Dynamic Multi-objective Co-evolutionary Optimization Theories And Methods

Dynamic multi-objective optimization problems(DMOPs)generally exist in the actual industrial production and daily life.DMOPs not only involve multiple conflicting objectives,and their Pareto sets may also vary with time,but the parameters of such problems may also be uncertain,the decision variables will changing over time or multi-modal exists in the objective functions,to say a few,raising a challenge for researchers to solve them.If a static multi-objective evolutionary algorithm is directly employed to tackle the DMOPs,the obtained results are not ideal due to the lack of strategies of problem detection and responding.In view of this,focusing on the different types of DMOPs,the decision variables are grouped based on the analysis of their characteristics and co-evolutionary.Furthermore,more targeted strategies for detecting problem change and responding are designed,respectively.Therefore,four dynamic multi-objective evolutionary optimization methods were proposed and applied to solve a practical problem.First,a cooperative co-evolutionary algorithm based on time-dependent was presented for solving DMOPs.In this algorithm,a new method that groups decision variables is proposed,in which all the decision variables are partitioned into two subcomponents according to their dependence with time,i.e.dependent or independent on time.Adopting two populations to cooperatively optimize the two subcomponents,two prediction methods,i.e.,differential prediction and Cauchy mutation,are then employed respectively to generate a higher quality initial solution set on the change of the problems.The proposed methods are compared with three state-of-the-art algorithms by applying to seven benchmark DMOPs.Experimental results reveal that the proposed algorithms significantly outperform the compared algorithms in terms of convergence and distribution on most DMOPs.Second,considering the DMOPs with dynamic interval parameters,a framework of dynamic interval multi-objective cooperative co-evolutionary optimization based on the interval similarity was proposed.In the framework,a strategy for decomposing decision variables is presented,through which all the decision variables are divided into two groups according to the interval similarity between each decision variable and interval parameters.Following that,two sub-populations are applied to cooperatively optimize decision variables in the two groups.Furthermore,two response strategies,i.e.,an adaptive step size based on the change intensity and a random mutation strategy,are utilized to rapidly track the changing Pareto front of the optimization problem.The proposed method is applied to eight benchmark optimization instances and compared with five state-of-the-art evolutionary algorithms.The experimental results indicate that the proposed method is not only capable of obtaining a solution set with good convergence and distribution,but also with lower imprecise.Then,a dynamic multimodal multi-objective evolutionary optimization algorithm is proposed.On the basis of the contribution of the decision variables to the objective functions,the decision variables are divided into convergent and distributed variables.The algorithm optimizes the above two types of variables by utilizing the existing multi-objective optimization algorithm,and saves the optimal solutions to convergence and distribution archive,respectively.When the problem changes are detected,a multi-direction prediction mechanism is employed to improve the convergence of the population in the objective space.At the same time,the diversity of population in the decision variable and objective space is maintained by niche method.Therefore,the aims of rapidly responding to the problem change and maintaining the diversity of obtained solutions can be reached.Next,for the DMOPs with changing decision variables,decision variables are divided into a number of groups using the maximum entropic epistasis,with decision variables in different groups generally having a weak correlation.Meanwhile,a subpopulation is generated to optimize decision variables in each group using a multiobjective evolutionary algorithm,and a complete solution including all the decision variables is achieved through the cooperation among subpopulations.When a new decision variable is added to the existing optimization problem,the grouping of decision variables will be dynamically adjusted based on the correlation between the added decision variable and existing groups.The proposed method is applied to eight benchmark optimization instances and compared with five state-of-the-art evolutionary algorithms.The experimental results demonstrate that the proposed method is very competitive on most optimization instances.Last but not least,the proposed dynamic interval multi-objective evolutionary algorithm is applied to solve a multi-period portfolio selection problem in emerging markets.To provide investors with more choices,the problem is formulated with uncertainties as a bi-objective optimization model with interval coefficients in the objectives.After that,the proposed algorithm and the comparative ones are employed to tackle the optimization problem.The experimental results reveal that,on the one hand,the proposed algorithm offers investors a larger return rate with the same risk loss rate than the comparative ones;on the other hand,the proposed algorithm provides investors more choices with different preferences.The obtained research results,on the one hand,have enriched the theories of dynamic multi-objective evolutionary optimization,and on the other hand,provided a more perfect solution to the practical DMOPs.Thereby,it has significant theoretical significance and practical value.