CN109460862B - Method for solving multi-objective optimization problem based on MAB (multi-object-based) hyperheuristic algorithm - Google Patents

Abstract

The invention relates to the field of a heuristic algorithm, in particular to a method for solving a multi-target optimization problem based on a MAB heuristic algorithm, which takes an MAB strategy as a learning strategy, uses four performance evaluation mechanisms to evaluate the performance of each low-level heuristic operator, and better combines the advantages of each low-level heuristic operator through the learning and selecting mechanisms; the algorithm is used for carrying out experiments on a continuous multi-objective optimization problem set WFG, and good experimental results are obtained. The invention aims to solve the problems that the universality is poor during the design of the traditional heuristic algorithm and the effect of a single heuristic algorithm on some problem examples is poor, and provides a method for solving a multi-target optimization problem based on a super-heuristic algorithm of an MAB (multi-object-based optimization).

Description

Method for solving multi-objective optimization problem based on MAB (multi-object-based) hyperheuristic algorithm

Technical Field

The invention relates to the field of a super-heuristic algorithm, in particular to a method for solving a multi-objective optimization problem based on a MAB (multi-object-based) super-heuristic algorithm.

Background

The multi-objective optimization problem is used as a class of NP difficult-to-solve problems, and a large number of algorithms are used for solving the class of problems. Researchers often resort to heuristic algorithms tailored to the problem to obtain an acceptable solution in a reasonable amount of time. The heuristic algorithm has the advantages that domain knowledge can be conveniently integrated according to problems, and the disadvantage that the universality is poor, and a specialized algorithm needs to be designed aiming at the problems. The design of the heuristic algorithm has strong problem correlation, so that the workload of algorithm design is increased. In addition, a single heuristic may work best on some problem instances, but poorly on others, i.e., it may not be guaranteed that a single heuristic will achieve high quality results on all problem instances.

The problem can be solved by taking a Hyper-heuristic algorithm (HH) as a search method for selecting a heuristic operator or constructing the heuristic operator. A search problem and a set of heuristics associated with the problem (referred to as a set of low-level heuristics) are given. The hyper-heuristic algorithm is not used for directly searching the neighborhood space of the problem, but is used as a high-level heuristic strategy to increase the searching level to search the neighborhood space of the low-level heuristic operator. In the searching process, the hyper-heuristic algorithm selects and applies a proper low-level heuristic operator from the low-level heuristic operator set according to different solving states, and finally the selected low-level heuristic operator forms a low-level heuristic operator sequence. The method can combine the advantages of the low-level heuristic operators and avoid the disadvantages of the low-level heuristic operators to a certain extent. The solution of the hyper-heuristic algorithm on the single-target optimization problem has achieved a good effect, however, the application of the hyper-heuristic algorithm on the multi-target optimization problem is less.

A search of prior art documents revealed that Burke et al published in Springer (2005, pp: 129- "Metaheuristics:Progress as Real Problem Solvers"a hyperheuristic algorithm using reinforcement learning and tabu search as selection strategies is proposed for solving the multi-objective spatial allocation and temporal planning problem. An article "A new discrete rule based genetic algorithm for the multi-objective job shop problem" published by Vazzez-Rodriguez and Petrovic in Journal of Hearistics (2010, Vol.16, No.6, pp: 771-793) proposes a hyper-heuristic algorithm based on scheduling rules and genetic algorithms for solving the multi-objective job shop problem. However, the above-proposed hyper-heuristic algorithm only solves the non-continuity multi-objective optimization problem. Mashael Maashi et al, Expert Systems with Applications (2014, Vol.41, pp:4475-4493), and proposes a hyper-heuristic algorithm HH _ CF based on a selection function for solving the continuity multi-objective optimization problem, and achieves better effect. But the algorithm uses the running interval (CPU time) of a low-level heuristic operator as a parameter of a selection mechanism, so that the running effect of the algorithm on different machines is not stable. The article "A Hyper-Heuristic for the Multi-Objective Integration and Test Order Problem" published by Giovani Guiizzo et al in GECCO (2015, Madrid, Spain, pp: 1343) proposes two Hyper-Heuristic algorithms HITO-CF and HITO-MAB for solving the search-based Multi-Objective software engineering method Problem. HITO-CF and HITO-MAB use various crossover operators and mutation operators as low-level heuristic operators, and respectively use CF and MAB as selection strategies. In the selection process, the return value of the low-level heuristic operator is calculated by using a method of accumulation summation according to the dominance relation between the parent individuals and the child individuals. However, this method of return value calculation focuses on convergence of the solution set, and ignores the distributivity of the solution set.

