task optimization for some applications such as shop floor scheduling (production optimization) and vehicle routing (logistics optimization) [3] and aircraft design [22]. Many practical problems need to optimize several related optimization tasks simultaneously, and effectively mining the potential connections between tasks will have important practical application value.

2.3 MFEA

Typically, traditional evolutionary algorithms can only solve a single optimization problem. This is the limitation of traditional evolutionary algorithms. In the real world, problems often do not exist in isolation. Inspired by the biocultural model of multi-factor inheritance, Gupta et al. [4] proposed a multi-factor evolutionary algorithm that allows the sharing of genetic information between tasks in the form of implicit genetic transfer [23].

MFEA is the earliest multitasking evolutionary algorithm that is influenced by multifactorial genetics. The developmental traits of offspring are not only influenced by genetic factors, but culture plays an important role in the formation of developmental traits, i.e. [24], both genetic and cultural transmission occurs during the process of population formation. Therefore, the two cannot be considered in separate treatments. For example, the phenotype of offspring may depend on the genotype, but acquired learning and cultural traits also play an important role in the formation of the phenotype. In other words, in addition to genetic factors, the habits and interests of the parental generation can influence the formation of the offspring. To solve multiple tasks, MFEA has created a multitasking environment. Tasks may have different properties from each other. Each task has its unique properties that affect the evolution of the population, and therefore different representations exist between individuals. A uniform representation is needed, by virtue of which the independent space of any task can be easily encoded into the same representation for search and evaluation.

In MFEA, a key step in MFEA is the encoding and decoding of chromosomes. The encoding is done to create a uniform search space Y. The uniform search space consists of k tasks, where the jth search space has dimension Dj. The unified search space is defined as D_multitask = max_j{ D_j }, where j∈ { 1, 2, …, k }. A chromosome y ∈ Y represents the solution of k tasks and each solution represents a task. Decoding is the solution that breaks down chromosome Y into k specific tasks. The maximum size of the chromosome has the maximum size of the task to determine. MFEA uses an implicit division of the group into k clusters of tasks, with different task groups studying different tasks, and the tasks are represented by skill factors with respect to their members. Thus, knowledge transfer between different tasks is achieved through categorical mating and vertical cultural communication [25].

Based on the above definition, Algorithm 1 provides a brief introduction to the basic framework of MFEA.

Algorithm 1
The basic framework of MFEA

Input: N, the population size

K, the number of tasks

Output: the best chromosome of tasks

1.Generate an initial population p1

2.Assign skill factors τ for each individual

3.Evaluate each individual in tasks

4.Calculate the factorial cost for each individual

5.While (stopping conditions are not satisfied) do

6.Generate offspring population p2 through assortative mating

7.Perform vertical cultural transmission

8.Assess individual offspring and generate new population p = p1 ∪ p2

9.Update scalar fitness φ and skill factors τ of each individual

10.Select the N fittest individuals from P to from P1

11.End While

First, n individuals need to be randomly generated in the uniform search space, and each individual is assigned a skill factor, evaluated as an individual and a coefficient cost is calculated to form a population after initialization.

Then, new populations are generated by categorical mating and vertical cultural propagation, and the best offspring in the population are selected to form a population, updating the scalar fitness φ and skill factors τ of each individual. Progeny is reproduced using taxonomic mating and vertical cultural transmission mechanisms based on simulated binary crosses and polynomial mutation operators [26].

Finally, an elite-based environment selection operator is used to generate the next generation. Unlike independent optimization algorithms, due to the synergy between tasks in a multi-task evolutionary algorithm, multiple tasks can be optimized simultaneously and a better solution can be obtained for each optimization task.

3 The proposed MFEA-VOM

3.1 The General framework of MFEA-VOM

In MFO, there are varying degrees of correlation between tasks. As evolution proceeds, the relevance of tasks to date decreases and the efficiency of knowledge transfer also decreases. To facilitate the improvement of knowledge transfer efficiency in the population, the MFEA-VOM algorithm expands the search scope in a multi-task environment and changes the similarity of individuals in the population. Algorithm 2 summarizes the pseudo-code of MFEA-VOM. The flow of MFEA-VOM algorithm operation is described below.

Algorithm 2
Basic structure of MEFA-VOM

Input: N, the population size

K, the number of tasks

Output: the best chromosome for tasks

1. Generate an initial population P

2. Assign skill factors τ for each individual

3. Calculate the factorial cost for each individual

4. While (stopping condition are not satisfied) do

5. For i = 1,2 . . . N/2

6. Randomly select two parent individuals p₁ and p₂

7. Obtain skill factor τ₁ and τ₂ of p₁ and p₂

8. If τ₁= = τ₂

9. p₁ and p₂ crossover to generate c₁

10. c₂=Opposition matrix (c₁)

11. Else if rand < RMP

12. c₁=Regulatory gene fusion strategy (p₁, p₂)

13. c₂=Opposition matrix(c₁)

14. Else

15. c₁=mutation(p₁)

16. c₂=mutation(p₂)

17. End If

18. End For

19. Evaluate offspring population C

20. New population PC = P∩C

21. Select fittest individuals from PC to form P

22. End while

First, MFEA-VOM performs an initialization operation to give birth to an initialized population.

During each iteration of evolution, two individuals are randomly selected as the selected parents, i.e., p₁ and p₂. If parents p₁ and p₂ have the same skill factor, then parents p₁ and p₂ produce offspring individuals c₁ by assorting mating mechanism. Then, opposition matrix strategy is applied to produce offspring individuals c₂. When the parents have different skill factors and the number of random numbers generated is less than the RMP, the parents produce offspring individual c₁ by regulatory gene fusion strategy, and then, the offspring individual c₂ by applying the opposition matrix strategy. Otherwise, they produce progeny individuals C1 and C2, respectively, by mutation operators. Note that the RMP is used to control and regulate the communication between tasks and serves to regulate the evolution of the population.

Finally, the new population is merged with the original population and the next population is generated by applying the elite-based environmental selection operator.

For the improved multi-task evolutionary algorithm MFEA-VOM, by introducing components, the time complexity of the algorithm increases, but the corresponding problem can be solved. Therefore, it is necessary to go within the appropriate scope to introduce components to solve the problem. For before the introduction of components, the time complexity of the multitasking evolutionary algorithm is o(n3), however, the time complexity level of the introduced components is o(n2), which does not result in a significant increase in the time complexity of the multitasking algorithm. Therefore, although the introduction of components increases the time complexity, it does not increase the time complexity of the algorithm significantly.

For the important parts of MFEA-VOM, i.e., implicit conversion strategy, opposition matrix strategy and regulatory gene fusion strategy, will be described below.
3.2 Implicit conversion strategy

EMT algorithm is vulnerable to negative knowledge transfer.

In order to maximize the knowledge transfer and reduce the impact caused by negative knowledge transfer. A strategy that can improve the knowledge transfer ability, i.e., the implicit conversion strategy, is proposed. During the early evolution of the population, task-to-task similarity was low. If transitioning to task-to-task communication, excellent solutions will not be achieved, but will result in a waste of resources. In [4], rmp was used as a genetic mating probability to regulate the communication between tasks. As a considered defined constant value, it cannot be changed adaptively according to the evolutionary process of the population. A value of rmp close to 1 indicates a greater likelihood of completely random mating between individuals for different tasks. In this case, the EMT algorithm falls into a local optimum because of the inability to eliminate the effect of negative transfer knowledge [4]. In contrast, a rmp close to 0 means that progeny will produce progeny individuals by mutation with a high probability.

