CN109460862A - The method that meta-heuristic algorithms based on MAB solve multi-objective optimization question - Google Patents

The method that meta-heuristic algorithms based on MAB solve multi-objective optimization question Download PDF

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CN109460862A
CN109460862A CN201811230929.8A CN201811230929A CN109460862A CN 109460862 A CN109460862 A CN 109460862A CN 201811230929 A CN201811230929 A CN 201811230929A CN 109460862 A CN109460862 A CN 109460862A
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张淑艳
杨太龙
郭
郭一
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Abstract

The present invention relates to meta-heuristic algorithms fields, a kind of more particularly to method that the meta-heuristic algorithms based on MAB solve multi-objective optimization question, this method is using MAB strategy as learning strategy, and the performance of the heuristic operator of each low layer is assessed using four kinds of Performance Evaluation mechanism, the advantages of by the mechanism that learns and select preferably to combine each low layer heuristic operator；The algorithm is tested on continuous multi-objective optimization question collection WFG, and achieves good experimental result.Versatility is poor when it is an object of the invention to overcome the design of traditional heuritic approach and single heuritic approach may in some problem-instances the poor problem of effect, propose a kind of method that the meta-heuristic algorithms based on MAB solve multi-objective optimization question.

Description

The method that meta-heuristic algorithms based on MAB solve multi-objective optimization question
Technical field
The present invention relates to meta-heuristic algorithms fields more particularly to a kind of meta-heuristic algorithms based on MAB to solve more mesh The method for marking optimization problem.
Background technique
Multi-objective optimization question is used as one kind NP double linear problems of difficulty for solving, and big quantity algorithm is for solving such problem.Researcher's warp The heuritic approach for often taking problem to customize, to obtain acceptable solution within reasonable time.Heuritic approach The advantages of domain knowledge is incorporated according to problem with being to can be convenient, disadvantage is that versatility is bad, it is often necessary to be directed to problem Design specialization algorithm.Since the design of heuritic approach has stronger problem correlation, which increases the works of algorithm design It measures.In addition to this, single heuritic approach may obtain more excellent effect in some problem-instances, and in other examples Upper effect is poor, i.e., it cannot be guaranteed that single heuritic approach can obtain high quality results in all problems example.
Meta-heuristic algorithms (Hyper-heuristic, HH) are as a kind of heuristic operator selection or heuristic operator The searching method of building can solve problem above.A given search problem and a heuristic operator relevant to problem Gather (the referred to as heuristic operator set of low layer).The neighborhood space of meta-heuristic algorithms not instead of direct search problem, conduct Search is risen a level by high-rise heuristic strategies, to search for the neighborhood space of the heuristic operator of low layer.In search process In, meta-heuristic algorithms select from the heuristic operator set of low layer according to different solving states and apply suitable low layer Heuristic operator, the heuristic operator of the low layer finally selected constitute the heuristic sequence of operators of low layer.This method can incite somebody to action The advantages of each low layer heuristic operator, combines, and the shortcomings that avoid the heuristic operator of low layer to a certain extent.Meta-heuristic Algorithm solves on single-object problem and has been achieved for preferable effect, however, meta-heuristic algorithms are excellent in multiple target In change problem using less.
Through the retrieval discovery to prior art document, Burke et al. is on " Springer " (2005, pp:129-158) Publish "Metaheuristics:Progress as Real Problem Solvers" propose using reinforce study and TABU search alternatively tactful meta-heuristic algorithms for solving the distribution of multiple target space and time planning problem. Vazquez-And Petrovic " Journal of Heuristics " (2010, Vol.16, No.6, pp: 771-793) article " the A new dispatching rule based genetic algorithm for the delivered on Multi-objective job shop problem ", proposes the meta-heuristic algorithms based on scheduling rule and genetic algorithm For solving multi-objective Job Shop problem.However, meta-heuristic algorithms set forth above are only in noncontinuity multiple-objection optimization It is solved in problem.Mashael Maashi et al. " Expert Systems with Applications " (2014, Vol.41, pp:4475-4493) on article " the A multi-objective hyper-heuristic based on that delivers Choice function " proposes a kind of meta-heuristic algorithms HH_CF based on selection function for solving the more mesh of continuity Optimization problem is marked, and achieves preferable effect.But the algorithm uses the interval of the heuristic operator of low layer (when CPU Between) parameter of alternatively mechanism, so that operational effect less stable on the different machine of algorithm.Giovani Article " the A Hyper- that Guizzo et al. is delivered on " GECCO " (2015, Madrid, Spain, pp:1343-1350) Heuristic for the Multi-Objective Integration and Test Order Problem " proposes two Kind meta-heuristic algorithms HITO-CF and HITO-MAB is for solving the multiple target software engineering method problem based on search. HITO-CF and HITO-MAB uses CF and MAB using a variety of crossover operators and mutation operator as the heuristic operator of low layer respectively It is alternatively tactful.In the selection process, cumulative summation is used according to the dominance relation between parent individuality and offspring individual The return value of the method calculating heuristic operator of low layer.However this return value calculating method focuses on the convergence of disaggregation, and neglect The distributivity of disaggregation is omited.
