CN117521788A

CN117521788A - Multi-mode and multi-target evolution method with double reference vector guidance

Info

Publication number: CN117521788A
Application number: CN202311539740.8A
Authority: CN
Inventors: 孙超利; 孙有为
Original assignee: Taiyuan University of Science and Technology
Current assignee: Taiyuan University of Science and Technology
Priority date: 2023-11-18
Filing date: 2023-11-18
Publication date: 2024-02-06

Abstract

The invention relates to the technical field of multi-mode multi-target optimization in evolutionary computation, in particular to a multi-mode multi-target evolutionary method with double reference vector guidance, which comprises the steps of dividing a decision space and a target space by using reference vectors, combining a crowded distance value in the decision space and an APD value in the target space, and selecting non-dominant solutions with higher diversity to put into an archive so as to improve the diversity of the decision space and the target space; meanwhile, the next generation population is generated based on the special crowding distance index, so that the diversity between the decision space and the target space can be balanced; the method of combining local convergence quality and special crowding distance and reserving potential boundary solutions is further adopted in environment selection, so that the optimization performance of the method is further improved, and a better result is obtained when the multi-mode multi-objective optimization problem is solved. Furthermore, the invention also provides a computer readable storage medium and a computer device.

Description

Multi-mode and multi-target evolution method with double reference vector guidance

Technical Field

The invention relates to the technical field of multi-mode multi-target optimization in evolutionary computation, in particular to a multi-mode multi-target evolutionary method guided by double reference vectors.

Background

Multi-objective optimization problems are often used in many practical applications, and these problems are typically composed of multiple conflicting optimization objectives, for example, minimization problems, which can be summarized as:

Min F(x)＝(f ₁ (x)，f ₂ (x)，...，f _m (x)) ^T

wherein x= (x) ₁ ,x ₂ ,…,x _D ) Representing decision variables, F (x) being m-dimensional target variables, the final objective of solving the multi-objective optimization problem is to obtain a set of uniformly distributed pareto solution sets, i.e. non-dominant solution sets, that converge as much as possible on the real pareto front.

However, multi-objective optimization also encounters a multi-modal situation, that is, the mapping relationship between the decision space and the target space is not necessarily one-to-one, and it may occur that, in some cases, the pareto optimal solution set PS is not unique, and such problems are called multi-modal multi-objective problem MMOPs, such as an address optimization problem, a flow shop scheduling problem, a bridge optimization problem, a food distribution network problem, a rocket engine task design problem, an aerospace task design problem, a brain function imaging problem, a feature selection problem, and the like, which all have typical multi-modal features.

For the multi-mode multi-objective optimization problem, it is very difficult to find as many pareto optimal solutions as possible, mainly because the conventional multi-objective optimization problem only considers knowing the distribution situation in the objective space, i.e. how to make the objective function value optimal, but ignores the decision space. In practice, the decision space has a great influence on the effect of the optimization algorithm, so in practical applications we need to pay attention to the change of the decision space in order to better select the appropriate optimization algorithm.

Disclosure of Invention

Aiming at the defects of the prior art, the invention provides a multi-mode multi-target evolution method with double reference vector guidance, which is used for solving the problem that only the distribution condition of solutions in a target space is considered and decision space is ignored in the prior art, further improving the diversity of the solutions in the decision space and the target space and accelerating the identification of the pareto optimal solution set.

Based on an RVEA algorithm, the invention divides a decision space and a target space by using a reference vector, combines a crowded distance value in the decision space and an APD value in the target space, selects non-dominant solutions with higher diversity, and puts the non-dominant solutions into an archive, thereby improving the diversity of the decision space and the target space; meanwhile, the next generation population is generated based on the special crowding distance index, so that the diversity between the decision space and the target space can be balanced, and more representative solutions can be selected; in addition, the method of combining local convergence quality and special crowding distance, retaining potential boundary solution and the like is adopted in environment selection, so that the optimization performance of the method is further improved, and a better result is obtained when the multi-mode multi-objective optimization problem is solved.

The invention provides a multi-mode multi-target evolution method guided by double reference vectors, which comprises the following specific steps:

s1, parameter setting: setting the population scale as N, the maximum iteration number as Maxgen, the current iteration number as t, the decision space dimension as D and the target space dimension as M;

s2, forming an initial population Pop through random initialization; in addition, creating an archive Arc with empty content;

s3, uniformly generating N reference vectors with the same scale as the initial population in a decision space and a target space by taking a lower boundary as an initial point;

s4, performing non-dominant ranking on the current population to obtain a non-dominant solution set; meanwhile, selecting non-dominant solutions with higher diversity from the obtained non-dominant solution sets, and storing the non-dominant solutions into an archive Arc;

s5, respectively calculating a special crowding distance SCD value of each non-dominant solution in the current population and the archive Arc, and selecting two solutions to carry out cross mutation by using a tournament selection method based on the calculated SCD value to generate a child population Off;

s6, selecting and forming a new generation population through the environment according to the generated offspring population Off;

s7, updating the archive Arc;

s8, judging whether a termination condition is met, if not, repeating the steps S4-S8 to continue iteration on the new generation population formed; if so, the iteration is terminated, and all non-dominant solutions in the archive Arc are output.

