On Some Aspects Of Duality And Stability In Vector Optimization

In this thesis, we study higher order Fritz John and Kuhn-Tucker type optimality conditions, higher order Mond-Weir and Wolfe type dual problems, and conjugate dual problems for constrained set-valued optimization problems, and we also study the upper, lower semicontinuity and continuity results on the solution mappings to parametric vector variational inequalities and vector equilibrium problems. This thesis is divided into nine chapters. It is organized as follows:In Chapter 1, we describe the development and current researches on the topic of vector optimization, including higher order optimality conditions and duality, conjugate duality and the semicontinuity of the solution mappings to parametric vector variational inequalities and vector equilibrium problems. We also give the motivation and the main research work.In Chapter 2, higher order generalized contingent epiderivative and higher order generalized adjacent epiderivative of set-valued mappings are introduced. Necessary and sufficient Fritz John type optimality conditions for Henig efficient solutions to a constrained set-valued optimization problem are given by employing the higher order generalized epiderivatives.In Chapter 3, the notions of higher order weak contingent epiderivative and higher order weak adjacent epiderivative for set-valued mappings are defined. By virtue of higher order weak epiderivatives and Henig efficiency, we introduce a higher order Mond-Weir type dual problem and a higher order Wolfe type dual problem for a constrained set-valued optimization problem (SOP) and discuss the corresponding weak duality, strong duality and converse duality properties. We also establish higher order Kuhn-Tucker type necessary and sufficient optimality conditions for (SOP).In Chapter 4, three conjugate dual problems are proposed by considering the different perturbations to a set-valued vector optimization problem with explicit constraints. Some concepts are introduced, such as the positive biconjugate mapping, positive subgradient and positive subdifferential. The weak duality, inclusion relations between the image sets of dual problems, strong duality and stability criteria are investigated. Some applications to so-called variational principles for a generalized vector equilibrium problem are shown.In Chapter 5, the concept of the parametric gap function is proposed and a key assumption is introduced by virtue of the parametric gap function. Then, by using the key assumption, sufficient conditions of the Hausdorff lower semicontinuity and continuity of the solution set map for a parametric (set-valued) weak vector variational inequality are obtained in Banach spaces.In Chapter 6, the parametric gap function for a parametric generalized vector quasivariational inequality is proposed by using a nonlinear scalarization function. By virtue of the parametric gap function and a key assumption, the Hausdorff lower semicontinuity of the solution set map is established in locally convex Hausdorff topological vector spaces. Simultaneously, we use an example to explain that the theorem concerning Hausdorff lower semicontinuity may not be a sufficient condition for the upper semicontinuity of the solution set map.In Chapter 7, we show that the uniform compactness assumptions used in proving the lower semicontinuity of the efficient solution set in [X.H. Gong and J.C. Yao, Lower semicontinuity of the set of efficient solutions for generalized systems, J. Optim. Theory Appl. 138 (2008) 197-205] and the weak efficient solution set in [X.H. Gong, Continuity of the solution set to parametric weak vector equilibrium problems, J. Optim. Theory Appl. 139 (2008) 35-46] are superfluous. Furthermore, we point out that under the assumptions of lower semicontinuity theorems, the solution set mappings are continuous actually. The upper semicontinuity of the solution set mappings are derived by scalarization methods. In addition, we also give some continuity results of various proper efficient solution sets to parametric generalized systems.In Chapter 8, based on a scalarization representation of the solution mapping and a property involving the union of a family of lower semicontinuous set-valued mappings, we establish the lower semicontinuity and continuity of the solution mapping to a parametric generalized vector equilibrium problem with set-valued mappings. Our consequences are new and include the corresponding results in [Y.H. Cheng and D.L. Zhu, Global stability results for the weak vector variational inequality, J. Global Optim. 32 (2005) 543-550] and [X.H. Gong, Continuity of the solution set to parametric weak vector equilibrium problems, J. Optim. Theory Appl. 139 (2008) 35-46] as special cases.In Chapter 9, we summarize the results of this thesis and make some discussions.