Summary of the invention
Versatility is poor when it is an object of the invention to overcome the design of traditional heuritic approach and single heuritic approach May in some problem-instances the poor problem of effect, it is excellent to propose that a kind of meta-heuristic algorithms based on MAB solve multiple target The method of change problem, this method are assessed each low layer using four kinds of Performance Evaluation mechanism and are inspired using MAB strategy as learning strategy The performance of formula operator, the advantages of by the mechanism that learns and select preferably to combine each low layer heuristic operator；The algorithm exists It is tested on continuous multi-objective optimization question collection WFG, and achieves good experimental result.
To achieve the goals above, the present invention adopts the following technical scheme that: a kind of meta-heuristic algorithms based on MAB The method for solving multi-objective optimization question, this method comprises:
(1) the set H={ h of the heuristic operator of low layer is provided1, h2...hnum, wherein num is the heuristic operator of low layer Number, h1, h2...hnumFor multi-objective optimization algorithm or operator；
(2) the random initial population P that construction size is N0, in decision space X, N number of solution is generated at random；
(3) performance number of each heuristic operator of low layer in four existing Performance Evaluation indexs is calculated.Each low layer opens Hairdo operator h1, h2...hnum(f=1,2...num) independently runs once same initial population P0, and each run is held Row hiTotal M generation, and calculate separately each heuristic operator hiFour performance evaluation indexes AE, RNI, SSC and UD；M is primary changes The number of the heuristic operator of low layer is executed in generation, M takes the integer in (0, ∞) range；
1) AE: assessment algorithm obtains the calculation amount of approximate disaggregation, and value range is (0, ∞), the smaller representative of AE value It can be better；
2) RNI: assessment algorithm obtain approximate solution concentrate non-domination solution ratio, value range be (0,1], RNI value It is bigger to indicate that approximate solution concentrates the ratio of non-domination solution higher；If RNI=1, all individuals for showing that approximate solution is concentrated all are Non-dominant；
3) SSC: assessment algorithm obtain approximate disaggregation institute covered object space size, value range for [0, ∞)；SSC value is higher, and the covering space for indicating seemingly nearly disaggregation is bigger, also indicates that performance is better on convergence and distributivity；
4) UD: assessing approximate disaggregation in the distribution situation of object space, value range be (0,1]；In non-dominant skill It is better to measure the higher distributivity for indicating approximate disaggregation on object space of UD value in identical situation；
(4) the return value r of the heuristic operator of each low layer is calculated using operator Performance Evaluation mechanismi, each heuristic calculation of low layer Sub- evaluation mechanism includes the following steps:
1) performance data obtained to the heuristic operator of each low layer is normalized, so that each Performance Evaluation index It is mapped within the same value range [0,1], the heuristic operator h of low layeriAfter the normalizing on four performance evaluation indexes Value is denoted as AE respectivelyi_nor、RNIi_nor、SSCi_norAnd UDi_nor；
2) according to the Performance Evaluation index value obtained after normalization, the heuristic operator of each low layer is 1. calculated using formula Return value ri:
ri=(1-AEi_nor)+2RNIi_nor+SSCi_nor+UDi_nor ①
Formula 1. in: because of AEi_norValue it is the smaller the better, and other three indexs are the bigger the better, therefore in return value riCalculating in used the heuristic operator h of low layeriAEi_norReverse value, i.e. 1-AEi_norValue；SSCi_norIt can reflect out The distributivity and convergence for the population that algorithm obtains after executing, UDi_norIt can reflect out the distribution that population is obtained after algorithm executes Property；Under Population Size unanimous circumstances, SSCi_norAnd UDi_norValue be the bigger the better；In order to avoid the variation of Population Size To SSCi_norAnd UDi_norThe influence of value, therefore in riCalculating in give SSCi_norAnd UDi_norArranged in pairs or groups respectively one it is non-dominant The ratio RNI of solutioni_nor；