The values of the mapping constants MM and NN are introduced, where MM has a value of 0.9 and NN has a value of 0.5. At the beginning of the iteration, the approximate value of RMP is MM. At this time, the intra-population communication dominates and the external communication of the population supplements it. As the iteration progresses, the magnitude of RMP gradually transitions from MM to NN, and at the end of the iteration, the value of RMP is close to NN. $RMP = MM - (MM - NN) * i / gen$ (2)

In (2), the value range of i is 1,2 . . . 800. gen represents the number of iterations. MM and NN are human-defined values of mapping constants.

3.3 Opposition matrix strategy

Based on opposition based learning [27], genetic transform strategy [28] have been proposed to solve problems with low correlation between tasks. Opposition-based learning is described in order to find the heel good candidate solution while estimating the inverse solution that is closer to the global optimal solution corresponding to it. Based on this, we propose a new strategy, namely the opposition matrix strategy. As the population evolves, the task-to-task correlation decreases. The resulting offspring are not well adapted to the requirements. That is, the efficiency of knowledge transfer between tasks gradually decreases as evolution proceeds, resulting in the generation of offspring that do not survive effectively. This not only causes a waste of resources, but also slows down the convergence of the population. To solve this problem, the opposition matrix strategy reduces the gap between task individuals and is given in pseudo-code in Algorithm 3.

First, the population is divided into two subpopulations, denoted as Po1 and Po2, based on the skill factor between individuals. Parents p₁ and p₂ are randomly selected from the subpopulation. Secondly, depending on the skill factor, two directions are selected for mapping.

Algorithm 3
Opposition matrix strategy

Input: p₁, p₂ the selected parents

OM₁, OM₂ the opposition matrix

Po1, Po2 the subpopulation

Output: the generated child c₂

1. obtain skill factors τ₁ and τ₂ of parents p₁ and p₂

2. p₁ and p₂ crossover to generate c₁

3. if τ₁ = 1 and τ₂ = 2

4. Form opposition matrix OM₁ basing on Po1

5. Generate c₂ based on opposition matrix OM₁

6. else if τ₁ = 2 and τ₂ = 1

7. Form opposition matrix OM₂ basing on Po2

8. Generate c₂ based on opposition matrix OM₂

When the skill factor of parent p₁ is 1 and the skill factor of parent p₂ is 2, the opposition matrix of subpopulation Po1 is applied to generate the progeny individual c₂. When the skill factor of parent p₁ is 2 and the skill factor of parent p₂ is 1, the opposition matrix of subpopulation Po2 is applied to generate the progeny individual c₂.

The operation rules of the opposition matrix are as follows: Four variance matrices are introduced according to the population division, mutate_a, mutate_b, mutate_c, mutate_d. The calculation rules are as follows: ${mutate}_{a} = {step}_{T} 1 [gen e_{1}, gen e_{2}, gen e_{3}]$ (3) ${mutate}_{b} = {step}_{T} 2 [gen e_{1}, gen e_{2}, gen e_{3}]$ (4) ${mutate}_{c} = {step}_{T} 1 [gen e_{n - 2}, gen e_{n - 1}, gen e_{n}]$ (5) ${mutate}_{d} = {step}_{T} 2 [gen e_{n - 2}, gen e_{n - 1}, gen e_{n}]$ (6) $task 1_{ma} = [{mutate}_{a}; {mutate}_{c}]$ (7) $task 2_{ma} = [{mutate}_{b}; {mutate}_{d}]$ (8) where step_T1 and step_T2 are vectors obtained based on the genetic factors of the task. step_T1 and step_T2 are obtained after population sorting. Select appropriate gene operators from the population and populate them into step_T1 and step_T2.

Based on the opposition matrix, the task is selected to establish a mapping relationship between Ackley and Sphere. As shown in Fig. 2, the rules for the usage of the opposition matrix are introduced. $c_{2} = k * O M_{2} - c_{1}$ (9) $c_{2} = k * O M_{1} - c_{1}$ (10)

Fig. 2
An example of opposition matrix transformation.

Selected from the task groups Ackley and Sphere, p₁ and p₂ denote individuals in the task groups Ackley and Sphere, respectively. Depending on the skill factor, subpopulation-based opposition matrices are created separately. By equations (9) and (10), the offspring individual C2 is generated. Because the opposition matrices OM₁ and OM₂ are built based on their respective subpopulations, the offspring individual C2 is mapped to the vicinity of the corresponding task group population according to the mapping relationship, thus tending to produce the better offspring individual.

Sphere individual may be located in the vicinity of Ackley population. Both tend to produce higher quality progeny individuals. The above process, called the transformation of the opposition matrix, can attenuate the effect of negative knowledge transfer and facilitate the evolution of the population and the retention of individuals.
3.4 Regulatory gene fusion strategy

In MFEA, EAs used a pair of genetic operators, i.e. crossover and mutation, to generate offspring [29, 30]. The disadvantage of this approach is that the position of offspring production is influenced by the parents during the evolution of the population [26]. Usually, offspring tend to be distributed close to the parents. It means that the uniform search space cannot be explored effectively. To solve this problem, a regulatory gene fusion strategy is proposed in this paper. By enhancing the search of uniform expression space, the search ability of the population is enhanced. The regulatory gene fusion strategy is described in detail in Algorithm 4.

Algorithm 4
Regulatory gene fusion strategy

Input: p₁, p₂ the selected parents

RU₁, RU₂ the regulatory vectors

P the defined constants

Output: the generated child c₁

1. obtain skill factors τ₁ and τ₂ of parents p₁ and p₂

2. if τ₁ = τ₂

3. Crossover p₁ and p₂ to generate c₁

4. else

5. if rand > p

6. p₁ produces through RU₂

7. Crossover and p₂ to generate c₁

8. else

9. p₂ produces through RU₁

10. Crossover and p₂ to generate c₁

The populations are divided into two categories based on the skill factor of each individual. The first judgment is made based on the skill factor τ. If the skill factors are the same, the offspring are produced directly by mating. If the skill factors are different, further judgments are made. An arbitrary direction is chosen for mapping to obtain the regulation vectors RU₁ and RU₂.The calculation rules for the regulation vectors are as follows. $R U_{1} = (1 - F U_{2}) / F U_{1}$ (11) $R U_{2} = (1 - F U_{1}) / F U_{2}$ (12)

Where FU₁ and FU₂ are the vectors selected according to the task. The individuals of subpopulations are selected and passed into FU1 and FU2 to take the mean value. p is a constant value of 0.5. Depending on the class rank of the subpopulation, the elite individuals of the subpopulation have a much higher probability of being selected than the average individuals. This also ensures that the population can keep evolving.

According to different cases, we will get different mapping vectors RU₁ and RU₂ and multiply the mapping vectors RU₁ and RU₂ with the parents p₁ and p₂ respectively to get the mapped parental individuals and . Since RU₁ and RU₂ are calculated based on their respective corresponding populations, the mapping vectors will tend to fuel higher genetic expression with the respective populations. For example, parent p₁ is transformed into mapping parent by RU₂. The mapped parent has higher similarity in genetic expression with parent p₂ and is more likely to produce elite individuals. At the same time, the effect of negative knowledge transfer was attenuated.
4 Experiments

4.1 Experiments on single-objective multi-factorial optimization problems

4.1.1 Purpose of the experiment

In this section, the proposed MFEA-VOM is compared and analyzed with other advanced EMT algorithms. The experimental results are described by performance score, parameter settings, experimental setup, comparison algorithm, and in-depth analysis.

4.1.2 Performance measures

In addition to the errors present in each task during the experiment, this experiment applies the simple performance metrics proposed in [18] to determine the performance of various algorithms.