Preferably, in the step S4, the selection of the non-dominant solution with higher diversity from the obtained non-dominant solution set to store in the archive Arc is based on the included angle between each non-dominant solution and each reference vector in the decision space and the target space, and the included angle is associated with the nearest reference vector in the corresponding space, so as to select the non-dominant solution with higher diversity to put in the archive Arc, which specifically includes the steps of:

s41, for each reference vector in the decision space, if only one non-dominant solution is associated with the reference vector, saving the non-dominant solution into an archive Arc; if there are multiple non-dominant solutions associated with it, then the congestion distance for each non-dominant solution is calculated and the non-dominant solution with the greatest congestion distance is saved to the archive Arc, with the congestion distance calculation formula as follows:

wherein x is _i And x _j It has been normalized that, ||x _j -x _i I represents individual x _i And x _j Euclidean distance between (CrowDis) _i Representing the crowded distance of individual i, N representing the population size;

s42, if only one non-dominant solution is associated with each reference vector in the target space, saving the non-dominant solution into an archive Arc; if there are multiple non-dominant solutions associated with them, then the angle penalty distance APD value for each non-dominant solution is calculated and the non-dominant solution with the smallest APD value is saved to the archive Arc, where the APD calculation formula is as follows:

d _t,i,j ＝(1+P(θ _t,i,j ))·||f′ _t，i ||

in θ _t,i,j Represents the angle between the ith individual at the t-th generation and the jth reference vector, and f%' _t,i The I is the transformed target vector f' _t,i Distance to ideal point, P (θ _t,i,j ) Is theta _t,i,j Is a penalty function of (1).

Preferably, before calculating the congestion distance of the individual, the step S41 needs to perform normalization processing, where the normalization processing formula is as follows:

wherein D is the number of decision variables, x' _i,d Represents the normalized value of the d-th variable,to normalize the solution x' _i Minimum value of the d-th variable of (a),/->To normalize the solution x' _i Maximum value of the d-th variable in (c).

Preferably, in the step S5, the cross operation adopts an analog binary cross algorithm SBX, and the mutation operation adopts a polynomial mutation method.

Preferably, the step S6 specifically includes the steps of:

s61, combining the current population and the offspring population Off into a combined population CP;

s62, performing non-dominant ranking on the combined population CP to obtain a non-dominant solution set; meanwhile, selecting non-dominant solutions with higher diversity from the obtained non-dominant solution sets, and storing the non-dominant solutions into an archive Arc;

s63, selecting boundary solutions from the non-dominant solution sets of the combined population CP, and marking the number of the boundary solutions as A;

s64, sorting the rest non-dominant solutions in the combined population CP according to the local convergence quality and the SCD, selecting N-A non-dominant solutions, and combining the N-A non-dominant solutions with A boundary solutions to form A new generation population.

Preferably, in the step S64, the ranking of the remaining non-dominant solutions in the combined population CP is mainly based on the local convergence quality and is assisted by SCD, and the specific steps are as follows:

s641, calculating the local convergence quality of the rest non-dominant solutions in the combined population CP, and sequencing the calculated local convergence quality from low to high in sequence, wherein the calculation formula of the local convergence quality is as follows:

in the formula, when solving for x _i Is solved for x _j B during the supporting process _i,j =1, whereas B _i,j ＝0；n _i Is x _i The number of neighborhood solutions;

s642, when the local convergence quality values of a plurality of non-dominant solutions are consistent, calculating SCD values of the batch of non-dominant solutions, and sequencing the batch of non-dominant solutions from big to small according to the calculated SCD values, wherein the SCD calculation formula is as follows:

in CD _i,x Representing the crowding distance of individual i in the decision space, CD _i,F Representing the crowded distance, CD, of individual i in target space _avg,x Representing average crowding distance, CD, in decision space _avg,F Representing an average crowding distance in the target space;

s643, selecting the first N-A solutions containing the N-A solutions from the non-dominant solution ranking table obtained according to the steps S641-S642, and combining the first N-A solutions with the A boundary solutions to form A new generation population.