Disclosure of Invention

The invention aims to overcome the problems that the universality is poor when the traditional heuristic algorithm is designed and the effect of a single heuristic algorithm on some problem examples is poor, and provides a method for solving a multi-target optimization problem based on a multi-objective heuristic algorithm of an MAB (multi-object-based) model, wherein the method takes an MAB strategy as a learning strategy, uses four performance evaluation mechanisms to evaluate the performance of each low-level heuristic operator, and better combines the advantages of each low-level heuristic operator through the learning and selecting mechanisms; the algorithm is used for carrying out experiments on a continuous multi-objective optimization problem set WFG, and good experimental results are obtained.

In order to achieve the purpose, the invention adopts the following technical scheme: a method for solving a multi-objective optimization problem based on a hyper-heuristic algorithm of MAB (multi-object-based B) comprises the following steps:

(1) giving the set H ═ H of the lower-level heuristic operator₁,h₂...h_numNum is the number of low-level heuristic operators, h₁,h₂...h_numOptimizing an algorithm or operator for multiple objectives;

(2) constructing a random initial population P of size N₀Randomly generating N solutions in a decision space X;

(3) calculating the performance values of the heuristic operators of each lower layer on the four existing performance evaluation indexes; each low-level heuristic operator h₁,h₂...h_numFor the same initial population P₀Respectively and independently run once, and each run is executed by h_iNum, and calculating each heuristic operator h respectively_iFour performance evaluation indices AE, RNI, SSC and UD; m is the number of times of executing the low-level heuristic operator in one iteration, and M is an integer in a range of (0, ∞);

1) AE: the evaluation algorithm obtains the calculated amount of the approximate solution set, the value range of the calculated amount is (0, infinity), and the smaller the AE value is, the better the performance is;

2) RNI: evaluating the proportion of the non-dominant solutions in the approximate solution set obtained by the algorithm, wherein the value range is (0, 1), the larger the RNI value is, the higher the proportion of the non-dominant solutions in the approximate solution set is, and if the RNI is 1, all individuals in the approximate solution set are non-dominant;

3) SSC: evaluating the size of a target space which can be covered by an approximate solution set obtained by an algorithm, wherein the value range is [0, ∞ ]; higher SSC values indicate greater coverage space like a solution set, which also indicates better performance in convergence and distribution;

4) UD: evaluating the distribution condition of the approximate solution set in the target space, wherein the value range is (0, 1), and the higher the UD value is, the better the distribution of the approximate solution set in the target space is, under the condition that the number of non-dominant solutions is the same;

(4) calculating the return value r of each low-level heuristic operator by using an operator performance evaluation mechanism_iEach low-level heuristic operator evaluation mechanism comprises the following steps:

1) normalizing the performance data acquired by the heuristic operators at each low layer to map each performance evaluation index to the same value range [0, 1]]Inner, lower heuristic operator h_iThe normalized values on the four performance evaluation indices were recorded as AE_{i_nor}、RNI_{i_nor}、SSC_{i_nor}And UD_{i_nor}；

2) According to the performance evaluation index value obtained after normalization, a formula is used for calculating the return value r of each low-level heuristic operator_i：

r_i＝(1-AE_{i_nor})+2RNI_{i_nor}+SSC_{i_nor}+UD _{i_nor} ①

In formula (i): because of AE_{i_nor}The smaller the value of (A) is, the better, and the larger the other three indexes are, the better, so that the return value r is_iIn the calculation of (a) uses a low-level heuristic operator h_iAE of_{i_nor}Reversal value, i.e. 1-AE_{i_nor}A value of (d); SSC_{i_nor}Can reflect the distribution and convergence of the population obtained after the algorithm execution, UD_{i_nor}The distributivity of the population obtained after the algorithm is executed can be reflected; SSC in case of uniform population size_{i_nor}And UD_{i_nor}The larger the value of (A) is, the better; SSC pairs to avoid variation in population size_{i_nor}And UD_{i_nor}Influence of the value, therefore at r_iIn the calculation of (A) to SSC_{i_nor}And UD_{i_nor}Respectively matched with a non-dominant solution ratio RNI_{i_nor}；

(5) According to the reported value r_iCalculating the MAB value of each low-level heuristic operator, and the steps are as follows:

1) for a set of num lower-level heuristics H ═ H₁,h₂...h_numFor each h_i(i ∈ { 1.,. num }) sets a performance tuple consisting of three variables

Wherein r is_iIs a reported value,

Is the average return value, k_iIs h_iThe selected times, and all values are initialized to 0 in the algorithm initial stage;

2) update h using a formula-_iPerformance tuple of

3) Heuristic operator h according to low layer_iPerformance tuple V of_iComputing MAB of low-level heuristic operator at current stage_iValue, formula (c) as follows:

wherein C is a control parameter, and controls the heuristic operator h of each lower layer_iAverage performance of

A balance between the number of times an operator executes in the framework operation;

(6) selecting the lower-layer heuristic operator h with the maximum MAB value_topI.e. top ═ argmax { MAB_iNum; if the algorithm with the maximum MAB value is multiple, one algorithm is randomly selected from the multiple algorithms;

(7) use of h_topEvolved population P_tExecute h_topObtaining a new population P' by M generations, and calculating an application low-level heuristic operator h_topObtaining performance values of the population P' on four performance evaluation indexes;

(8) updating population P with improved acceptance strategy_t+1Execution h using SSC determination_topWhether the later population has improvement or not, if not, the population P of the next iteration (t +1 generation)_t+1By the t-th generation, i.e. P_t+1＝P_t(ii) a If there is improvement, the population P of the t +1 th generation_t+1Is replaced by the just acquired population P', i.e. P_t+1＝P′；

(9) Judging whether the iteration times exceed a set maximum value W, if so, finishing the HH-MAB algorithm; if not, entering the next iteration and returning to the step (4); w is the maximum number of iterations of the algorithm, taking an integer in the range of (0, ∞).

The technical scheme of the invention has the following beneficial effects:

1. a novel MAB-based hyper-heuristic algorithm HH-MAB is provided for solving a multi-target optimization problem, and no low-level heuristic algorithm is learned in the aspects of calculated quantity, non-dominated solution-to-occupation ratio, solution set distribution and solution set convergence by using a MAB strategy in the hyper-heuristic algorithm at present.

2. A new return value algorithm is designed, and the performance of a low-level heuristic operator on the calculated amount, the non-dominated solution occupation ratio, the solution set distribution and the solution set convergence is comprehensively considered from multiple aspects by utilizing a method of normalizing and summing four performance evaluation indexes (AE, RNI, SSC and UD).

3. And normalizing each evaluation index value to enable the performance evaluation indexes with four different value intervals to be normalized in the same interval, so that the calculation of the return value has no preference for the four performance evaluation indexes, and a proper bottom layer heuristic operator is better selected at each decision point to improve the performance of the algorithm.

4. Compared with the existing hyper-heuristic methods (HH-RAND and HH-CF) for solving the multi-objective optimization problem, the HH-MAB selects the low-level heuristic operators according to the return values by using the MAB strategy, and the advantages of the low-level heuristic operators can be better combined by selecting the proper low-level heuristic operators at different solving stages, so that the algorithm performance is improved, and a better approximate solution set is obtained.

5. The HH-MAB adopts an improved acceptance strategy to update the population, and the method not only improves the performance of the algorithm, but also improves the stability of the algorithm.

Drawings

FIG. 1 is a flow chart of the HH-MAB algorithm framework of the present invention. FIG. 2 is a graph showing the average usage of the HH-RAND, HH-CF, and HH-MAB for three lower level heuristics (NSGA-II, SPEA2, MOGA) in the example, independently run 30 times on the WFG1-WFG9 test question.

FIG. 3 is a schematic diagram of the box plots on the evaluation index RNI of the HH-MAB and comparison algorithm of the example run independently 30 times on the WFG1-WFG9 test problems.

FIG. 4 is a schematic diagram of the box plot on the evaluation index SSC for 30 independent runs of the HH-MAB and comparison algorithm on WFG1-WFG9 test questions in the example.

FIG. 5 is a schematic diagram of the box map on the evaluation index UD of the HH-MAB and the comparison algorithm in the example, independently run 30 times on the WFG1-WFG9 test questions.

FIG. 6 is a schematic representation of the approximate Pareto front of the example HH-RAND, HH-CF, and HH-MAB, run independently 30 times, 50% on WFG1 test questions.

FIG. 7 is a schematic representation of the approximate Pareto front of the example HH-RAND, HH-CF, and HH-MAB, run independently 30 times, 50% on WFG2 test questions.

