Duality in Mathematical Programming Involving Class of E-Convex Functions

Abstract

The present thesis is organised into four chapters. The additional objective is to clarify the structure underlying generalized convex functions by presenting analogoues to the properties of convex functions. The material on convex functions and their generalizations is intensely large. The overview of the chapters are described below: The first chapter of the dissertation is the introduction with the brief description of basic concepts, definitions of convex functions and and other definitions to be used in subsequent chapter are given, that are used throughout work which is useful for understanding the general concept of convexity. A brief account of the related studies made by various authors in the field and a summary of the thesis has also presented in this chapter. Chapter 2 is devoted to a class of E-convex sets and E-convex functions are introduced by relaxing the definitions of convex sets and convex functions. This kind of generalized convexity is based on the effect of an operator E on the sets and domain of definition of the functions. The optimality results for E-convex programming problems are established. In Chapter 3, we have reviewed a class of functions, F-convex functions and Second order F-convex functions as a generalization of convex functions. Under F-convexity, F-concavity, F-pseudoconvexity, F-pseudoconcavity, duality results for pair of Wolfe and Mond-Weir type symmetric dual nonlinear programming problems and second order symmetric duality are established. Chapter 4, we discussed a class of functions called F-E-convex functions which are generalizations of F-convex and E-convex function. We proved the weak duality and strong duality results for the second order Mond-Weir type dual problem with cone constraints.