Assuming that there are k algorithms, A₁, …, A_k for a problem with M minimization tasks T₁, …, T_M, and Each algorithm is run independently L times. Assume that B (i, j) _l denotes the best result of the algorithm on the lth run by algorithm A_k on Task T_m and μ_m and σ_m be the mean and standard deviation (std) of B (i, j) _l, k = 1,2 . . . K, l = 1,2 . . . L. The normalized performance B’ (i, j) _l is computed as: $B’ {(i, j)}_{l} = (B {(i, j)}_{l} - μ_{m}) / σ_{m}$ (13) and the performance score:

4.1.3 Parameter settings

To allow for a fair comparison of the experiments, Table 2 summarizes the settings of the common parameters used during the experiments.

Table 2
Parameter setting in single target MFO

Parameter Value

Population size 100

Crossover probability 1

Distribution index of crossover 2

Maximum number of evaluations 200,000

Distribution index of mutation 5

Mutation probability 1/max_dim

Parameter	Value
Population size	100
Crossover probability	1
Distribution index of crossover	2
Maximum number of evaluations	200,000
Distribution index of mutation	5
Mutation probability	1/max_dim

Where the task components in the MFO problem may have different dimension sizes (i.e., 25D and 50D), where dim denotes the maximum dimension of all tasks on the benchmark. sex and polynomial variation are used to reproduce offspring individuals. The algorithms in the experiments uniformly use BFG fitted Newton’s method and Lamarck’s principle [31]. The experimental results were obtained on the basis of 30 independent runs. The other parameters of GMFEA [31], MFEALBS [32], MFPSO [33], LDA-MFEA [5], MFEARR [13], MFDE [33] and MFEA [4] are set according to the original publication. The scaling rate sr range in MFEA-VOM is set to [0.5,1.5].

4.1.4 Experimental setup

The single-objective MFO test suite proposed by Da et al. [34] was used to evaluate the performance of MFEA-VOM in this paper. These include nine consecutive MFO benchmarks, each consisting of two optimization problems, the Griewank, Rastrigin, Schwefel, Ackley, Rosenbrock, Schwefel and Weierstrass functions. According to the intersection of decision variables of the global optimal solution of the task, we can further divide it into three sub-categories, namely complete intersection (CI), partial intersection (PI) and no intersection (NI). In this experiment, we choose nine questions to demonstrate the effectiveness of MFEA-VOM. The similarity measure between tasks is based on the Spearman correlation similarity measure. Question with R_s < 0.2 are called low similarity. Question with 0.2 < R_s<0.8 are called medium similarity, and R_s≥0.8 means high similarity. Table 3 summarizes the nature of the MFO problem.

Table 3
Benchmark problem sets

Category Task D_T Landscape R _s

1 CI+HS Griewank (T1) 50 multimodal, nonseparable 1.0000

Rastrigin (T2) 50 multimodal, nonseparable

2 CI+MS Ackley (T1) 50 multimodal, nonseparable 0.2261

Rastrigin (T2) 50 multimodal, nonseparable

3 CI+LS Ackley (T1) 50 multimodal, nonseparable 0.0002

Schwefel (T2) 50 unimodal, separable

4 PI+HS Rastrigin (T1) 50 multimodal, nonseparable 0.8670

Sphere (T2) 50 unimodal, separable

5 PI+MS Ackley (T1) 50 multimodal, nonseparable 0.2154

Rosenbrock (T2) 50 multimodal, nonseparable

6 PI+LS Ackley (T1) 50 multimodal, nonseparable 0.0725

Weierstrass (T2) 25 multimodal, nonseparable

7 NI+HS Rosenbrock (T1) 50 multimodal, nonseparable 0.9434

Rastrigin (T2) 50 multimodal, nonseparable

8 NI+MS Griewank (T1) 50 multimodal, nonseparable 0.3669

Weierstrass (T2) 50 multimodal, nonseparable

9 NI+MS Rastrigin (T1) 50 multimodal, nonseparable 0.0016

Schwefel (T2) 50 unimodal, separable

Category	Task	D_T	Landscape	R _s
1 CI+HS	Griewank (T1)	50	multimodal, nonseparable	1.0000
	Rastrigin (T2)	50	multimodal, nonseparable
2 CI+MS	Ackley (T1)	50	multimodal, nonseparable	0.2261
	Rastrigin (T2)	50	multimodal, nonseparable
3 CI+LS	Ackley (T1)	50	multimodal, nonseparable	0.0002
	Schwefel (T2)	50	unimodal, separable
4 PI+HS	Rastrigin (T1)	50	multimodal, nonseparable	0.8670
	Sphere (T2)	50	unimodal, separable
5 PI+MS	Ackley (T1)	50	multimodal, nonseparable	0.2154
	Rosenbrock (T2)	50	multimodal, nonseparable
6 PI+LS	Ackley (T1)	50	multimodal, nonseparable	0.0725
	Weierstrass (T2)	25	multimodal, nonseparable
7 NI+HS	Rosenbrock (T1)	50	multimodal, nonseparable	0.9434
	Rastrigin (T2)	50	multimodal, nonseparable
8 NI+MS	Griewank (T1)	50	multimodal, nonseparable	0.3669
	Weierstrass (T2)	50	multimodal, nonseparable
9 NI+MS	Rastrigin (T1)	50	multimodal, nonseparable	0.0016
	Schwefel (T2)	50	unimodal, separable

4.1.5 Compared algorithms

GMFEA uses a decision variable shifting strategy and a decision variable shuffling strategy, aiming to enhance population diversity, accelerate population convergence and improve the knowledge transfer effectiveness.

MFPSO applies the particle swarm algorithm to the multitask evolutionary algorithm to update the particle motion by velocity vector to obtain the best position to update the particles.

LDA-MFEA converts the search space of a simple task into a new search space by applying LDA for the purpose of solving the task efficiently.

MFEALBS enhances the original MFEA by employing a uniform alignment-based representation and level-based selection.

MFEARR establishes a resource redistribution mechanism that is used to facilitate the discovery and exploitation of synergies between tasks.

MFDE changes the allocation strategy through a popular population-based optimization algorithm called differential evolution.

The MFEA. We used the MFEA code provided in the WCCI2018 competition evolutionary multi-task optimization.

4.1.6 Comparison to existing techniques

The experimental results of the classical single target MFO test suite are shown in Tables 4 5. Specifically, Tables 4 5 summarize the mean and standard deviation of the target values of all algorithms for the 30 independent runs. The results of the composite performance indicator S-values are shown in Table 6. To make the experimental results more visible, the better experimental results in terms of average target and S-values will be marked in bold. The proposed MFEA-VOM is analyzed in comparison with other EMT algorithms by performing Wilcoxon rank sum test at 5% significance level. The significantly better, significantly worse, or significantly similar results are indicated as “+”, “-”, and “≈” in Tables 4 5, respectively.

