Some aspects of optimality and duality in vector optimization

Abstract

The main objective of the thesis is to study optimality conditions and duality results for vector optimization problems. The thesis is divided into four chapters, which are further subdivided into sections. Chapter I is introductory and aims to provide a necessary background by presenting briefly various fundamental concepts related to optimization problems. This chapter concludes with a brief summary of the work presented in the thesis. Chapter 2 talks about some generalizations of cone convex functions and proves optimality conditions and duality results for vector optimization problems. This chapter is divided into four sections. Section 2.1 introduces the concept of cone semistrictly convex functions on topological vector spaces. Some properties and interrelations of cone convex and cone semistrictly convex functions are studied. Section 2.2 derives sufficient optimality conditions for a vector valued minimization problem over topological vector spaces, involving Gâteaux derivatives. A Mond-Weir type dual is associated with the primal problem and weak and strong duality results are established. Section 2.3 introduces cone semilocally preinvex and related functions. Necessary and sufficient optimality conditions and duality results are established for a vector optimization problem with equality and inequality constraints over cones. Section 2.4 focuses on nonsmooth vector optimization. In this section generalized type-I, generalized quasi type-I, generalized pseudo type-I, generalized quasi pseudo type-I and generalized pseudo quasi type-I functions over cones are introduced, for a nonsmooth vector optimization problem. Various optimality and duality results are proved under cone generalized type-I assumptions, using Clarke s generalized gradients of locally Lipschitz functions. Chapter 3 studies second order symmetric duality in vector optimization and is divided into two sections. Section 3.1 aspires to examine pairs of second order Wolfe type and Mond-Weir type symmetric duals.