Preferably, the step S7 specifically includes:

s71: receiving and storing a new generation population;

s72: performing non-dominant sorting on the archive Arc to obtain a non-dominant solution set, and deleting all dominant solutions;

s73: screening repeated non-dominant solutions from all non-dominant solutions of the archive Arc at present, wherein each group of repeated non-dominant solutions only keeps one non-dominant solution and deletes the rest repeated non-dominant solutions, so that redundancy is prevented;

s74: judging whether the number of all non-dominant solutions which are not repeated in the archive Arc is less than or equal to N, if not, calculating the crowding distance of each non-dominant solution, reserving N solutions with the largest crowding distance, and deleting other solutions.

Preferably, the iteration termination condition is that the execution is terminated when the number of iterations is maximum.

Preferably, before the multi-mode multi-objective optimization problem is solved by using the multi-mode multi-objective evolution method with dual reference vector guidance, different training sets are loaded and preprocessed by using matlab to ensure that the training set can be used for the multi-mode multi-objective optimization problem.

Preferably, the proximity PSP index and the over-volume HV index of the pareto collection are used to verify the effectiveness of MMEA-DSRV in solving the multi-modal, multi-objective problem MMOPs.

The present invention provides a computer readable storage medium having a computer program stored thereon, which when executed by a processor, implements a multi-modal multi-objective evolutionary method with dual reference vector guidance.

The invention also provides a computer device comprising a memory, a processor and a computer program stored on the memory and capable of running on the processor, when the processor executes, the multi-mode multi-target evolution method with double reference vector guidance can be realized.

Compared with the prior art, the invention has the following beneficial effects:

1. according to the invention, the decision space and the target space are divided by using the reference vector, and the non-dominant solution with higher diversity is selected and put into the archive by combining the crowded distance value in the decision space and the APD value in the target space, so that the diversity of the decision space and the target space is improved.

2. The invention generates the next generation population based on the special crowding distance index, can balance the diversity between the decision space and the target space, and is favorable for selecting more representative solutions; in addition, the method of combining local convergence quality and special crowding distance, retaining potential boundary solution and the like is adopted in environment selection, so that the optimization performance of the method is further improved, and a better result is obtained when the multi-mode multi-objective optimization problem is solved.

Drawings

In order to more clearly illustrate the embodiments of the invention or the technical solutions of the prior art, the drawings which are used in the description of the embodiments or the prior art will be briefly described, it being obvious that the drawings in the description below are only some embodiments of the invention, and that other drawings can be obtained according to these drawings without inventive faculty for a person skilled in the art.

FIG. 1 is a flow chart of a multi-mode multi-objective evolutionary method with dual reference vector guidance in an embodiment of the invention.

FIG. 2 is a diagram of a multi-modal multi-objective optimization problem in an embodiment of the present invention;

FIG. 3 is a schematic diagram of selecting non-dominant solutions in a decision space and a target space in an embodiment of the present invention;

FIG. 4 is a schematic diagram of a different algorithm tested on the MMF4 problem in an embodiment of the present invention;

FIG. 5 is a schematic diagram of a different algorithm tested on the MMF5 problem in an embodiment of the present invention;

FIG. 6 is a schematic diagram of a different algorithm tested on the MMF6 problem in an embodiment of the present invention;

FIG. 7 is a schematic diagram of a test on MMF8 problem for different algorithms in an embodiment of the present invention;

FIG. 8 is a schematic diagram of PSP and IGD in MM4, MM5, MM6, and MM8 test questions for different algorithms provided by embodiments of the present invention _x A schematic of the trend as a function of increasing iteration number.

Detailed Description

The following description of the embodiments of the present invention will be made clearly and completely with reference to the accompanying drawings, in which it is apparent that the embodiments described are only some embodiments of the present invention, but not all embodiments. All other embodiments, which can be made by those skilled in the art based on the embodiments of the invention without making any inventive effort, are intended to be within the scope of the invention.

When solving the problem of multi-mode multi-objective optimization, the traditional optimization algorithm mainly focuses on the distribution condition of the objective space, namely, how to optimize the objective function value. However, this optimization method has relatively few considerations in terms of decision space. In practice, the decision space has a great influence on the effect of the optimization algorithm, so in practical applications we need to pay attention to the change of the decision space in order to better select the appropriate optimization algorithm.

The reference vector directs a multi-objective optimized evolutionary algorithm, the RVEA algorithm, to decompose the original multi-objective optimization problem into a plurality of single-objective sub-problems by the reference vector and clarify user preferences based on the reference vector in order to generate uniformly distributed pareto optimal solutions in a preference region of the objective space. However, the algorithm is equally focused on the distribution of the target space, without taking into account the decision space.