Table 4
Task1 average standard target value of MFEA-VOM and other EMT algorithms on 9 sets of benchmark functions for 30 independent runs

Category Task1

MFEA-VOM MFPSO GMFEA MFEA

CI + HS 0(0) 0.940(0.965)+ 0.382(0.492)+ 0.432(0.558)+

CI + MS 1.6198E-10(1.0E-10) 7.214(1.242)+ 3.904(0.682)+ 3.966(0.621)+

CI + LS 8.139(2.370) 21.270(0.716)+ 20.111(0.060)+ 20.145(0.072)+

PI + HS 52.271(72.711) 960.982(150.656)+ 481.798(88.209)+ 453.796(64.539)+

PI + MS 2.594(0.308) 6.079(1.028)+ 3.563(0.434) ≈ 3.678(0.427) ≈

PI + LS 3.358(7.299) 13.338(1.624)+ 14.060(9.322)+ 19.953(0.103)+

NI + HS 2.145(3.573) 4.456E+5(2.083E+5)+ 1.257E+3(626.884)+ 1.205E+3(543.163)+

NI + MS 1.970E-2(0.371) 1.126(0.074)+ 0.437(0.066)+ 0.518(0.069)+

NI + LS 1.14E-08(4.002E-4) 3.100E+3(699.178)+ 508.763(109.688)+ 527.172(85.326)+

Category Task1

LDA-MFEA MFEALBS MFEARR MFDE

CI + HS 0.307(0.084)+ 0.455(0.065)+ 0.322(0.054)+ 0.0017(63.301)+

CI + MS 5.160(0.871)+ 4.332(0.559)+ 12.656(8.378)+ 0.065(0.162)+

CI + LS 21.100(0.147)+ 20.132(0.050)+ 20.101(0.055)+ 21.203(0.036)+

PI + HS 623.692(107.541)+ 516.275(100.640)+ 329.958(61.741)+ 78.563(17.572)+

PI + MS 3.110(0.456) ≈ 3.581(0.404) ≈ 3.319(0.278) ≈ 0.010(0.005)-

PI + LS 4.155(0.920)≈ 19.286(3.163)+ 7.804(7.203)≈ 0.720(0.706)-

NI + HS 1.044E+3(649.597)+ 1.336E+3(567.961)+ 1.801E+3(855.837)+ 114.974(33.723)+

NI + MS 0.551(0.166)+ 0.448(0.082)+ 0.355(0.098)+ 2.250E-3(0.002)+

NI + LS 1.388E+3(637.660)+ 525.726(88.586)+ 365.940(63.301)+ 198.105(106.468)+

Category	Task1
CI + HS	0(0)	0.940(0.965)+	0.382(0.492)+	0.432(0.558)+
CI + MS	1.6198E-10(1.0E-10)	7.214(1.242)+	3.904(0.682)+	3.966(0.621)+
CI + LS	8.139(2.370)	21.270(0.716)+	20.111(0.060)+	20.145(0.072)+
PI + HS	52.271(72.711)	960.982(150.656)+	481.798(88.209)+	453.796(64.539)+
PI + MS	2.594(0.308)	6.079(1.028)+	3.563(0.434) ≈	3.678(0.427) ≈
PI + LS	3.358(7.299)	13.338(1.624)+	14.060(9.322)+	19.953(0.103)+
NI + HS	2.145(3.573)	4.456E+5(2.083E+5)+	1.257E+3(626.884)+	1.205E+3(543.163)+
NI + MS	1.970E-2(0.371)	1.126(0.074)+	0.437(0.066)+	0.518(0.069)+
NI + LS	1.14E-08(4.002E-4)	3.100E+3(699.178)+	508.763(109.688)+	527.172(85.326)+
Category	Task1
	LDA-MFEA	MFEALBS	MFEARR	MFDE
CI + HS	0.307(0.084)+	0.455(0.065)+	0.322(0.054)+	0.0017(63.301)+
CI + MS	5.160(0.871)+	4.332(0.559)+	12.656(8.378)+	0.065(0.162)+
CI + LS	21.100(0.147)+	20.132(0.050)+	20.101(0.055)+	21.203(0.036)+
PI + HS	623.692(107.541)+	516.275(100.640)+	329.958(61.741)+	78.563(17.572)+
PI + MS	3.110(0.456) ≈	3.581(0.404) ≈	3.319(0.278) ≈	0.010(0.005)-
PI + LS	4.155(0.920)≈	19.286(3.163)+	7.804(7.203)≈	0.720(0.706)-
NI + HS	1.044E+3(649.597)+	1.336E+3(567.961)+	1.801E+3(855.837)+	114.974(33.723)+
NI + MS	0.551(0.166)+	0.448(0.082)+	0.355(0.098)+	2.250E-3(0.002)+
NI + LS	1.388E+3(637.660)+	525.726(88.586)+	365.940(63.301)+	198.105(106.468)+

Table 5

Task2 average standard target value of MFEA-VOM and other EMT algorithms on 9 sets of benchmark functions for 30 independent runs

Category	Task2
	MFEA-VOM	MFPSO	GMFEA	MFEA
CI + HS	0(0)	373.191(31.354)+	186.298(42.806)+	187.964(43.171)+
CI + MS	0(0)	526.217(106.996)+	218.624(52.008)+	211.159(40.048)+
CI + LS	340.423(391.427)	1.565E+4(839.095)+	2.749E+3(328.399)+	3.461E+3(412.005)+
PI + HS	36.635(7.037)	4.939E+3(1.240E+3)+	14.174(3.017)-	14.514(3.434)-
PI + MS	74.298(111.088)	8.524E+4(5.632E+4)+	840.604(202.712)+	973.401(412.420)+
PI + LS	2.98E-3(5.97E-3)	10.888(2.885)+	11.474(7.689)+	18.348(3.171)+
NI + HS	1.923E-11(2.623E-12)	564.385(109.456)+	247.941(76.009)+	277.053(91.078)+
NI + MS	1.183E-2(0.033)	32.448(1.475)+	25.260(2.986)+	24.486(2.993)+
NI + LS	1.308(1.801)	1.5658E+4(1.498E+4)+	2.868E+3(498.105)+	3.374E+3(465.137)+
Category	Task2
	LDA-MFEA	MFEALBS	MFEARR	MFDE
CI+HS	222.742(74.363)+	229.424(40.233)+	330.873(61.925)+	3.773(4.703)+
CI + MS	272.591(75.563)+	232.084(37.093)+	339.392(84.738)+	0.427(0.850)+
CI + LS	8.925E+3(2.431E+3)+	3.525E+3(438.432)+	3.306E+3(431.239)+	1.240E+4(1.638E+3)+
PI + HS	2.456(1.441)-	16.048(3.778) ≈	8.743(4.860)-	1.27E-3(1.094E-3)-
PI + MS	236.266(74.018)+	930.875(281.718)+	1.135E+3(319.167)+	82.852(19.033) ≈
PI + LS	4.499(1.011)+	17.490(4.023)+	14.726(2.385)+	0.406(0.951)+
NI + HS	285.909(99.202)+	259.196(68.410)+	1.801E+3(855.837)+	28.362(10.442)+
NI + MS	14.635(1.803)+	25.075(2.940)+	38.130(4.136)+	3.603(0.946)+
NI + LS	6.635E+3(733.921)+	3.339E+3(488.185)+	3.303E+3(523.665)+	4.247E+3(1.090E+3)+

Table 6

Score values of MFEA-VOM, MFPSO, GMFEA, MFEALBS, MFDE, LDA-MFEA, MFEARR and MFEA on 30 independent runs on a single-objective classical test suite

Problem	MFEA-VOM	MFPSO	GMFEA	MFEALBS	MFDE	LDA-MFEA	MFEARR	MFEA
CI + HS	-13.821	16.043	0.515	2.910	-13.645	0.561	4.041	1.04179
CI + MS	-11.405	10.157	-1.002	-0.380	-11.376	1.484	9.837	-0.744
CI + LS	-20.176	9.748	-2.220	-1.317	8.133	4.915	-1.528	-1.373
PI + HS	-8.296	21.949	-1.251	-0.083	-8.053	1.062	-4.251	-1.359
PI + MS	-3.483	18.273	-0.557	-0.939	-11.551	-2.302	-1.273	0.328
PI + LS	-8.484	2.829	0.571	10.451	-11.677	-7.039	3.340	11.768
NI + HS	-5.213	13.224	-3.249	-3.188	-4.983	-3.318	13.169	-3.181
NI + MS	-14.184	15.183	1.958	1.495	-12.866	-1.067	6.083	2.663
NI + LS	-9.762	22.973	-3.786	-3.524	-4.294	5.605	-4.196	-3.010

As known from Tables 4 5, MFEA-VOM shows significantly better solution quality in 17 out of 18 test tasks on the classical single-target test suite compared to the traditional MFEA, indicating a significant improvement in algorithm performance. Compared to GMFEA, MFEA-VOM showed superior performance to GMFEA in 14 out of 18 test tasks, while two test tasks showed similar performance. Compared to MFDE, 14 out of 18 test tasks indicate superior performance to MFDE, while one of the test tasks exhibits similar performance results. 3 test tasks show MFDE outperforms MFEA-VOM. Compared to MFPSO, MFEA-VOM showed better performance than MFPSO in 15 of the 18 test tasks, while the remaining three test tasks showed similar performance. Compared to LDA-MFEA, 15 of the 18 test tasks showed better performance than LDA-MFEA, 2 test tasks showed similar results, and 1 test task showed better performance results than MFEA-VOM. Compared to MFEALBS, 16 test tasks showed better performance than MFEALBS, and 2 test tasks showed similar performance results to MFEA-VOM. Compared to MFEARR, 15 of the 18 test tasks showed better performance results than MFEARR, 2 test tasks showed similar performance results, and 1 test task showed better performance results than MFEA-VOM.