(5) according to return value ri, the MAB value of the heuristic operator of each low layer is calculated, steps are as follows:
1) for there is the set H={ h of the heuristic operator of num low layer1, h2...hnum, it is each hi (i∈ { 1...num }) one performance tuple being made of three variables of settingWherein ri be return value,It is average Return value, kiFor hiThe number selected, in the algorithm initial stage, all values are initialized as 0；
2) h is 2. updated using formulaiPerformance tuple
3) according to the heuristic operator h of low layeriPerformance tuple ViCalculate the MAB of the heuristic operator of current generation low layeriValue, Following formula is 3.:
Wherein C is control parameter, controls the heuristic operator h of each low layeriAverage behaviorIt is held with operator in frame operation A kind of balance between capable number；
(6) the heuristic operator h of the maximum low layer of MAB value is selectedtoP, i.e. top=argmax { MABi, i=1,2 ..., num；If the algorithm for possessing maximum MAB value has multiple, therefrom random selection one；
(7) h is usedtopAdvanced group species Pt, execute htopIn total M generation, new population P ' is obtained, and calculated heuristic using low layer Operator htopPerformance number of the obtained population P ' on four performance evaluation indexes；
(8) using improvement reception strategy Population Regeneration Pt+1, judge to execute h using SSCtopWhether population has improvement afterwards, if Without improvement, then the population P of next iteration (t+1 generation)t+1It is replaced by the population in t generation, i.e. Pt+1=Pt；If there is improvement, the The population P in t+1 generationt+1Population P ' the replacement just obtained, i.e. Pt+1=P '；
(9) judge the number of iterations whether be more than setting maximum value W, if it does, HH-MAB algorithm terminates；If not yet It has more than, into next round iteration, returns (4)；W is the maximum number of iterations of algorithm, takes the integer in (0, ∞) range.
What technical solution of the present invention generated has the beneficial effect that:
1, a kind of novel meta-heuristic algorithms HH-MAB solution multi-objective optimization question based on MAB is proposed, at present There has been no meta-heuristic algorithms using MAB strategy in calculation amount, non-domination solution accounting, disaggregation distributivity and disaggregation convergence side Face learns low layer heuritic approach.
2, new return value calculating method is devised, is normalized using four performance evaluation indexes (AE, RNI, SSC and UD) The method of summation, comprehensively consider from many aspects the heuristic operator of low layer calculation amount, non-domination solution accounting, disaggregation distributivity and Performance in disaggregation convergence.
3, each evaluation index value is normalized so that the Performance Evaluation index of different value intervals is normalized to together there are four gathering around In one section so that the calculating of return value to four performance evaluation indexes without preference, thus preferably each decision point select close The heuristic operator of suitable bottom, with the performance of boosting algorithm.
4, relative to the existing meta-heuristic method (HH-RAND and HH- CF) for solving multi-objective optimization question, HH-MAB The heuristic operator of low layer is selected according to return value using MAB strategy, by selecting suitable low layer to inspire in the different solution stages Formula operator can preferably combine the advantages of operator that each low layer is heuristic, improve algorithm performance, obtain preferably close Like disaggregation.
5, HH-MAB not only increases the performance of algorithm, is also promoted using reception strategy Population Regeneration, this method is improved The stability of algorithm.
Detailed description of the invention
Fig. 1 is HH-MAB algorithm frame flow chart of the present invention.Fig. 2 is that HH-RAND, HH-CF and HH-MAB exist in embodiment Independent operating 30 times is to the flat of three heuristic operators of low layer (NSGA-II, SPEA2, MOGA) on WFG1-WFG9 test problem The schematic diagram of equal utilization rate.
Fig. 3 is that HH-MAB is being assessed for independent operating 30 times on WFG1-WFG9 test problem with comparison algorithm in embodiment The schematic diagram of box figure on index RNI.