The multi-mode multi-target evolution method with double reference vector guidance is improved on the basis of the prior RVEA algorithm aiming at the problem that the traditional optimization algorithm does not consider decision space, and the improved algorithm is used for optimizing the multi-mode multi-target problem. The key of the method for solving the multi-mode and multi-target problems is that the algorithm maintains the diversity of the decision space and the target space in the evolution process, firstly, the decision space and the target space are divided by a reference vector, and the congestion distance value and the APD value are combined for selection, so that the diversity of the decision space and the target space is improved; secondly, generating the next generation population based on the special crowding distance index, so that the diversity between the decision space and the target space can be balanced, and the selection of a more representative solution is facilitated; finally, the method of combining local convergence quality and special crowding distance and reserving potential boundary solutions is adopted in environment selection, so that the optimization performance of the algorithm is further improved, and better performance is obtained when the multi-mode multi-objective optimization problem is solved.

Referring to fig. 1-8, the present invention provides a multi-mode multi-objective evolutionary algorithm with dual reference vector guidance, which is called as MMEA-DSRV for short, comprising the following specific steps:

s1, parameter setting: setting the population scale as N, the maximum iteration number as Maxgen, the current iteration number as t, the decision space dimension as D and the target space dimension as M.

In the embodiment of the application, the English of the Multi-mode Multi-target evolutionary algorithm with double reference vector guidance is called A Multi-mode Multi-objective Evolutionary Algorithm with Dual Sets of Reference Vectors, so that the English abbreviation of the algorithm is called MMEA-DSRV for short.

S2, forming an initial population Pop through random initialization; in addition, an archive Arc with empty content is created.

It should be noted that, the archive Arc generated in step S2 in the present application is an empty archive, and no solution or individual is stored in the archive, which is formed by continuously adding and updating in steps S4-S8.

S3, uniformly generating N reference vectors with the same scale as the initial population in a decision space and a target space by taking a lower boundary as an initial point.

In the embodiment of the present application, a set of reference vectors vd= { V is uniformly generated in a decision space with the lower boundary as an initial point _D,1 ，V _D,2 ，······，V _D,N Simultaneously, a group of reference vectors VM= { V is uniformly generated in a target space by taking a lower boundary as an initial point _M,1 ，V _M,2 ，······，V _M,N }。

FIG. 2 in the embodiment of the present application shows a schematic diagram of a multi-modal multi-objective optimization problem in the present method, where PS represents a Patero solution set, which is a non-dominant solution set in a decision space; PF represents the Patero front and is the mapping vector of PS in target space.

S4, performing non-dominant ranking on the current population to obtain a non-dominant solution set; and simultaneously, selecting non-dominant solutions with higher diversity from the obtained non-dominant solution sets, and storing the non-dominant solutions into an archive Arc.

Referring to fig. 3, in the embodiment of the present application, in step S4, the selection of the non-dominant solution with higher diversity from the obtained non-dominant solution set for storing in the archive Arc is based on the included angle between each non-dominant solution and each reference vector in the decision space and the target space, and the included angle is associated with the nearest reference vector in the corresponding space, so as to select the non-dominant solution with higher diversity for storing in the archive Arc, which specifically includes the following steps:

wherein x is _i And x _j It has been normalized that, ||x _j -x _i I represents individual x _i And x _j Euclidean distance between (CrowDis) _i The crowding distance of the individual i is represented, and N represents the population size or population number.

In this embodiment, before calculating the congestion distance of the individual, the step S41 needs to perform normalization processing, where the normalization processing formula is as follows:

d _t,i,j ＝(1+P(θ _t,i,j ))·||f′ _t，i ||

Wherein P (θ) _t,i,j ) The calculation formula is as follows:

where M is the number of targets,representing the reference vector V _t,j Minimum included angle with other reference vectors in the currently generated target space, α is a user-defined parameter for controlling the rate of change P (θ _t,i,j )，t _max The maximum number of iterations is represented, and t represents the current number of iterations.

It should be noted that, in the embodiment of the present application, the steps S41 and S42 are not sequentially separated, and may be performed simultaneously, the step S41 may be performed first, the step S42 may be performed first, or the step S42 may be performed first, and then the step S41 may be performed first.