FIG. 8 is a graphical representation of the approximate Pareto front of the example HH-RAND, HH-CF, and HH-MAB, run independently 30 times, 50% on WFG3 test questions.

FIG. 9 is a schematic representation of the approximate Pareto front of the example HH-RAND, HH-CF, and HH-MAB, run independently 30 times, 50% on WFG4 test questions.

FIG. 10 is a schematic representation of the approximate Pareto front of the example HH-RAND, HH-CF, and HH-MAB running 30 times independently on WFG5 test questions, 50%.

FIG. 11 is a graphical representation of the approximate Pareto front of the example HH-RAND, HH-CF, and HH-MAB, run independently 30 times, 50% on WFG6 test questions.

FIG. 12 is a schematic representation of the approximate Pareto front of the example HH-RAND, HH-CF, and HH-MAB running 30 times independently on WFG7 test questions, 50%.

FIG. 13 is a schematic representation of the approximate Pareto front of the example HH-RAND, HH-CF, and HH-MAB running 30 times independently on WFG8 test questions, 50%.

FIG. 14 is a schematic representation of the approximate Pareto front of the example HH-RAND, HH-CF, and HH-MAB running 30 times independently on WFG9 test questions, 50%.

Fig. 15 is a schematic diagram of the original value and the normalized value of the performance data obtained by the lower-layer heuristic operator in step four 1).

Detailed Description

The invention will be further elucidated and described with reference to the drawings and specific embodiments.

For the multi-target optimization problem to be solved, firstly, a set H ═ H of the existing low-level heuristic operator aiming at the problem to be solved is given₁,h₂...h_num}. These low-level heuristics have strong problem dependencies. HH-MAB is used as a high-level heuristic strategy, and the space of a low-level heuristic operator is searched without directly solving the problem. The HH-MAB designs and uses an operator evaluation mechanism and a MAB selection mechanism to learn the performance of each low-level heuristic operator and uses an improved acceptance mechanism to decide whether to accept the generated new population.

Without loss of generality, assuming that a multi-objective optimization problem has n decision variables and m objective functions, the multi-objective optimization problem is defined as min F (x) ═ F₁(x),f₂(x),...,f_m(x))^TWherein x ═ x₁,..,x_n) Belongs to a decision vector (solution for short) with X as n dimension, wherein X is a decision space with n dimension and X_iRepresents the ith decision component of x, i ∈ {1, 2.., n }; f (x) ═ f₁(x),f₂(x),...,f_m(x))^TAnd E is a target vector with m dimensions of Y, and Y is a target space with m dimensions of Y. f. of₁(x)，...，f_m(x) M mutually conflicting objective functions; for the multi-objective optimization problem, if there are two solutions x^AAnd x^BWhen f is satisfied for all i ═ 1,2_i(x^A)≤f_i(x^B) And f is present on at least one target_j(x^A)<f_j(x^B) Then call x^ADominating x^B(ii) a If no other solution domination solution x exists, the solution x is called a Pareto optimal solution; a set formed by all Pareto optimal solutions is called a Pareto optimal solution set, and the mapping of the Pareto optimal solutions in a target space is called a Pareto front surface; for the multi-objective optimization problem, the objective of algorithm design is to obtain an approximate solution set that is uniformly distributed over the target space and converges towards the Pareto frontier.

As shown in fig. 1-15, the method of the present embodiment includes the following steps:

step one, a set H ═ H of a low-level heuristic operator is given₁,h₂...h_numNum is the number of the low-level heuristic operators, and in the invention, 3 low-level heuristic operators are selected, namely num is 3; h is₁,h₂,h₃The method is an existing classical multi-objective optimization algorithm;

h ₁:NSGAII(Deb,K.；Pratap,A.；Agarwal,S.；and Meyarivan,T.A fast and elitist multi-objective genetic algorithm:NSGA-II.IEEE Trans.on Evolutionary Computation,2002,6(2):182-197)

h₂:SPEA2(Zitzler,E.；Laumanns,M.；and Thiele,L.SPEA2:Improving the Strength Pareto Evolutionary Algorithm.In:Proceedings of Evolutionary Methods for Design,Optimization and Control with Applications to Industrial Problems,Berlin,Germany:Springer-Verlag,2001.95-100)

h₃:MOGA(Fonseca CM,Fleming PJ.Genetic algorithm for multiobjective optimization:Formulation,discussion and generation.In:Proceedings of the 5th Int’l Conf.on Genetic Algorithms,San Mateo:Morgan Kauffman Publishers,1993.416-423)。

the three algorithms have advantages and disadvantages and are more suitable for verifying the capability of HH-MAB in combining the advantages of low-level heuristic operators, the probability of crossover and mutation in each low-level heuristic operator is the same as that of a classical algorithm NSGA-II and most multi-objective evolutionary algorithms, the probability of crossover and mutation is respectively set to be 0.9 and 1/24, and the crossover and mutation distribution indexes are respectively 10 and 20.