Fig. 4 is that HH-MAB is being assessed for independent operating 30 times on WFG1-WFG9 test problem with comparison algorithm in embodiment The schematic diagram of box figure on index S SC.
Fig. 5 is that HH-MAB is being assessed for independent operating 30 times on WFG1-WFG9 test problem with comparison algorithm in embodiment The schematic diagram of box figure on index UD.
Fig. 6 is HH-RAND, HH-CF and HH-MAB independent operating 30 times, 50% on WFG1 test problem in embodiment Approximate Pareto leading surface schematic diagram.
Fig. 7 is HH-RAND, HH-CF and HH-MAB independent operating 30 times, 50% on WFG2 test problem in embodiment Approximate Pareto leading surface schematic diagram.
Fig. 8 is HH-RAND, HH-CF and HH-MAB independent operating 30 times, 50% on WFG3 test problem in embodiment Approximate Pareto leading surface schematic diagram.
Fig. 9 is HH-RAND, HH-CF and HH-MAB independent operating 30 times, 50% on WFG4 test problem in embodiment Approximate Pareto leading surface schematic diagram.
Figure 10 is HH-RAND, HH-CF and HH-MAB independent operating 30 times, 50% on WFG5 test problem in embodiment Approximate Pareto leading surface schematic diagram.
Figure 11 is HH-RAND, HH-CF and HH-MAB independent operating 30 times, 50% on WFG6 test problem in embodiment Approximate Pareto leading surface schematic diagram.
Figure 12 is HH-RAND, HH-CF and HH-MAB independent operating 30 times, 50% on WFG7 test problem in embodiment Approximate Pareto leading surface schematic diagram.
Figure 13 is HH-RAND, HH-CF and HH-MAB independent operating 30 times, 50% on WFG8 test problem in embodiment Approximate Pareto leading surface schematic diagram.
Figure 14 is HH-RAND, HH-CF and HH-MAB independent operating 30 times, 50% on WFG9 test problem in embodiment Approximate Pareto leading surface schematic diagram.
Figure 15 be step 4 1) in the heuristic operator of each low layer obtain performance data original value and normalization after numerical value Schematic diagram.
Specific embodiment
With reference to the accompanying drawing and specific embodiment the invention will be further elaborated.
A kind of meta-heuristic algorithms based on MAB solve the method HH-MAB of multi-objective optimization question, for be solved Multi-objective optimization question provides the set H={ h of the existing heuristic operator of low layer for problem to be solved first1, h2...hnum}.The heuristic operator of these low layers has stronger problem correlation.HH-MAB is as high-rise heuristic strategies, no With to problem direct solution, and the space of the heuristic operator of low layer is scanned for.HH-MAB is designed and operator is used to assess machine It makes with MAB selection mechanism and learns the performance of the heuristic operator of each low layer, and decide whether to receive using acceptance mechanism is improved The new population of generation.
Without loss of generality, it is assumed that multi-objective optimization question has a n decision variable, m objective function, then more mesh Mark optimization problem is defined as min F (x)=(f1(x), f2(x) ..., fm(x))T, wherein x=(x1.., xn) ∈ X be n dimension Decision vector (referred to as solves), and X is the decision space of n dimension, xiIndicate i-th of decision component of x, i ∈ { 1,2 ..., n }；F(x) =(f1(x), f2(x) ..., fm(x))T∈ Y is the object vector of m dimension, and Y is the object space of m dimension.f1(x) ..., fm(x) For m conflicting objective functions；For multi-objective optimization question, if there are two solve xAAnd xB, when to all i=1, 2 ..., m, meets fi(xA)≤fi(xB), and there are f at least one targetj(xA) < fj(xB), then claim xADominate xB； Other solutions dominate solution x if it does not exist, then x is referred to as Pareto optimal solution；The optimal collection being deconstructed into of all Pareto is collectively referred to as Pareto optimal solution set is known as Pareto leading surface in the mapping of object space；For multi-objective optimization question, algorithm is set The target of meter is to obtain being uniformly distributed on object space and to the convergent approximate disaggregation of Pareto leading surface.