FIG. 3 in the present embodiment gives a simple example of how non-dominant solutions are selected in the decision space and target space to update the archive Arc, where N is the population size, circles represent non-dominant solutions in the current population, diamonds represent dominant solutions in the current population, V _D,k 、V _D,k+1 、V _D,k+2 For the decision spatial reference vector, θ, enumerated herein ₁ For non-dominant solution x in decision space ₂ With the nearest reference vector V _D,k+1 Included angle theta of ₂ For non-dominant solution x in decision space ₃ With the nearest reference vector V _D,k+1 Is included in the plane of the first part; v (V) _M,k 、V _M,k+1 、V _M,k+2 For the target spatial reference vector listed in this application, θ ₃ For non-dominant solution x in target space ₁ With the nearest reference vector V _M,k+1 Included angle theta of ₄ For non-dominant solution x in target space ₄ With the nearest reference vector V _M,k+1 Is included in the bearing.

S5, calculating a special crowding distance SCD value of each non-dominant solution in the current population and the archive Arc respectively, and selecting two solutions to carry out cross mutation by using a tournament selection method based on the calculated SCD value to generate a offspring population Off.

In the embodiment of the application, the archive Arc is used as a basic optimizer for assisting the method, and the data generated by the current population are continuously selected and stored in iteration, so that the convergence speed is increased. In addition, the next generation population is generated by carrying out cross mutation according to the archive Arc and the current population, so that solutions with better diversity can be reserved, and the most optimal pareto solutions can be found as much as possible.

It should be noted that the cross mutation method in step S5 is common knowledge in the art, and thus will not be described in detail in the embodiments of the present application.

In the embodiment of the application, the offspring population is generated based on the special crowding distance index, so that the diversity of the decision space and the target space is effectively balanced, and the selection of more representative solutions is facilitated.

S6, forming a new generation population through environmental selection according to the generated offspring population Off.

In this embodiment of the present application, the specific step of step S6 is:

s61, combining the current population and the offspring population Off into a combined population CP.

It should be noted that, in the embodiment of the present application, during the first iteration, the current population mentioned in the steps S4-S6 is the initial population Pop, and after the subsequent iteration update, the current population mentioned in the steps S4-S6 is the new generation population formed after the last iteration.

S62, performing non-dominant ranking on the combined population CP to obtain a non-dominant solution set; and simultaneously, selecting non-dominant solutions with higher diversity from the obtained non-dominant solution sets, and storing the non-dominant solutions into an archive Arc.

In the embodiment of the present application, in step S62, a method for selecting a non-dominant solution with a higher diversity from the non-dominant solution sets of the combined population CP to be put into the archive Arc is detailed in step S4.

S63, selecting boundary solutions from the non-dominant solution sets of the combined population CP, and recording the number of the boundary solutions as A.

Since the PF obtained is incomplete and non-uniform at the early stages of evolution, boundary solutions are chosen in the embodiments of the present application to promote a more uniformly distributed PF, i.e., a pareto front, during evolution.

It should be noted that, in the embodiment of the present application, the remaining non-dominant solutions in the combined population CP in step S64 refer to non-dominant solutions other than the boundary solution in step S63.

In the embodiment of the application, the diversity of non-dominant solutions is ensured, meanwhile, the local convergence quality and the special crowding distance are comprehensively considered, and better solutions are found and put into new populations when new solutions are selected each time, so that the solutions in the new-generation populations and the archive Arc are effectively ensured to have better convergence.

in the formula, when solving for x _i Is solved for x _j B during the supporting process _i,j =1, whereas B _i,j ＝0；n _i Is x _i Is a number of neighborhood solutions.

Preferably, when calculating the number of the neighborhood solutions, firstly, calculating the average distance of the decision space according to the neighborhood definition formula of each solution, then, calculating the Euclidean distance between the solution and other solutions, and finally, selecting the solution with the Euclidean distance smaller than or equal to the average distance of the decision space, wherein the solution is called as the neighborhood solution of the solution, correspondingly, the number of the neighborhood solutions of the solution is the number of the solutions in the part of solution, wherein the neighborhood definition of each solution can be expressed by the following formula:

wherein V represents the average distance of the decision space, the parameter eta is set according to priori knowledge and is used for controlling the average radius of the neighborhood, in the embodiment of the application, the parameter eta is set to be 0.2, D represents the number of decision variables,is the minimum of the d decision variable, < ->Is the maximum value of the d decision variable.

in CD _i,x Representing the crowding distance of individual i in the decision space, CD _i,F Representing the crowded distance, CD, of individual i in target space _avg,x Representing average crowding distance, CD, in decision space _avg,F Representing the average crowding distance in the target space.

S7, updating the archive Arc, wherein the concrete steps are as follows:

s71: receiving and storing a new generation population;

It should be noted that in the embodiment of the present application, the number of all solutions in the updated archive Arc is consistent with the population size, that is, the number of all solutions in the updated archive Arc is less than or equal to N.