Step two, constructing a random initial population P containing N solutions₀(the starting population is designated P because it is the starting population, i.e., the 0 th generation population₀(ii) a In this experiment, N is 100); in the decision space X, for the problem to be solved, each decision component X is set_iThe value range is set as [ x ]_il,x_iu],i∈{1,2,...,n}，x_ilAnd x_iuRespectively representing the i-th decision component x of the solution x in the problem definition_iThe values of (1) are lower bound and upper bound, N values are randomly generated in the decision space XRandom initial population P of solutions₀＝{x¹,x²,...,x^NThe method comprises the following steps: for each random solution x^j(j ∈ {1, 2...., N }), the ith decision component x of which^j _iIs its value range [ x ]_il,x_iu](i ∈ {1, 2.,. n }).

Calculating initial performance values of the heuristic operators at the lower layers on the four performance evaluation indexes so as to obtain the performance of a solution set through a multi-aspect measurement algorithm; each low-level heuristic operator h₁,h₂,h₃Respectively for the same initial population P₀Independently run once, and execute h every time_iFor M generations, i ∈ {1,2,3} (M is a user-defined parameter, which is the number of times of executing the low-level heuristic operator in one iteration, M is an integer in the range of (0, ∞), M is set to be 250 in this embodiment), and then each heuristic operator h is calculated separately_iFour performance evaluation indexes (AE, RNI, SSC and UD), i ∈ {1,2,3 };

h₁the four performance evaluation indexes of (1) are respectively recorded as AE₁、RNI₁、SSC₁And UD₁；

h₂The four performance evaluation indexes of (1) are respectively recorded as AE₂、RNI₂、SSC₂And UD₂；

h₃The four performance evaluation indexes of (1) are respectively recorded as AE₃、RNI₃、SSC₃And UD₃；

1) Algorithm Effort (AE): the evaluation algorithm obtains the computation of the approximate solution set by dividing the fixed time step set by the user by the number of function evaluations performed within that step, over a range of (0, ∞), with smaller AE values indicating better Performance (Tan, K.C., Lee, T.H., & Khor, E.F. (2002)

2) Ratio of non-doped indicviduals (RNI): evaluating the proportion of the non-dominant solutions in the approximate solution set obtained by the algorithm, wherein the value is the number of the non-dominant solutions in the current population divided by the initial population size N, the value range is (0, 1), the larger the RNI value, the higher the proportion of the non-dominant solutions in the approximate solution set, and the better the Performance (Tan, K.C., Lee, T.H., & Khor, E.F. (2002). evolution algorithms for multi-object optimization: Performance assessment and compliance

3) Size of space converted or so-called S _ metric Hypervolume (SSC): the approximate solution set obtained by the evaluation algorithm is referenced to a reference point ref (ref)₁,ref₂,...,ref_m) The range of the target space covered by the space between the two is [0, ∞); reference point ref ═ ref₁,ref₂,...,ref_m) Assuming a solution in the target space dominated by all solutions in the set of approximate solutions, for minimizing the multi-objective optimization problem, generally ref_iA maximum value represented on the ith objective function; for the two target WFG problem, the common reference point is set to ref ═ 4, 4; higher SSC values indicate a more similar solution set with more coverage, indicating better performance in convergence and distribution (ziegler, e.,&Thiele,L.(1999).Multiobjective evolutionary algorithms:A comparative case study and the strength)

4) a uniformity distribution of a non-doped position (UD): evaluating the distribution of the non-dominant solution set in the target space, wherein the value range is (0, 1); in the case of the same number of non-dominant solutions, the higher the UD value indicates the better distribution of the approximate solution set in the target space (Srinivas, N., & Deb, K. (1994). Multi object optimization using non-dominant analysis in genetic algorithm, 2, 221-248.)