As represented in figures 1 through 14, the method for the present embodiment includes the following steps:
Step 1: providing the set H={ h of the heuristic operator of low layer1, h2...hnum, wherein num is the heuristic calculation of low layer The number of son selects 3 heuristic operators of low layer, i.e. num=3 in the present invention；h1, h2, h3For the already present more mesh of classics Mark optimization algorithm；
h1: 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)。
These three algorithms respectively have advantage and disadvantage, more suitable for verifying HH-MAB on combining the calculation heuristic operator advantage of low layer Ability, intersection and mutation probability and classic algorithm NSGA-II and most of multiple targets in each heuristic operator of low layer into It is identical to change algorithm, is respectively set as 0.9 and 1/24, intersects and mutation profile exponent is respectively 10 and 20.
Step 2: construction contains the random initial population P of N number of solution0(due to being initial population, i.e. the 0th generation population, therefore Initial population is denoted as P0；In this experiment, N=100)；In decision space X, for problem to be solved, if each decision component xi The value range set is [xil, xiu], i ∈ { 1,2 ..., n }, xilAnd xiuIt is illustrated respectively in i-th that x is solved in problem definition Decision component xiValue lower bound and the upper bound, then generated at random in decision space X and possess the random initial population P of N number of solution0= {x1, x2..., xNMethod are as follows: for each RANDOM SOLUTION xj(j ∈ { 1,2 ..., N }), i-th of decision component xj i's Value is its value range [xil, xiu] random value in (i ∈ { 1,2 ..., n }).
Step 3: calculate initial performance values of the heuristic operator of each low layer on four performance evaluation indexes, with from multi-party Planar survey algorithm obtains the performance of disaggregation；Each heuristic operator h of low layer1,h2,h3Respectively to same initial population P0Independent fortune Row is primary, and each run executes hiIn total M generation, (M is user's custom parameter to i ∈ { 1,2,3 }, to execute low layer in an iteration The number of heuristic operator, M take the integer in (0, ∞) range, M=250 are arranged in the present embodiment), it then calculates separately each Heuristic operator hiFour performance evaluation indexes (AE, RNI, SSC and UD), i ∈ { 1,2,3 }；
h1Four performance evaluation indexes be denoted as AE respectively1、RNI1、SSC1And UD1；
h2Four performance evaluation indexes be denoted as AE respectively2、RNI2、SSC2And UD2；
h3Four performance evaluation indexes be denoted as AE respectively3、RNI3、SSC3And UD3；
1) Algorithm effort (AE): assessment algorithm obtains the calculation amount of approximate disaggregation, and value is set by user For set time step-length divided by the function evaluation quantity executed in the step-length, value range is (0, ∞), and lesser AE value indicates more Good performance (Tan, K.C., Lee, T.H., &Khor, E.F. (2002) Evolutionary algorithms for multi-objective optimization:Performance assessments and comparisons.Artificial Intelligence Review,17,253–290)
2) Ratio of non-dominated individuals (RNI): the approximate solution that assessment algorithm obtains is concentrated non- Dominate the ratio of solution, value is the number of non-domination solution in current population divided by initial population size N, value range be (0, 1], RNI value is bigger indicates that approximate solution concentrates the ratio of non-domination solution higher, performance better (Tan, K.C., Lee, T.H., & Khor,E.F.(2002).Evolutionary algorithms for multi- objective optimization: Performance assessments and comparisons.Artificial Intelligence Review,17, 253–290.)
3) Size of space covered or so-called S_metric Hypervolume (SSC): assessment is calculated The approximate disaggregation and reference point ref=(ref that method obtains1,ref2,...,refm) between covered object space it is big It is small, value range be [0, ∞)；Reference point ref=(ref1,ref2,...,refm) be set as in object space by approximate disaggregation In the solution that all dominates of all solutions, for minimizing multi-objective optimization question, usual refiIt indicates on the i-th objective function most Big value；For two target WFG problems, common reference point is set as ref=(4,4)；SSC value is higher to indicate covering like nearly disaggregation Lid space is bigger, also indicates that performance better (Zitzler, E. , &Thiele, L. (1999) on convergence and distributivity .Multiobjective evolutionary algorithms:A comparative case study and the strength)
4) Uniform distribution of a non-dominated population (UD): assessment is non-dominant Disaggregation object space distribution situation, value range be (0,1]；In the identical situation of non-domination solution number, UD value is got over Height indicate distributivity of the approximate disaggregation on object space it is better (Srinivas, N., K. (1994) .Multiobjective optimization using nondominated sorting in genetic algorithms.Evolutionary Computation,2,221–248.)