In the embodiment of the present application, the iteration termination condition is that the execution is terminated when the maximum number of iterations is reached.

In the embodiment of the application, before the multi-mode multi-target optimization problem is solved by using the multi-mode multi-target evolution method with dual reference vector guidance, different training sets are loaded and preprocessed by using matlab so as to ensure that the training sets can be used for the multi-mode multi-target optimization problem.

The effectiveness of MMEA-DSRV in solving the multi-mode multi-target problem MMOPs is verified through 22 test functions of CEC2019, and compared with experimental results of 5 most advanced multi-mode multi-target optimization algorithms, wherein the experimental results comprise multi-mode multi-target optimization problem HREA with local pareto fronts based on hierarchical ordering, particle swarm optimization algorithm MO_ring_PSO_SCD based on a Ring topology structure and a special crowding distance index for multi-mode multi-target optimization MMEAWI based on a weighted index, improved non-dominant ordering genetic algorithm DN-NSGAII and multi-target particle swarm optimizer based on self-organizing species differentiation in solving the multi-mode multi-target problem SSMOPSO.

The 22 test functions in CEC2019 as used herein include 19 minimum maximum fairness series test questions, MMF, and another 3 test functions, including SYM-PART 1, SYM-PART 2 and Omni-test functions. Wherein MMFs 1-8 are conventional multi-modal multi-objective test questions, MMFs 4, 5, 6 and 8 are test questions with 4 (at most) pareto subsets in the MMF family test question set (MMFs 1-8). It should be noted that MMF4, MMF5 and MMF8 test questions do not have overlapping pareto subsets, while MMF6 has overlapping pareto subsets. MMF9-mmf15_a is a test problem with different reference points, where the dimensions of the decision space and the target space of MMF13 are 2, 3, MMF14, MMF15, mmf14_a and mmf15_a are 3, respectively, and the dimensions of the decision space and the target space of the remaining MMF series are 2, respectively. The SYM-Part 1 and SYM-Part 2 test questions have 9 pareto subsets, while the Omni-test function has 27 pareto subsets.

In the experiments of the present application, the population size was set to min {100×d,300}, where D is the number of decision variables, the maximum number of function evaluations MaxFEs was set to 5000×d, and the maximum number of iterations Maxgen was set to MaxFEs/N. Specifically, for HREA, we set e=0.3, p=0.5 in the experiments of this application. For MO Ring PSO SCD and SSMOPSO, c1=c2=2.05 and w= 0.7298 are set in the experiments of the present application. Offspring of all algorithms were generated using simulated binary crossover and Polynomial Mutation (PM) operators, except MO Ring PSO SCD and SSMOPSO. Notably, the experimental parameter settings for all comparison algorithms are consistent with those provided in the respective original papers. Furthermore, each algorithm was run independently 20 times.

Fig. 4-7 are schematic diagrams of different algorithms tested on MMF4, MMF5, MMF6 and MMF8 problems, respectively, and as can be seen from fig. 4-7, for the multi-modal multi-objective problem, the solution of the real pareto optimal set can be found more by the provided MMEA-DSRV than by other five algorithms. Furthermore, the distribution of the pareto optimal set by the MMEA-DSRV in the decision space and the target space appears to be more uniform and dense than other algorithms.

In experiments, the present application uses the proximity PSP index of the pareto set and the hypervolume HV index to evaluate the quality of the obtained solution, the PSP index is used to evaluate the diversity of the solution in the target space and the decision space, and the HV index evaluates its performance from convergence and the diffusion of the pareto front set in the target space. Wherein, a kind of IGD is introduced into the PSP index _x For quantifying the quality of the solution obtained in the decision space, measuring the diversity and convergence of the solution obtained in the decision space.

In the embodiment of the present application, the PSP index may be defined by the following formula:

in the formula, IGD _x Represents inversion iteration distance, is a convergence index, CR represents convergence ratio, S is obtained PS, S ^* For true PS, d (c, S) represents the Euclidean distance between c and the nearest point in S, |S ^* | is denoted at S ^* The number of intermediate solutions is determined by the number of intermediate solutions,represents the maximum value of the ith decision variable in PS,/->Representing the minimum value of the ith decision variable in PS. When (when)When sigma _i =1; conversely, sigma _i ＝0。

The PSP and IGD in the MM4, MM5, MM6 and MM8 test questions for the different algorithms are given in FIG. 8 _x Trend as the number of iterations increases. From FIG. 8 we can see that MMEA-DSRV achieves superior convergence in PSP, while at IGD, compared to the other five algorithms _x Aspects also provide a powerful competitiveness.