The above results confirm the competitiveness of MFEA-VOM. MFEA-VOM shows better performance results on high similarity benchmark problems, i.e., CI+HS, PI+HS, NI+HS, than the rest of EMT algorithms. This mainly depends on the ability of MFEA-VOM to weaken the effect of negative knowledge migration. Compared with MFEA-VOM, MFEA cannot mitigate the effect of negative knowledge migration, so it leads to low performance results. On benchmark problems with low similarity, i.e., CI+LS, NI+LS, and PI+LS. MFEA-VOM shows similar excellent solutions to other EMT algorithms. This mainly depends on the effectiveness of GMFEA when benchmarking between different dimensional tasks through a mixed variables strategy. MFDE exploits the individual differences of the current population and changes the classification assignment strategy. On benchmark problems with moderate similarity, i.e., CI+MS, PI+MS, and NI+MS.MFEA-VOM outperforms most of the solutions compared with other EMT algorithms. MFEALBS uses a uniformly aligned representation and a level-based selection strategy that enables MFEA to be enhanced. MFEARR promotes synergy between tasks through a resource-in-allocation mechanism. MFEA-VOM effectively solves the problem of knowledge transfer by constructing a genetic expression space for populations with different tasks, and effectively solves the communication effect between populations poorly. Accelerate population fusion by regulating vectors.

For a clear analysis of the evolutionary process, Fig. 3 depicts the average convergence plot for the first 800 generations.

Fig. 3

MFEA-VOM, GMFEA, MFDE, LDAMFEA, MFEALBS, MFEARR, MFPSO and MFEA in single target MFO test components obtained for 30 average target value tasks1.

It is clear from Fig. 3 that the MFEA-VOM algorithm converges significantly faster than the other EMT algorithms in the vast majority of tasks of the evolutionary process, i.e., the factorial cost in MFEA-VOM decreases at a large rate under the same conditions. It is shown that the proposed strategies, i.e. implicit conversion strategy, opposition matrix strategy and regulatory gene fusion strategy are well used for the tasks and obtain better solutions by improving the efficiency of knowledge transfer and establishing new genetic expression spaces. In CI+MS and CI+LS, MFEA-VOM is able to quickly reduce the stratum cost and converge to an excellent solution, while other EMT algorithms, in this process, not only converge slowly, but also some barely converge. This means that other EMT algorithms are prone to fall into local optimum on CI+MS, CI+LS. In the test component of Task 1, the MFEA-VOM algorithm significantly outperforms the other EMT algorithms, especially in PI+LS, CI+LS.

Experimental analysis was conducted for the problems in Task 2, namely CI+HS, CI+MS, CI+LS, PI+HS, PI+MS, PI+LS, NI+HS, NI+MS, NI+LS. from the experimental results, we can conclude that MFEA-VOM converges significantly faster than other EMT algorithms. This means that MFEA-VOM can effectively solve the single-target MFO problem. From Fig. 4, we can see that the convergence of the sub-generation process of GMFEA and MFEA slows down or even does not converge in PI+LS and NI+MS. This indicates that GMFEA and MFEA easily fall into local optimum. In contrast, MFEA-VOM and MFDE converge faster and the stratification cost reflects that the obtained solutions are much better than GMFEA and MFEA. The convergence of MFEA-VOM is better compared to MFDE. In NI+LS and CI+LS, the convergence speed as well as the convergence results of MFEA-VOM are significantly better than other EMT algorithms, reflecting the well-documented performance of MFEA-VOM algorithm.

Fig. 4

MFEA-VOM, GMFEA, MFDE, LDAMFEA, MFEALBS, MFEARR, MFPSO and MFEA in single target MFO test components obtained for 30 average target value tasks.

As shown in Fig. 6, the performance score S value of MFEA-VOM is larger than the performance score S value of MFDE algorithm for only one test set, PI+MS, compared with other EMT algorithms. For all other test sets, the performance score S-value of MFEA-VOM is significantly smaller than other EMT algorithms, which proves that MFEA-VOM shows excellent performance results overall.

4.17 Experimental comparison of MFEA-VOM and MFEA-II

To further illustrate, the effectiveness of the MFEA-VOM algorithm, we analyzed the results of the data obtained including MFEA-VOM and MFEA-II on the composite benchmark problem as shown in Table 7.

Table 7
Comparison of the means and standard deviations of the continuous minimization benchmark problem MFEA-VOM and MFEA-II over 30 independent runs

Problem set Task1 Task2

MFEA-VOM MFEA-II MFEA-VOM MFEA-II

CI+HS 0(0) 0(0) 0(0) 0(0)

CI + MS 1.61E-1(1.0E-1) 0(0) 0(0) 0(0)

CI + LS 8.13(2.37) 19.98(0.04) 340.42(391.42) 1.67E+03(5.18E+02)

NI + HS 2.14(3.57) 55.42(29.91) 1.92E-11(2.62E-12) 4.08(9.73)

NI + MS 1.97E-2(0.37) 0(0) 1.18E-2(0.03) 20.61(1.19)

Problem set	Task1	Task2
CI+HS	0(0)	0(0)	0(0)	0(0)
CI + MS	1.61E-1(1.0E-1)	0(0)	0(0)	0(0)
CI + LS	8.13(2.37)	19.98(0.04)	340.42(391.42)	1.67E+03(5.18E+02)
NI + HS	2.14(3.57)	55.42(29.91)	1.92E-11(2.62E-12)	4.08(9.73)
NI + MS	1.97E-2(0.37)	0(0)	1.18E-2(0.03)	20.61(1.19)

It can be seen that MFEA-VOM shows superior performance overall. MFEA-VOM not only performs better at high similarity, but also outperforms MFEA-II for low task similarity (CI+LS) problems. on Tasks2, MFEA-VOM shows far better performance than MFEA-II on NI+HS and NI+MS. This is due to the fact that MFEA-VOM incorporates three optimization strategies, which can significantly reduce the occurrence of negative knowledge migration, facilitate the exploration of search space, and promote knowledge migration.

4.18 Deeper insight of MFEA-VOM

For the single target problem PI+HS, PI+MS and PI+LS were studied and analyzed and used to investigate the effect of the opposition matrix strategy and the regulatory gene fusion strategy, respectively. It should be noted that three variants of MFEA-VOM were considered, namely one using only the opposition matrix strategy, denoted by MFEA-VOM1, one using only the regulatory gene fusion strategy, denoted by MFEA- VOM2 and one using only the implicit conversion strategy, denoted by MFEA-VOM3. In the experiments, MFEA-VOM1, MEFA-VOM2 and MFEA-VOM3 were analyzed in comparison with MFEA-VOM and MFEA. The average convergence plots represented in Fig. 5 are based on the factor cost of 30 independent runs of MFEA, MFEA-VOM1, MFEA-VOM2, MFEA-VOM3, and MFEA-VOM.