Step 4: calculating the return value r of the heuristic operator of each low layer using operator Performance Evaluation mechanismi；Each low layer inspires Formula operator evaluation mechanism includes the following steps:
1) performance data obtained to the heuristic operator of each low layer is normalized；By four performance evaluation indexes AE, RNI, SSC and UD are normalized within the same value range [0,1] from different value intervals；Four performances are obtained respectively Minimum value and maximum value of the evaluation index in three heuristic operators of low layer of current generation, are denoted as AE respectivelymin、AEmax、 RNImin、RNImax、SSCmin、SSCmax、UDminAnd UDmax, wherein AEmin=argmin { AE1,AE2,AE3, AEmax=argmax {AE1,AE2,AE3, RNImin=argmin { RNI1,RNI2,RNI3, RNImax=argmax { RNI1,RNI2,RNI3, SSCmin =argmin { SSC1,SSC2,SSC3, SSCmax=argmax { SSC1,SSC2,SSC3, UDmin=argmin { UD1,UD2, UD3, UDmax=argmax { UD1,UD2,UD3}；If hiThe normalization of (i ∈ { 1,2,3 }) on four performance evaluation indexes it Preceding original value is denoted as AE respectivelyi、RNIi、SSCiAnd UDi, then hiNormalized value AE on four performance evaluation indexesi_nor, RNIi_nor, SSCi_nor,UD i_norIt calculates as follows:
By taking Figure 15 as an example, (a) is original of the heuristic operator of first three low layer of normalization on four performance evaluation indexes Initial value (b) is normalized value of three heuristic operators of low layer on four performance evaluation indexes；It, can from (a) by taking AE as an example Know AEmin=0.000108, AEmax=0.000430；Then h1The normalized value AE on AE1_nor=(0.000259- 0.000108)/(0.000430-0.000108)=0.468944.
2) according to the Performance Evaluation index value after the normalization 1) obtained, the heuristic calculation of each low layer is 1. calculated using formula Sub- hiReturn value ri, i ∈ { 1,2,3 }:
ri=(1-AEi_nor)+2RNIi_nor+SSCi_nor+UDi_nor ①
Wherein AEi_norIt represents and executes the heuristic operator h of low layeriCalculation amount, SSCi_norIt is heuristic to represent execution low layer Operator hiThe distributivity and convergence for the population obtained afterwards, UDi_norIt represents and executes the heuristic operator h of low layeriPoint of population is obtained afterwards Cloth, RNIi_norThe heuristic operator h of low layer is held in expressioniAfterwards in population non-domination solution ratio；Due to RNIi_nor、SSCi_norWith UDi_norIt is that the bigger performance of value is better, and AEi_norIt is that the smaller performance of value is better.Herein by AEi_norTurn from minimization problem Turn to maximization problems, i.e., formula 1. in use 1-AEi_norValue；But formula 1. in, UDi_norValue and SSCi_norWhat is be worth is big The small influence that will receive non-domination solution number in population；For example, when population at individual is less, even if the distributivity of current population It is poor, UDi_norThe more population of non-domination solution number in still possible larger or even preferable than the distributivity but population of value UDi_norValue wants high.Algorithm in the process of implementation, it is contemplated that Population Size may change, return value riCalculating process In we be SSCi_norAnd UDi_norThe RNI for an equal proportion of having arranged in pairs or groups respectivelyi_nor。
1. formula mixes four performance evaluation indexes, after normalization, due to the value of each Performance Evaluation index Range is identical, so the heuristic operator h of each low layeriReturn value riIt is four performances to four performance evaluation indexes without preference The comprehensive of evaluation index embodies；1. according to formula, each heuristic operator h of low layer after normalizationiReturn value riAs shown in table 2； With h1For, r1=(1-0.468944)+2 × 1.000000+ 1.000000+1.000000=4.531056.