In the embodiment of the present application, the HV indicator may be defined by the following formula:

where Vol (-) is the lebegre metric, x represents the decision variable, P represents the obtained pareto solution set, vr= (vr ₁ ，...，vr _m ) Representing a reference point in the target space.

In the experiments of the application, the significance of the MMEA-DSRV and the results of other algorithms is evaluated under the significance level of 0.05 by adopting Wilcoxon rank sum test, and signs of "+", "-" and "=" indicate that the proposed MMEA-DSRV algorithm has obviously better performance, obviously worse performance or no obvious difference compared with the other algorithms, and particularly refer to the table 1 and the table 2, wherein the table 1 is the comparison result of the MMEA-DSRV with PSP indexes of the other 5 algorithms on a CEC2019 test function, and the table 2 is the comparison result of the MMEA-DSRV with HV indexes of the other 5 algorithms on a CEC2019 test function.

TABLE 1 MMEA-DSRV results of comparison with other 5 algorithm PSP indicators on CEC2019 test function

As can be seen from Table 1, MMEA-DSRV achieved better PSP results over 14 of the 22 test questions, significantly better than the other 5 comparison algorithms. Notably, MMEA-DSRV performs better on MMF1-10, MMF13, MMF1_z, MMF1_e, and MMF15_a test problems and worse on MMF11, SYM_PART1, and Omni_test problems than HREA. Compared to MMEAWI, MMEA-DSRV performed better on MMF1, MMF3-12, MMF14-15, MMF1_z- -MMF15_a and SYM_PART2 test questions, and performed worse on no other test questions. Compared to MO_RING_PSO_SCD, MMEA-DSRV performed better on MMF1-9, MMF1_z-MMF14_a and SYM_Part1-SYM_Part2 test problems, and worse on MMF10, MMF12-14 and Omni_test problems. Compared to DNNSGAII, MMEA-DSRV performs better on MMF1- -SYM_PART2 test problems, and worse on Omni_test problems. Compared to SSMOPSO, MMEA-DSRV performed better on MMF1, MMF4-9, MMF14, MMF1_z- -MMF14_a and SYM_Part1- -SYM_Part2 test problems, and performed worse on MMF2-3, MMF11-13 and Omni_test problems. .

To further analyze the behavior of the algorithm of the present application, the present application in turn selects the HV indicator to evaluate its performance from convergence and the expansion of the pareto front set in the target space.

TABLE 2 comparison of MMEA-DSRV with other 5 algorithm HV indicators on CEC2019 test function

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As seen in FIG. 2, MMEA-DSRV performed better on MMF1-3, MMF5-7, MMF9-11 and MMF1_z test problems and performed worse on MMF14-15, MMF1_e- -SYM_PART1 test problems than HREA. Compared to MMEAWI, MMEA-DSRV performs better on MMF1, MMF4-10, MMF14, MMF1_z- -MMF1_e and SYM_ParT2 test problems, and worse on MMF2-3, MMF12, MMF15, MMF14_a- -MMF15_a test problems. Compared with MO_RING_PSO_SCD, MMEA-DSRV performs better on MMF1-6, MMF 8-Omni_test test problems, and performs worse on no other test problems. Compared to DNNSGAII, MMEA-DSRV performs better on MMF 2-MMF 4, MMF7, MMF15, MMF14_a, and sym_part2 test problems, and worse on MMF9, MMF11-12, MMF14, mmf1_e, mmf15_a-sym_part1 test problems. Compared to SSMOPSO, MMEA-DSRV performs better on MMF3-4, MMF7-10, MMF12-15, MMF1_z- -MMF14_a and SYM_Part1- -SYM_Part2 test problems, and performs worse on MMF11 test problems.

In summary, the above experimental results demonstrate the effectiveness of the multi-modal multi-objective evolutionary algorithm with dual reference vector guidance to solve the multi-modal multi-objective problem MMOPs. From the PSP index, the algorithm has better performance in a decision space, and can find as many pareto optimal solutions as possible; from the HV index, the algorithm has better performance in the target space, and the obtained PF can be converged to the real PF.

It will be appreciated by those skilled in the art that embodiments of the present invention may be provided as a method, system, or computer program product. Accordingly, the present invention may take the form of a hardware embodiment, a software embodiment, or an embodiment combining software and hardware aspects. Furthermore, the present invention may take the form of a computer program product on one or more computer-usable storage media having computer-usable program code embodied therein, including but not limited to disk storage and optical storage devices, and the like.