Fourthly, calculating the return value r of each low-level heuristic operator by using an operator performance evaluation mechanism_i(ii) a Each low-level heuristic operator evaluation mechanism comprises the following steps:

1) normalizing the performance data acquired by the heuristic operators of each lower layer; the four performance evaluation indexes AE, RNI, SSC and UD are normalized from different value intervals to the same value range [0, 1%]Within; respectively obtaining the minimum value and the maximum value of the four performance evaluation indexes in the three low-level heuristic operators at the current stage, and respectively recording the minimum value and the maximum value as AE_min、AE_max、RNI_min、RNI_max、SSC_min、SSC_max、UD_minAnd UD_maxWherein AE_min＝argmin{AE₁,AE₂,AE₃}，AE_max＝argmax{AE₁,AE₂,AE₃}，RNI_min＝argmin{RNI₁,RNI₂,RNI₃}，RNI_max＝argmax{RNI₁,RNI₂,RNI₃}，SSC_min＝argmin{SSC₁,SSC₂,SSC₃}，SSC_max＝argmax{SSC₁,SSC₂,SSC₃}，UD_min＝argmin{UD₁,UD₂,UD₃}，UD_max＝argmax{UD₁,UD₂,UD₃}; if h_iThe raw values before normalization of (i ∈ {1,2,3}) on the four performance evaluation indices are denoted as AE_i、RNI_i、SSC_iAnd UD_iThen h is_iNormalized value AE on four Performance evaluation indices_{i_nor}，RNI_{i_nor}，SSC_{i_nor}，UD _{i_nor}The calculation is as follows:

taking fig. 15 as an example, (a) is the original values of the three low-level heuristics on the four performance evaluation indexes before normalization, and (b) is the normalized values of the three low-level heuristics on the four performance evaluation indexes; taking AE as an example, AE can be known from (a)_min＝0.000108，AE_max0.000430; then h is₁Normalization of value AE on AE_{1_nor}＝(0.000259-0.000108)/(0.000430-0.000108)＝0.468944。

2) Calculating a heuristic index value of each low layer by using a formula (i) according to the normalized performance evaluation index value obtained in the step 1)Operator h_iIs given a return value r_i，i∈{1，2，3}：

r_i＝(1-AE_{i_nor})+2RNI_{i_nor}+SSC_{i_nor}+UD _{i_nor} ①

Wherein AE_{i_nor}Representing the execution of a low-level heuristic operator h_iCalculated amount of (SSC)_{i_nor}Representing execution of a low-level heuristic h_iThe distribution and convergence of the later acquired population, UD_{i_nor}Representing execution of a low-level heuristic h_iThe distribution of the population, RNI, is then obtained_{i_nor}Representing low-level heuristic operator h_iThe proportion of non-dominant solutions in the later population; due to RNI_{i_nor}、SSC_{i_nor}And UD_{i_nor}Is that the larger the value the better the performance, and AE_{i_nor}The smaller the value, the better the performance. Herein will be aE_{i_nor}From the minimization problem to the maximization problem, i.e. 1-AE is adopted in the formula (r)_{i_nor}A value of (d); however, in formula (I), UD_{i_nor}Value and SSC_{i_nor}The value is influenced by the number of non-dominant solutions in the population; for example, UD when population individuals are few, even if the distribution of the current population is poor_{i_nor}The values may still be larger, even than UDs for populations that are better distributed but have more non-dominant solutions in the population_{i_nor}The value is high. In the execution process of the algorithm, the return value r is considered to be changed in the size of the population_iIn the calculation process we are SSCs_{i_nor}And UD_{i_nor}Respectively matched with an RNI with equal proportion_{i_nor}。

The formula is that four performance evaluation indexes are mixed, and after normalization, the value ranges of the performance evaluation indexes are the same, so that the heuristic operators h of the lower layers are all the same_iIs given a return value r_iThe four performance evaluation indexes are not preferred, and are comprehensively embodied; according to a formula, normalizing each low-level heuristic operator h_iIs given a return value r_iAs shown in table 1; by h₁For example, r₁＝(1-0.468944)+2×1.000000+1.000000+1.000000＝4.531056。

TABLE 1 reporting values

	h1	h2	h3
				r	4.531056	4.406907	0.000000

Step five, according to the return value r_iCalculating the MAB value of each low-level heuristic operator, and the steps are as follows:

1) for the set of low-level heuristics H ═ H₁，h₂，h₃H, for each low-level heuristic operator h_iSetting a performance tuple consisting of three variables