Table 2- return value
h1 h2 h3
r 4.531056 4.406907 0.000000
Step 5: according to return value ri, the MAB value of the heuristic operator of each low layer is calculated, steps are as follows:
1) the set H={ h of operator heuristic for low layer1, h2, h3, it is the heuristic operator h of each low layeriSetting one The performance tuple being made of three variablesWherein riIt is hiStep 4 obtain return value,It is from the beginning of Up to the present hiAverage value, the k of obtained return valueiFor hiUp to the present the number selected, i ∈ { 1,2,3 }；In algorithm Initial stage, three variables were initialized to ri=0,ki=0；
2) the return value r obtained according to step 4i, the heuristic operator h of low layer is 2. updated using formulaiCorresponding performance tupleIn latter two variableAnd ki,
3) according to the heuristic operator h of low layeriPerformance tuple Vi, calculate the heuristic operator h of current generation low layeriMABi It is worth (MABi> 0), as formula 3. shown in:
Wherein C is control parameter (2 are set as in this experiment), for controlling the heuristic operator h of each low layeriAverage return ValueWith the heuristic operator h of low layer in algorithm operationiExecution number kiBetween a kind of balance；
Step 6: the selection heuristic operator h of the maximum low layer of MAB valuetop, i.e. top=argmax { MAB1, MAB2, MAB3}；If have the heuristic operator of multiple low layers while possessing maximum MAB value, i.e. MABi=MABj> MABk, wherein i, j, k ∈ { 1,2,3 }, then from hiAnd hjMiddle random selection one is used as htop；
Step 7: using the heuristic operator h of low layertopEvolve current population Pt, execute htopIn total M=250 generation, obtains new Population P '；It is calculated according to Performance Evaluation index AE, RNI, SSC and UD of step 3 introduction using the heuristic operator h of low layertop Performance index value AE of the obtained population P ' on four performance evaluation indexestop、RNItop、SSCtopAnd UDtop。
Step 8: using reception strategy Population Regeneration P is improvedt+1；Due to the four performance evaluation indexes used at us In, SSC is that only one can assess convergence in population but also assess the index of Species structure, therefore be held using SSC judgement Row htopWhether population has improvement afterwards (i.e. whether SSC value becomes larger)；If the more last round of iterative value of SSC evaluation index value does not become Greatly, then the population P of next iteration (t+1 generation)t+1By the population P in t generationtReplacement, i.e. Pt+1=Pt；If value becomes larger, t+1 The population P in generationt+1Population P ' the replacement just obtained, i.e. Pt+1=P '.
Step 9: judging whether the number of iterations is more than the iterations max W that sets (W as user's custom parameter, to calculate The maximum number of iterations of method takes the integer in (0, ∞) range, W=25 is arranged in the present embodiment)；If it does, HH-MAB is calculated Method terminates；If be not above, into next round iteration, return step four.
Fig. 6-14 is HH-RAND, HH-CF and HH-MAB independent operating 30 times, 50% on WFG1-WFG9 test problem Approximate Pareto leading surface schematic diagram.Approximate Pareto leading surface is before true Pareto acquired in three kinds of algorithms Along face, (PF) is closer, illustrates that convergence is better.From Fig. 6-14 as can be seen that the constringency performance of HH-MAB is compared with HH-RAND It is more preferable with HH-CF effect.
The above is only embodiments of the present invention.It should be pointed out that the present embodiment is enterprising in two target WFG test sets Row experiment, which shares 9 test cases, cover Pareto leading surface it is continuous discontinuous, Pareto leading surface convex surface Concave surface it is linear, multi-modal many aspects such as single mode, fraud problems, be to be widely used for verifying multi-objective optimization algorithm The test set of energy.The characteristics of according to meta-heuristic algorithms, only needs change or replacement low layer to inspire if the problem that replacement is to be solved The number and type of formula operator, without changing high-rise heuristic strategies.Therefore, the heuristic operator number of change low layer and Type is still the protection scope of this patent.