Fig. 5

Illustrate the performance of each adaptation of the three strategies.

Table 7

Task1 average standard target value of MFEA-VOM on 3 sets of benchmark functions for 30 independent runs

Category	Task1
MFEA-VOM	MFEA-VOM1	MFEA-VOM2	MFEA-VOM3	MFEA
PI + HS	52.271(72.711)	532.1767(96.845)	790.782(181.111)	957.366(144.125)	453.796(64.539)
PI + MS	2.594(0.308)	4.166(0.879)	4.025(0.837)	4.853(1.153)	3.678(0.427)
PI + LS	3.358(7.299)	3.929(5.761)	18.366(3.711)	19.686(3.029)	19.953(0.103)

As can be seen in Fig. 5, MFEA-VOM achieved better experimental results than both MFEA-VOM1 and MFEA-VOM2 in task groups PI+HS, PI+MS, and PI+LS. As can be seen in Fig. 5, MFEA-VOM achieves better experimental results than MFEA-VOM1 and MFEA-VOM2 in task groups PI+HS, PI+MS, and PI+LS. Due to the limitation of the SBX operator, MFEA can only search on the local landscape. In contrast, MFEA-VOM1 (opposition matrix strategy) was able to expand the search capacity of the population in a uniform search space as well as in each task. The population is able to enter more uncharted territories and explore more unknown solutions. However, expanding the search capacity of the population sometimes does not work well due to negative knowledge transfer. For example, the effect plot of PI+MS in task 1 in the test component shows the limitation of MFEA-VOM1. MFEA-VOM2 (regulatory gene fusion strategy) can enhance the similarity of population individuals in the genetic expression space to obtain higher quality solutions, but it can easily fall into local optimum. For example, MFEA-VOM2 falls into a local optimum in the later stages of the iteration in the PI+LS T1 and PI+LS T2 tasks, which slows down the population convergence. Since the opposition matrix strategy and the regulatory gene fusion strategy can complement each other and make up for each other’s deficiencies, the MFEA-VOM algorithm tends to outperform other algorithms of the same type on most test problems. MFEA-VOM3 (implicit conversion strategy) can regulate the evolutionary direction of the population to favor the iterative process, but the implicit conversion strategy alone cannot be relied on well to the convergence speed of MFEA- VOM3 is slow, as shown in Fig. 5. If combined with other strategies, excellent solutions will be obtained more quickly.

The following conclusions can be drawn from Tables 7 8, MFEA-VOM achieves better optimization experimental results than MFEA-VOM1, MFEA-VOM2 and MFEA-VOM3 for all optimization tasks, i.e. PI+HS, PI+MS and PI+LS. The three strategies complement each other and promote the formation of excellent solutions for the populations.

Table 8

Task2 average standard target value of MFEA-VOM on 3 sets of benchmark functions for 30 independent runs

Category	Task2
MFEA-VOM	MFEA-VOM1	MFEA-VOM2	MFEA-VOM3	MFEA
PI + HS	36.635(7.037)	242.203(48.624)	47.997(12.565)	77.297(17.077)	14.514(3.434)-
PI + MS	74.298(111.088)	3.487E+3(7.361E+3)	1.166E+3(469.414)	4.151E+3(2165E+3)	973.401(412.420)+
PI + LS	2.98E-3(5.97E-3)	2.741(5.20)	19.710(4.875)	20.052(3.283)	18.348(3.171)+

4.2 Experiments on multi-objective multi-factorial optimization problems

4.2.1 Performance indicators

In MOO, the inverse intergenerational distance (IGD) [35] is the most widely used metric to evaluate the quality of the solutions obtained by the algorithm in terms of convergence and diversity; however, IGD has the disadvantage of not meeting the Pareto criterion. Therefore, in this paper, a more reliable metric based on IGD, Improved IGD [36] (IGD+), is used to compare the quality of solutions of various algorithms. IGD + is used to evaluate the performance of the algorithm on a single optimization task in MFO. To evaluate the performance of the algorithm on each benchmark in a comprehensive manner, another metric called Mean and Standard Score (MSS) [37] is used. Suppose an MFO benchmark test consisting of k tasks, denoted as t₁,t₂,...,t_k, denotes the IGD+ value of I_k for the kth task of a single run of the algorithm, where k = 1,2,...,k. The mean and standard deviation of IGD+for all comparisons of the EMT algorithm on the kth task will be denoted as μ_k and σ_k. The MSS equation is as follows. $MS S_{i} = \frac{1}{k} \sum_{k = 1}^{k} \frac{I_{l} - μ_{k}}{σ_{k}}$ (14)

4.2.2 Parameter settings

In order to experimentally allow for fair comparisons, all algorithms used the same genetic operators, i.e., SBX crossover and polynomial mutation. Table 9 summarizes the settings of some common parameters, where max_dim indicates the maximum dimensionality for all tasks in the MFO benchmark test. other parameters in MOMFEA, TMOMFEA and NSGA-II are consistent with the original reference settings. The overall size of each task is set to 100. for all EMT algorithms, the maximum number of fitness evaluations is 200000.

Table 9
Multi-objective experiments often have parameter settings

Parameter Value

Population size 100

Crossover probability 0.9

Distribution index of crossover 20

Maximum number of evaluations 200,000

Distribution index of mutation 20

Mutation probability 1/max_dim

Parameter	Value
Population size	100
Crossover probability	0.9
Distribution index of crossover	20
Maximum number of evaluations	200,000
Distribution index of mutation	20
Mutation probability	1/max_dim

4.2.3 Experimental setup

The multi-target experiments were conducted using the MFO test suite introduced by Yuan et al [37]. The test suite consists of nine benchmark tests with different properties. Among them, each benchmark test contains two multi-objective problems and two or three objective functions. Similar to the single-objective test suite, the tests are classified into high similarity (HS), medium similarity (MS), and low similarity (LS) based on the similarity in the fitness landscape. Each category is further divided into three subcategories of complete intersection (CI), partial intersection (PI) and no intersection (NI). To demonstrate the effectiveness of MFEA-VOM in solving different types of multi-objective MFO problems, nine problems are used in this experiment, and this experiment uses a multi-objective MFO complex test suite to test the effectiveness of MFEA-VOM.

4.2.4 Compared algorithms

The proposed MFEA-VOM is compared with other state-of-the-art EMT algorithms, including TMOMFEA [38], MOMFEA [39] and NSGA-II [40], on a multi-objective task test set. TMOMFEA performs setting multiple rmp values based on different types of decision variables. the NSGA-II algorithm uses a fast non-dominated sorting algorithm with significantly lower computational complexity, citing elite strategy, expands the sampling space, prevents the loss of the best individuals, and improves the computational speed and robustness of the algorithm.

4.2.5 Experimental results

The experimental results of the classical multi-objective MFO test suite are shown in Tables 10 11. The results of the MFEA-VOM and other multi-objective EMT comparison algorithms obtained IGD+mean and standard deviation after 30 independent runs are shown in Tables 10 11. The experimental results of the MSS values are shown in Table z. The optimal average IGD+and MSS values are highlighted in bold. The proposed MFEA-VOM algorithm is compared with other excellent EMT algorithms, and those that are significantly better, significantly worse, or significantly similar are noted as+, -, and ≈, respectively.