Wherein r is_iIs h_iThe reported value obtained in step four,

Is from the beginning h_iAverage value of the obtained reported values, k_iIs h_iThe number of times selected so far, i ∈ {1,2,3 }; three variables are initialized to r in the initial stage of the algorithm_i＝0，

k_i＝0；

2) Obtaining the return value r according to the step four_iUpdating the heuristic operator h of the lower layer by using a formula 2_iCorresponding performance tuple

The last two variables in

And k_i，

3) Heuristic operator h according to low layer_iPerformance tuple V of_iCalculating the heuristic operator h of the current stage low layer_iMAB of (A)_iValue (MAB)_i>0) As shown in formula (c):

wherein C is a control parameter (set to 2 in the experiment) for controlling each low-level heuristic operator h_iAverage return value of

And algorithm operation low-layer heuristic operator h_iIs performed a number of times k_iA balance is kept between;

step six, selecting a low-level heuristic operator h with the maximum MAB value_topI.e. top ═ argmax { MAB₁，MAB₂，MAB₃}; if a plurality of low-level heuristic operators simultaneously have the maximum MAB value, namely MAB_i＝MAB_j>MAB_kWhere i, j, k ∈ {1,2,3}, then h is counted from_iAnd h_jRandomly selects one as h_top；

Step seven, using a low-level heuristic operator h_topEvolution of the Current population P_tExecute h_top250 generations altogetherObtaining a new population P'; calculating an application low-level heuristic operator h according to the performance evaluation indexes AE, RNI, SSC and UD introduced in the step three_topPerformance index value AE of obtained population P' on four performance evaluation indexes_top、RNI_top、SSC_topAnd UD_top。

Step eight, adopting an improved acceptance strategy to update the population P_t+1(ii) a Since SSC is the only one of the four performance evaluation indexes we use to evaluate both the population convergence and the population distribution, the execution of h is determined using SSC_topWhether the later population has improved (i.e., whether the SSC value is large); if the SSC evaluation index value is not larger than the last iteration value, the population P of the next iteration (t +1 generation)_t+1The t generation of the population P_tReplacement, i.e. P_t+1＝P_t(ii) a If the value becomes larger, the population P of the t +1 th generation_t+1Is replaced by the just acquired population P', i.e. P_t+1＝P′。

Step nine, judging whether the iteration number exceeds a set iteration maximum value W (W is a user-defined parameter and is the maximum iteration number of the algorithm, and an integer in a range of (0, ∞) is taken, wherein in the embodiment, W is set to be 25); if so, the HH-MAB algorithm ends; if not, entering the next iteration and returning to the step four.

FIGS. 6-14 are schematic representations of the HH-RAND, HH-CF, and HH-MAB approximate Pareto front for 30 independent runs on WFG1-WFG9 test questions, 50%. The closer the approximate Pareto front surface obtained by the three algorithms is to the real Pareto front surface (PF), the better the convergence of the algorithms is. As can be seen from FIGS. 6 to 14, the convergence properties of HH-MAB are more excellent than those of HH-RAND and HH-CF.

The above description is only an embodiment of the present invention. It should be noted that, in the present embodiment, an experiment is performed on two target WFG test sets, and the test set has 9 test cases, covers multiple aspects of Pareto front surface continuity \ discontinuity, Pareto front surface convexity \ concavity \ linearity, multiple modes \ single mode, fraud problem, and the like, and is a test set widely used for verifying the performance of the multi-target optimization algorithm. According to the characteristics of the hyper-heuristic algorithm, if the problem to be solved is changed, only the number and the types of the low-level heuristic operators need to be changed or replaced, and the high-level heuristic strategy does not need to be changed. Therefore, changing the number and the types of the heuristic operators of the lower layer still remains the protection scope of the patent.

Claims (1)

1. A method for solving a multi-objective optimization problem based on a MAB (multi-object-based) hyperheuristic algorithm is characterized by comprising the following steps: the method comprises the following steps:

(1) giving the set H ═ H of the lower-level heuristic operator₁,h₂...h_numNum is the number of low-level heuristic operators, h₁,h₂...h_numOptimizing an algorithm or operator for multiple objectives;

(2) constructing a random initial population P of size N₀Randomly generating N solutions in a decision space X;

(3) calculating the performance values of the heuristic operators of each lower layer on the four existing performance evaluation indexes; each low-level heuristic operator h₁,h₂...h_numFor the same initial population P₀Respectively and independently run once, and each run is executed by h_iNum, and calculating each heuristic operator h respectively_iFour performance evaluation indices AE, RNI, SSC and UD; m is the number of times of executing the low-level heuristic operator in one iteration, and M is an integer in a range of (0, ∞);