Claims (1)

1. a kind of method that the meta-heuristic algorithms based on MAB solve multi-objective optimization question, it is characterised in that: this method packet It includes:
(1) the set H={ h of the heuristic operator of low layer is provided1,h2…hnum, wherein num is the number of the heuristic operator of low layer, h1,h2…hnumFor multi-objective optimization algorithm or operator；
(2) the random initial population P that construction size is N0, in decision space X, N number of solution is generated at random；
(3) performance number of each heuristic operator of low layer in four existing Performance Evaluation indexs is calculated.Each low layer is heuristic Operator h1,h2…hnum(i=1,2 ... num) is to same initial population P0Independently operation is primary, and each run executes hiTotal M Generation, and calculate separately each heuristic operator hiFour performance evaluation indexes AE, RNI, SSC and UD；M is to execute in an iteration The number of the heuristic operator of low layer, M take the integer in (0, ∞) range；
1) AE: assessment algorithm obtains the calculation amount of approximate disaggregation, and value range is (0, ∞), and the smaller expression performance of AE value is got over It is good；
2) RNI: assessment algorithm obtain approximate solution concentrate non-domination solution ratio, value range be (0,1], RNI value is bigger Indicate that approximate solution concentrates the ratio of non-domination solution higher；If RNI=1, all individuals for showing that approximate solution is concentrated all are non-branch Match；
3) SSC: assessment algorithm obtain approximate disaggregation institute covered object space size, value range for [0, ∞)； SSC value is higher, and the covering space for indicating seemingly nearly disaggregation is bigger, also indicates that performance is better on convergence and distributivity；
4) UD: assessing approximate disaggregation in the distribution situation of object space, value range be (0,1]；In non-domination solution quantity phase The higher distributivity for indicating approximate disaggregation on object space of UD value is better in the case where；
(4) the return value r of the heuristic operator of each low layer is calculated using operator Performance Evaluation mechanismi, each heuristic operator assessment of low layer Mechanism includes the following steps:
1) performance data obtained to the heuristic operator of each low layer is normalized, so that each Performance Evaluation index is mapped to Within the same value range [0,1], the heuristic operator h of low layeriValue difference after the normalizing on four performance evaluation indexes It is denoted as AEi_nor、RNIi_nor、SSCi_norAnd UDi_nor；
2) according to the Performance Evaluation index value obtained after normalization, the return value of the heuristic operator of each low layer is 1. calculated using formula ri:
ri=(1-AEi_nor)+2RNIi_nor+SSCi_nor+UDi_nor ①
Formula 1. in: because of AEi_norValue it is the smaller the better, and other three indexs are the bigger the better, therefore in return value ri's The heuristic operator h of low layer has been used in calculatingiAEi_norReverse value, i.e. 1-AEi_norValue；SSCi_norIt can reflect out algorithm The distributivity and convergence of the population obtained after execution, UDi_norIt can reflect out the distributivity that population is obtained after algorithm executes；? Under Population Size unanimous circumstances, SSCi_norAnd UDi_norValue be the bigger the better；In order to avoid the variation pair of Population Size SSCi_norAnd UDi_norThe influence of value, therefore in riCalculating in give SSCi_norAnd UDi_norIt has arranged in pairs or groups respectively a non-domination solution Ratio RNIi_nor；
(5) according to return value ri, the MAB value of the heuristic operator of each low layer is calculated, steps are as follows:
1) for there is the set H={ h of the heuristic operator of num low layer1,h2…hnum, it is each hi(i ∈ { 1 ... num }) setting One performance tuple being made of three variablesWherein riBe return value,It is average return value, kiFor hiIt is selected The number selected, in the algorithm initial stage, all values are initialized as 0；
2) h is 2. updated using formulaiPerformance tuple
3) according to the heuristic operator h of low layeriPerformance tuple ViCalculate the MAB of the heuristic operator of current generation low layeriValue is as follows Formula is 3.:
Wherein C is control parameter, controls the heuristic operator h of each low layeriAverage behaviorWith time that operator executes in frame operation A kind of balance between number；
(6) the heuristic operator h of the maximum low layer of MAB value is selectedtop, i.e. top=argmax { MABi, i=1,2 ..., num；If The algorithm for possessing maximum MAB value has multiple, then therefrom random selection one；
(7) h is usedtopAdvanced group species Pt, execute htopIn total M generation, new population P ' is obtained, and calculated using the heuristic operator h of low layertop Performance number of the obtained population P ' on four performance evaluation indexes；
(8) using improvement reception strategy Population Regeneration Pt+1, judge to execute h using SSCtopWhether population has improvement afterwards, if without changing Into the then population P of next iteration (t+1 generation)t+1It is replaced by the population in t generation, i.e. Pt+1=Pt；If there is improvement, t+1 generation Population Pt+1Population P ' the replacement just obtained, i.e. Pt+1=P '；
(9) judge the number of iterations whether be more than setting maximum value W, if it does, HH-MAB algorithm terminates；If do not surpassed It crosses, into next round iteration, returns (4)；W is the maximum number of iterations of algorithm, takes the integer in (0, ∞) range.
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