Table 10
The averaged and standard IGD+value of MFEA-VOM, TMOMFEA, MOMFEA, NSGA-II over 30 independent runs on the classical multi–objective test suite of task1

Problem set Task1

MFEA-VOM TMOMFEA MOMFEA NSGA-II

CI + HS 2.49E-3(3.79E-5) 0.020(7.002E-3)+ 0.011(3.205E-3)+ 0.029(6.166E-3)+

CI + MS 7.37E-3(3.46E-3) 1.409(2.364)+ 1.224(1.979)+ 3.660(2.117)+

CI + LS 0.855(1.671) 0.036(7.287E-3)- 0.011(1.453E-3)- 23.351(4.166)+

PI + HS 2.90E-3(6.94E-5) 0.423(0.151)+ 0.021(0.014)+ 0.027(5.296E-3)+

PI + MS 11.47(2.88) 0.332(0.068)- 0.228(0.076)- 0.217(0.045)-

PI + LS 0.842(0.519) 0.035(8.001E-3)- 0.011(5.405E-3)- 9.051(5.25E-3)+

NI + HS 2.149(1.318) 53.166(1.695)+ 48.962(0.538)+ 709.866(1653.921)+

NI + MS 82.731(33.64) 16.559(0.086)- 27.577(29.732)- 43.141(37.082)-

NI + LS 0.179(0.038) 1.064(4.208E-3)+ 0.059(8.615E-3)- 0.048(0.012)-

Problem set	Task1
CI + HS	2.49E-3(3.79E-5)	0.020(7.002E-3)+	0.011(3.205E-3)+	0.029(6.166E-3)+
CI + MS	7.37E-3(3.46E-3)	1.409(2.364)+	1.224(1.979)+	3.660(2.117)+
CI + LS	0.855(1.671)	0.036(7.287E-3)-	0.011(1.453E-3)-	23.351(4.166)+
PI + HS	2.90E-3(6.94E-5)	0.423(0.151)+	0.021(0.014)+	0.027(5.296E-3)+
PI + MS	11.47(2.88)	0.332(0.068)-	0.228(0.076)-	0.217(0.045)-
PI + LS	0.842(0.519)	0.035(8.001E-3)-	0.011(5.405E-3)-	9.051(5.25E-3)+
NI + HS	2.149(1.318)	53.166(1.695)+	48.962(0.538)+	709.866(1653.921)+
NI + MS	82.731(33.64)	16.559(0.086)-	27.577(29.732)-	43.141(37.082)-
NI + LS	0.179(0.038)	1.064(4.208E-3)+	0.059(8.615E-3)-	0.048(0.012)-

Table 11

The averaged and standard IGD+value of MFEA-VOM, TMOMFEA, MOMFEA, NSGA-II over 30 independent runs on the classical multi–objective test suite of task2

Problem set	Task2
MFEA-VOM	TMOMFEA	MOMFEA	NSGA-II
CI+HS	4.545E-3(3.0E-4)	0.105(0.020)+	0.079(0.016)+	0.091(0.011)+
CI + MS	0.075(1.231)	0.098(0.095)≈	0.296(0.484)+	1.191(0.782)+
CI + LS	4.14E-3(3.09E-3)	4.65E-3(1.7E-4)≈	3.755E-3(8.254E-3)-	7.28E-3(5.1E-3)+
PI + HS	0.092(0.058)	22.396(4.073)+	1.128(0.437)+	0.685(0.313)+
PI + MS	466.33(169.39)	537.621(62.601)+	557.931(80.270)+	747.592(240.492)+
PI + LS	17.639(1.159)	1.969(0.248)-	0.227(0.029)-	20.080(0.029)+
NI + HS	3.04E-3(1.68E-4)	0.038(0.011)+	0.016(2.7E-3)+	0.023(3.863E-3)+
NI + MS	4.538(3.446)	0.051(0.086)-	0.476(0.536)-	1.475(1.076)-
NI + LS	5.733(4.542)	21.074(0.054)+	20.306(8.373E-3)+	20.294(4.264E-4)+

In comparison with the basic MOMFEA algorithm, MFEA-VOM obtains superior solutions for 8 tasks in the classical test set. In addition, compared to other additional EMT algorithms, MFEA-VOM achieves good performance on 14 out of 18 benchmarking tasks compared to TMOMFEA. MFEA-VOM achieves good performance on 14 out of 18 benchmarking tasks compared to NSGA-II for MFEAV-VOM. The above results lead to the conclusion that MFEA-VOM can effectively solve the multi-target MFO problem and shows the superior performance of MFEA-VOM compared to other EMT algorithms.

For the classical MFO test set, as shown in Table 12, the MSS values of MFEA-VOM are significantly better than other EMT algorithms on the benchmarks of CI+HS, CI+MS, PI+HS, NI+HS and NI+LS, specifically because, during the evolutionary process, individuals of different tasks are separated into different locations in different landscapes, which makes the communication between tasks gradually become inefficient MFEA-VOM drives the knowledge transfer through the mapping relationship between tasks, and even transforms the negative knowledge transfer into positive knowledge transfer. Finally, on the benchmark of complete intersection of global optima such as CI+HS and CI+MS, the result of obtaining smaller MSS values compared with other EMT algorithms is achieved, which confirms the effectiveness of MFEA-VOM algorithm.

Table 12

The MSS value of MFEA-VOM, TMOMFEA, MOMFEA, NSGA-II over 30 independent runs on the\\ multi-objective classical test suit

Problem	MFEA-VOM	MOMFEA	TMOMFEA	NSGA-II
CI+HS	-3.00734	1.348418	-0.23035	1.897074
CI + MS	-1.93791	-0.82457	-0.53083	3.278417
CI + LS	-1.11629	-0.82837	-1.47643	3.413837
PI + HS	-1.28882	3.462779	-1.07614	-1.08904
PI + MS	0.668194	-0.94058	-0.76671	1.054705
PI + LS	0.425945	-1.53815	-1.73923	2.855869
NI + HS	-2.03321	0.912008	-0.84617	1.964213
NI + MS	3.256841	-1.94058	-1.25883	-0.06851
NI + LS	-1.05605	1.972107	-0.45289	-0.47965

4.3 Practical study

Multitasking evolution with a wide range of application scenarios, e.g., point cloud registration [41] and double cart-pole balancing problem [42]. In this part of the experiment, to confirm the efficiency of our proposed MFEA-VOM algorithm in a real environment. Following the setup in [42], this experiment is able to solve several neuroevolution-based tasks simultaneously, i.e., the double cart-pole balancing problem. During the task, each task requires the consideration of training a neural network to balance the two poles in the cart. The unique feature of MFO is the ability to achieve knowledge transfer between complex and simple tasks.

The double cart-pole balancing problem is usually considered as a standard benchmark for artificial control systems. This problem is solved by training a neural network-based controller as a way to balance two rods of different lengths hinged on a cart (imaging as two inverted pendulums). One pole is a fixed 1.0 m, while the other is variable 0.0 to 1.0 m. But when the length of the short pole approaches 1.0 m, the balancing task becomes increasingly difficult at that point. This study is mainly to show that both simple and complex problems can be solved very effectively when the knowledge of problem solving is properly utilized.

In this experiment, the main value of interest is the Markov case of the double cart-pole balancing problem, where the speed of the car is used as an input to the controller. That is, the neural controller has a total of six inputs, which are the cart position x, the cart velocity $\dot{x}$ , the rotation angle θ₁ of the long pole, the rotation angle θ₂ of the short pole, the angular velocity of the long pole, and the angular velocity of the short pole (all inputs are normalized before being fed into the neural network). For other parameter settings and environment settings refer to [42] and [43]. The length of the long pole is fixed at 1.0 m, while the length of the short pole is set to 0.6 m, 0.65 m and 0.7 m, respectively. The experimental results in Table 13 show the efficiency of the proposed MFEA-VOM algorithm during the actual processing.