1) AE: the evaluation algorithm obtains the calculated amount of the approximate solution set, the value range of the calculated amount is (0, infinity), and the smaller the AE value is, the better the performance is;

2) RNI: evaluating the proportion of the non-dominant solutions in the approximate solution set obtained by the algorithm, wherein the value range is (0, 1), the larger the RNI value is, the higher the proportion of the non-dominant solutions in the approximate solution set is, and if the RNI is 1, all individuals in the approximate solution set are non-dominant;

3) SSC: evaluating the size of a target space which can be covered by an approximate solution set obtained by an algorithm, wherein the value range is [0, ∞ ]; higher SSC values indicate greater coverage space like a solution set, which also indicates better performance in convergence and distribution;

4) UD: evaluating the distribution condition of the approximate solution set in the target space, wherein the value range is (0, 1), and the higher the UD value is, the better the distribution of the approximate solution set in the target space is, under the condition that the number of non-dominant solutions is the same;

(4) calculating the return value r of each low-level heuristic operator by using an operator performance evaluation mechanism_iEach low-level heuristic operator evaluation mechanism comprises the following steps:

1) normalizing the performance data acquired by the heuristic operators at each low layer to map each performance evaluation index to the same value range [0, 1]]Inner, lower heuristic operator h_iThe normalized values on the four performance evaluation indices were recorded as AE_{i_nor}、RNI_{i_nor}、SSC_{i_nor}And UD_{i_nor}；

2) According to the performance evaluation index value obtained after normalization, a formula is used for calculating the return value r of each low-level heuristic operator_i：

r_i＝(1-AE_{i_nor})+2RNI_{i_nor}+SSC_{i_nor}+UD_{i_nor} ①

In formula (i): because of AE_{i_nor}The smaller the value of (A) is, the better, and the larger the other three indexes are, the better, so that the return value r is_iIn the calculation of (a) uses a low-level heuristic operator h_iAE of_{i_nor}Reversal value, i.e. 1-AE_{i_nor}A value of (d); SSC_{i_nor}Can reflect the distribution and convergence of the population obtained after the algorithm execution, UD_{i_nor}The distributivity of the population obtained after the algorithm is executed can be reflected; SSC in case of uniform population size_{i_nor}And UD_{i_nor}The larger the value of (A) is, the better; SSC pairs to avoid variation in population size_{i_nor}And UD_{i_nor}Influence of the value, therefore at r_iIn the calculation of (A) to SSC_{i_nor}And UD_{i_nor}Respectively matched with a non-dominant solution ratio RNI_{i_nor}；

(5) According to the reported value r_iCalculating the MAB value of each low-level heuristic operator, and the steps are as follows:

1) for a set of num lower-level heuristics H ═ H₁,h₂...h_numFor each h_i(i ∈ { 1.,. num }) setSetting a performance tuple consisting of three variables

Wherein r is_iIs a reported value,

Is the average return value, k_iIs h_iThe selected times, and all values are initialized to 0 in the algorithm initial stage;

2) update h using a formula-_iPerformance tuple of

3) Heuristic operator h according to low layer_iPerformance tuple V of_iComputing MAB of low-level heuristic operator at current stage_iValue, formula (c) as follows:

wherein C is a control parameter, and controls the heuristic operator h of each lower layer_iAverage performance of

A balance between the number of times an operator executes in the framework operation;

(6) selecting the lower-layer heuristic operator h with the maximum MAB value_topI.e. top ═ argmax { MAB_iNum; if the algorithm with the maximum MAB value is multiple, one algorithm is randomly selected from the multiple algorithms;

(7) use of h_topEvolved population P_tExecute h_topObtaining a new population P' by M generations, and calculating an application low-level heuristic operator h_topObtaining performance values of the population P' on four performance evaluation indexes;

(8) updating population P with improved acceptance strategy_t+1Execution h using SSC determination_topWhether the later population has improvement or not, if not, the population P of the next iteration (t +1 generation)_t+1By the t-th generation, i.e. P_t+1＝P_t(ii) a If there is improvement, the population P of the t +1 th generation_t+1Is replaced by the just acquired population P', i.e. P_t+1＝P′；

(9) Judging whether the iteration times exceed a set maximum value W, if so, finishing the HH-MAB algorithm; if not, entering the next iteration and returning to the step (4); w is the maximum number of iterations of the algorithm, taking an integer in the range of (0, ∞).

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