Study On Convergence And Duality Of Set-valued Optimization Problems

In this paper,we study the convergence of the solution set of a nonconvex vector optimization problem and a set-valued optimization problem under the perturbation of feasible set,objective function and ordering cone,and study the high-order optimality conditions and duality theory of the weakly efficient solution of a set-valued optimization problem.Firstly,we introduce the research significance and development of vector and set-valued optimization problems,describe the contributions made by different scholars in recent years,and describe the motivation and main research work of this paper.Then we mainly discuss the convergence of the solution sets of a nonconvex vector optimization problem,which is a quasi-connected vector optimization problem with respect to the perturbation of feasible set,objective function and ordering cone.To obtain the convergence results of this problem,we first study the KuratowskiPainlevé convergence of(weak)minimal point set for a sequence of cone-sectionwise connected sets with variable order structures to that of a given cone-sectionwise connected set,and then investigate the Kuratowski-Painlevé convergence of the sets of(weak)minimal points and(weak)efficient points for perturbed optimization problems to those of a given problem.Several numerical examples are also given to illustrate the main results and compare these results with the corresponding ones of the recent references.Then we establish the stability of the solution set of the set-valued optimization problem in terms of a sequence of solution sets of perturbed set optimization problems,which perturbation of feasible set,objective map and ordering cone,convergence to the solution set of the original set optimization problem both in the given space and the image space.External stability is established for weak minimal solution sets under certain compactness and continuity assumptions which leads to the upper Kuratowski-Painlevé convergence of the sequence of weak efficient and efficient solution sets in the given space.Internal stability is established for minimal solution sets under certain continuity,compactness and domination assumptions which leads to the lower Kuratowski-Painlevé convergence of the sequence of efficient and weak efficient solution sets in the given space.External stability for minimal solution sets and internal stability for weak minimal solution sets are deduced under strict quasi-convexity assumption.Finally,we introduce the notion of higher-order generalized(incident)epiderivatives of set-valued mapping,and obtain a crucial result of the epiderivatives.Then we use this crucial result to establish the higher-order sufficient and necessary optimality conditions of the constrained set-valued optimization problem.By virtue of the epiderivatives and the optimality conditions,we establish the higher-order mixed duality problem for the set-valued optimization problem and obtain the corresponding duality theorems.