Table 13
Results of double cart-pole balancing problem

Task T₁, T₂ T₁, T₃ T₂, T₃

l_s(m) 0.60,0.65 0.60,0.70 0.65,0.70

MFEA-VOM 59%,62% 78%,14% 58%,30%

MFDE 55%,45% 40%,10% 15%,0%

MTO-AE 35%,25% 45%,5% 10%,5%

SREMTO 36%,36% 10%,3% 13%,7%

MFEA-II 30%,27% 30%,7% 27%,27%

DE 35%,5% 35%,0% 5%,0%

Task	T₁, T₂	T₁, T₃	T₂, T₃
l_s(m)	0.60,0.65	0.60,0.70	0.65,0.70
MFEA-VOM	59%,62%	78%,14%	58%,30%
MFDE	55%,45%	40%,10%	15%,0%
MTO-AE	35%,25%	45%,5%	10%,5%
SREMTO	36%,36%	10%,3%	13%,7%
MFEA-II	30%,27%	30%,7%	27%,27%
DE	35%,5%	35%,0%	5%,0%

The experimental results show that migration between tasks can help the MFO algorithms, i.e., MFAE-VOM, MFDE, MTO-AE, SREMTO, MFEA-II, and DE, to find available controllers for the balancing task with a higher success rate, while for the common single-task optimizer DE has only 35% of trial successes in the relatively simple task T1, only 5% in task T2 The success rate of MFEA-VOM is twice as high as that of MFEA-II. Moreover, the proposed MFEA-VOM outperforms various MFOs, such as MFDE, MTO-AE, and SREMTO, and MFEA-VOM outperforms all the other MFO algorithms among the six methods compared.

4.4 Time complexity and overall results analysis

The overall time complexity of the multitask evolutionary algorithm is o(n3). By introducing the components, MFEA-VOM increases the computational overhead problem to some extent, but does not have a significant effect on the time complexity. Through the experiments of single objective, the comparison experiments between MFEA-VOM and MFEA-II can be seen that MFEA-VOM not only exhibits performance results approximating those of MFEA-II for high similarity problems, but also outperforms MFEA-II for low similarity (CI+LS) problems. through the experiments of multi-objective benchmarking tasks, MFEA-VOM has a better performance than MFEA-II in overall performance is better than other EMT algorithms, namely TMOMFEA, MOMFEA, and NSGA-II. especially on high similarity problems, MFEA-VOM effectively promotes knowledge transfer and reduces the occurrence of negative knowledge transfer by incorporating three strategies.

5 Discussion

The unique feature of the multitasking evolutionary algorithm is that it can solve multiple tasks at once. In the process of multi-task algorithm evolution, the individuals in the population are able to solve the current task well, and more importantly, the genes in the current task can effectively work on another task, which is the meaning of the existence of EMT algorithm. Negative knowledge transfer often accompanies communication between unrelated tasks. Negative knowledge transfer can slow down population convergence and waste resources. Negative knowledge transfer is a difficult problem in multi-task evolutionary algorithms.

MFEA-VOM is composed of an implicit conversion strategy, an opposition matrix strategy, and a regulatory gene fusion strategy. The opposition matrix strategy is used to attenuate the effect of negative knowledge transfer by constructing a highly similar gene expression space in individuals with different tasks. From Fig. 5(c) and 5(f), it can be seen that the opposition matrix strategy is applicable to low similarity problems, e.g., PI+LS T1 and PI+LS T2. Comparing MFEA-VOM1 (opposition matrix strategy) and MFEA-VOM2 (regulatory gene fusion strategy), it can be seen that the convergence rate of MFEA-VOM1 is significantly higher than that of MFEA-VOM2, confirming that the effectiveness of the opposition matrix strategy on the low-similarity problem. Although the regulatory gene fusion strategy cannot construct a genetic expression space to suppress the occurrence of negative knowledge transfer, it can improve the similarity between different populations and enhance the search for a uniform search space. As shown in Fig. 5(b) and 5(e), MFEA-VOM2 is slightly more effective than MFEA-VOM1. Overall, MFEA-VOM is the most effective in the benchmarking task, demonstrating that the strategies proposed in this paper can complement each other, make up for each other’s shortcomings, and lead to a significant improvement in algorithm performance.

The experimental results demonstrate the effectiveness of MFEA-VOM’s on the MFO benchmarking task, but the algorithm still has some shortcomings. For PI+MS in Table 6, MFEA-VOM is defeated by the MFDE algorithm. This is due to the fact that MFEA-VOM tends to lead to a gradual homogenization of populations, resulting in a loss of diversity among populations and not well adapted to the task.

MFEA-VOM has a wide range of promising applications. These include weighting strategies to guide multi-objective evolutionary algorithms for multi-UAV mission planning, applied to UAV path planning. Traditional evolutionary algorithms do not solve such problems well. However, a better solution can be obtained by EMT algorithm. Another typical application of EMT evolutionary algorithms is supply chain networks. For example, how to jointly optimize route planning and base station siting.

6 Conclusions and future work

In this paper, three strategies are proposed for the evolutionary multitasking algorithm MFEA-VOM, namely, the implicit conversion strategy, the opposition matrix strategy and the regulatory gene fusion strategy. MFEA-VOM achieves effective communication and exchange between tasks and improves the efficiency of knowledge transfer. The implicit conversion strategy is used to regulate the evolutionary process of populations. The use of the regulatory gene fusion strategy enhances the population search ability and strengthens the search for a uniform search space. Comprehensive experiments were conducted on the single-objective MFO problem to evaluate the effectiveness of MFEA-VOM compared with other EMT algorithms.

MFEA-VOM shows good performance results on many benchmarking problems, but there are still some unresolved issues. How to extend the task to higher latitudes and integrate more complex multi-factors. In some multitasking environments, knowledge transfer can become confusing due to the nature between tasks, so but class cost convergence becomes slow. It remains a challenge to maintain the stable performance of the EMT algorithm in a multitasking environment.

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An interactive opposition matrix and regulatory gene fusion strategy for evolutionary multitasking

Abstract

Keywords

1 Introduction

2.1 Multifactorial optimization

2.3 MFEA

3.1 The General framework of MFEA-VOM

4.1 Experiments on single-objective multi-factorial optimization problems

4.1.1 Purpose of the experiment

4.1.2 Performance measures

Table 2 Parameter setting in single target MFO Parameter Value Population size 100 Crossover probability 1 Distribution index of crossover 2 Maximum number of evaluations 200,000 Distribution index of mutation 5 Mutation probability 1/max d im

4.1.6 Comparison to existing techniques

4.2.1 Performance indicators

Table 9 Multi-objective experiments often have parameter settings Parameter Value Population size 100 Crossover probability 0.9 Distribution index of crossover 20 Maximum number of evaluations 200,000 Distribution index of mutation 20 Mutation probability 1/max d im

4.2.4 Compared algorithms

4.2.5 Experimental results

Table 13 Results of double cart-pole balancing problem Task T1, T2 T1, T3 T2, T3 l s (m) 0.60,0.65 0.60,0.70 0.65,0.70 MFEA-VOM 59%,62% 78%,14% 58%,30% MFDE 55%,45% 40%,10% 15%,0% MTO-AE 35%,25% 45%,5% 10%,5% SREMTO 36%,36% 10%,3% 13%,7% MFEA-II 30%,27% 30%,7% 27%,27% DE 35%,5% 35%,0% 5%,0%

5 Discussion

6 Conclusions and future work

References

Table 2
Parameter setting in single target MFO

Parameter Value

Population size 100

Crossover probability 1

Distribution index of crossover 2

Maximum number of evaluations 200,000

Distribution index of mutation 5

Mutation probability 1/max_dim

Table 9
Multi-objective experiments often have parameter settings

Parameter Value

Population size 100

Crossover probability 0.9

Distribution index of crossover 20

Maximum number of evaluations 200,000

Distribution index of mutation 20

Mutation probability 1/max_dim