Research On Evolutionary Multi-Objective Optimization With Complex Constraints

No matter in large-scale engineering problems or military command and control,there are many constrained multi-objective optimization problems(CMOPs),which require to find a set of well-distributed and well-convergent solutions to optimize multiple objectives on the premise of satisfying certain constraints.The difficulties of this kind of problems are as follows:1)Constraints will make part of the search space become infeasible regions,which will hinder the search for optimal solutions;2)Objective functions are often mutually affected,and optimization of one of them requires degradation of at least another objective as a cost.Evolutionary computation,as a non-gradient iterative optimization technique based on population search,has become a mainstream method to solve CMOPs because of its simplicity and efficiency.However,the existing algorithms still face great challenges when handling constraints with irregular constraint boundary,small feasible region and a large number of constraints,such as poor diversity of solution distribution,difficulty in crossing infeasible region and difficulty in optimizing.In view of this,based on the design idea of multi-population,multi-stage and self-adaptation,this paper carries out a systematic and in-depth study on the above problems,mainly including the following contributions and innovations:(1)A co-evolutionary multi-objective optimization algorithm based on two populations is proposed for CMOPs with irregular constraints.When the feasible regions of CMOPs have irregular distribution and shape,traditional algorithms will be affected by irregular constraint boundary during evolution,and tend to find the partial region with small obstacle and easy to search,which is easy to fall into local optimal.However,decision-makers often hope to obtain a set of optimal solutions with good convergence,wide distribution and high feasibility.How to effectively balance convergence,diversity and feasibility in search is a big challenge to solve this kind of problem.In view of this,this paper proposes a two-population coevolutionary algorithm named c-DPEA.c-DPEA maintains two cooperative and complementary populations during the whole evolutionary process,which tend to retain infeasible and feasible solutions respectively,so as to balance the search in infeasible and feasible regions.In addition,an adaptive fitness function is proposed to keep a balance between convergence and diversity.Experimental results on three test suites and practical engineering cases show that compared with the existing algorithms,the proposed c-DEPA can achieve a better balance between convergence,diversity and feasibility,and show better performance.(2)A co-evolutionary multi-objective optimization algorithm based on two stages and two populations is proposed for multi-objective optimization problems with small feasible regions.In existing algorithms,the proportion of search in infeasible regions is far less than the proportion of search in feasible regions,causing that the useful information in infeasible solutions has not been effectively utilized.Therefore,most algorithm will show poor performance when the optimization problem has small feasible regions.In view of this,this paper proposes a new algorithm called DD-CMOEA,which has dual stages(i.e.exploration and exploitation stages)and dual populations.In the exploration stage,two populations are used to explore feasible and infeasible regions respectively.Then,in the exploitation stage,the population used to explore the feasible region provides information about the location of the feasible region to help the other population locate the constraint boundary and exploit the surrounding infeasible solutions.At the same time,the useful information in the infeasible solutions obtained by the latter in turn helps the former to converge better.Experimental results on three test suites and a hybrid renewable energy system(HRES)show that DD-CMOEA can quickly cross the infeasible region and localize constraint boundary,and can fully explore and exploit the useful information in feasible and infeasible regions.(3)An adaptive multi-stage evolutionary multi-objective optimization algorithm is proposed for multi-objective optimization problems with large-scale constraints.Existing constraint-handling methods treat all constraints as a whole at the same time.When the number of constraints is large,it is difficult to quickly find the optimal solution set satisfying all constraints.This paper proposes a priority-ranking mechanism,and an adaptive cascaded constraint-handling(ACCH)framework,and further combines ACCH with an existing evolutionary algorithm as an adaptive multi-stage multi-objective evolutionary algorithm.This algorithm does not consider any constraints in the first stage,and then prioritizes the unconsidered constraints when entering each new stage,automatically selects those with the highest priority to join the constraint set,and adaptively allocates the evolutionary generations for the new stage according to the current evolutionary status until all constraints are processed.In the early stage,the number of constraints handled by this algorithm is small and the priority is high.Since it is free from the interference of other constraints,the population can deal with these constraints more easily.Then,considering the remaining constraints step by step,the population can gradually converge to the optimal solutions.Experimental results on two test suites and a grid-connected hybrid renewable energy system show that the ACCH framework can adaptively divide the evolutionary process into different stages and deal with the constraints hierarchically,which can solve the optimization problem with large-scale constraints well.The mentioned works lay a good foundation for solving CMOPs with complex constraints,effectively integrate the evolutionary multi-objective optimization method and constraint-handling mechanism in theory,and provide a new means for improving the performance of constrained multi-objective evolutionary algorithm to find the optimal solution set.Moreover,they provide strong support for solving practical engineering problems such as optimal design of hybrid renewable energy systems.This paper finally summarizes some interesting and meaningful directions worth studying on in the future.