The present invention is described with reference to flowchart illustrations and/or block diagrams of methods, apparatus/systems, and computer program products according to embodiments of the invention. It will be understood that each flow and/or block of the flowchart illustrations and/or block diagrams, and combinations of flows and/or blocks in the flowchart illustrations and/or block diagrams, can be implemented by computer program instructions. These computer program instructions may be provided to a processor of a general purpose computer, special purpose computer, embedded processor, or other programmable data processing apparatus to produce a machine, such that the instructions, which execute via the processor of the computer or other programmable data processing apparatus, create means for implementing the functions specified in the flowchart flow or flows and/or block diagram block or blocks.

These computer program instructions may also be stored in a computer-readable memory that can direct a computer or other programmable data processing apparatus to function in a particular manner, such that the instructions stored in the computer-readable memory produce an article of manufacture including instruction means which implement the function specified in the flowchart flow or flows and/or block diagram block or blocks.

These computer program instructions may also be loaded onto a computer or other programmable data processing apparatus to cause a series of operational steps to be performed on the computer or other programmable apparatus to produce a computer implemented process such that the instructions which execute on the computer or other programmable apparatus provide steps for implementing the functions specified in the flowchart flow or flows and/or block diagram block or blocks.

Those skilled in the art will appreciate that implementing all or part of the above-described methods in accordance with the embodiments may be accomplished by way of a computer program stored on a computer readable storage medium, which when executed may comprise the steps of the embodiments of the methods described above. The storage medium may be a magnetic disk, an optical disk, a read-only memory ROM or a random access memory RAM.

Finally, it should be noted that: the foregoing description is only illustrative of the preferred embodiments of the present invention, and although the present invention has been described in detail with reference to the foregoing embodiments, it will be apparent to those skilled in the art that modifications may be made to the embodiments described, or equivalents may be substituted for elements thereof, and any modifications, equivalents, improvements or changes may be made without departing from the spirit and principles of the present invention.

Claims

1. The multi-mode and multi-target evolution method with double reference vector guidance is characterized by comprising the following specific steps:

s7, updating the archive Arc;

2. The multi-modal multi-objective evolutionary method with dual reference vector guidance according to claim 1, wherein the step S4 of selecting the non-dominant solution with higher diversity from the obtained non-dominant solution set to store in the archive Arc is to associate each non-dominant solution with the nearest reference vector in the corresponding space according to the included angle between each non-dominant solution and each reference vector in the decision space and the target space, thereby selecting the non-dominant solution with higher diversity to put in the archive Arc, comprising the following specific steps:

d _t,i,j ＝(1+P(θ _t,i,j ))·||f′ _t,i ||

3. The multi-mode and multi-objective evolutionary method with dual reference vector guidance of claim 2, wherein the step S41 requires normalization processing before calculating the crowding distance of the individual, and the normalization processing formula is as follows:

4. A multi-modal multi-objective evolutionary method with dual reference vector guidance as claimed in claim 3, wherein the crossover operation in step S5 employs a simulated binary crossover algorithm SBX and the mutation operation employs a polynomial mutation method.

5. The method for multi-modal and multi-objective evolutionary approach with dual reference vector guidance of claim 4, wherein step S6 comprises the specific steps of:

6. The multi-modal multi-objective evolutionary method with dual reference vector guidance of claim 5, wherein the ranking of the remaining non-dominant solutions in the combined population CP in step S64 is based on local convergence quality and SCD is aided by the steps of:

7. The method for multi-modal and multi-objective evolutionary approach with dual reference vector guidance of claim 6, wherein the step S7 comprises the following specific steps:

s71: receiving and storing a new generation population;

8. The multi-modal multi-objective evolutionary method with dual reference vector guidance of claim 7, wherein the iteration termination condition is that the execution is terminated up to a maximum number of iterations.

9. The multi-modal multi-objective evolutionary method with dual reference vector guidance of claim 8, wherein prior to solving the multi-modal multi-objective optimization problem using the multi-modal multi-objective evolutionary method with dual reference vector guidance, different training sets are also loaded and preprocessed with matlab to ensure that the training sets can be used for the multi-modal multi-objective optimization problem.

10. The multi-modal multi-objective evolutionary method with dual reference vector guidance of claim 9, wherein the effectiveness of the mmoa-DSRV in solving the multi-modal multi-objective problem MMOPs is verified using the proximity PSP index and the over-volume HV index of the pareto set.

11. A computer readable storage medium, wherein a computer program is stored on the storage medium, and when the computer program is executed by a processor, a multi-modal multi-objective evolutionary method with dual reference vector guidance as claimed in any one of claims 1-10 is achieved.

12. A computer device comprising a memory, a processor and a computer program stored on the memory and executable on the processor, which when executed by the processor, implements a multi-modal multi-objective evolutionary method with dual reference vector guidance as claimed in any one of claims 1 to 10.