Some Aspects of Duality in Mathematical Programming Problems

Abstract

The work exhibited in this thesis is an endeavor to achieve various duality results for minimax fractional programming and multiobjective programming problems. The proposed work encapsulates these results which are weaved into six chapters. The present thesis is assembled into chapters as described below: Chapter 1 is introductory and consists of definitions, notations and prerequisites of the present work. A brief account of the related work studied by various authors in the field and a summary of the thesis are also presented. Chapter 2 presents a parametric dual model for nondifferentiable minimax fractional programming (NMFP) problems. Optimality conditions and duality relations are acquired using (p, r)-ρ-(η, θ)-invex suppositions. Two types of second-order dual models are proposed for NMFP problem and usual duality results are developed under second-order B- (p, r)-invex functions. In Chapter 3, we present a novel concept of higher-order B-(p, r)-invex functions. we construct a higher-order dual for NMFP problem and achieve duality results under higher-order B-(p, r)-invexity. A numerical example is solved for finding optimal solution of NMFP problem. In Chapter 4, we develop second-order duality results for nondifferentiable multiobjective fractional variational problem under second-order (F, α, ρ, d)-pseudoconvexity suppositions. An illustration showing the existence of second-order (F, α, ρ, d)-pseudoconvex functions is provided. An example is obtained to validate the theoretical results of weak duality. Chapter 5 presents a new pair of higher-order symmetric dual for multiobjective programming problems involving support functions over arbitrary cones. We construct an example of a non trivial function that shows the existence of higher-order K-η-convex functions. Various duality relations are explored under aforesaid assumptions. Some special cases are also examined to show that this work extends known results of the literature. In Chapter 6, we propose a mixed type higher-order symmetric dual model for multiobjective programming problems. Weak, strong and converse duality theorems are established under higher-order K-(F, α, ρ, d)-convexity assumptions.