Sufficiency and Duality in Nonsmooth Multiobjective Programming Problem under Generalized Univex Functions

Pallavi Kharbanda,Deepa Sinha,Divya Agarwal

doi:10.1155/2014/904640

Abstract

We consider a nonsmooth multiobjective programming problem where the functions involved are nondifferentiable. The class of univex functions is generalized to a far wider class of (φ,α,ρ,σ)-dI-V-type I univex functions. Then, through various nontrivial examples, we illustrate that the class introduced is new and extends several known classes existing in the literature. Based upon these generalized functions, Karush-Kuhn-Tucker type sufficient optimality conditions are established. Further, we derive weak, strong, converse, and strict converse duality theorems for Mond-Weir type multiobjective dual program.

Highlights

Generalizations of convexity related to optimality conditions and duality for nonlinear single objective or multiobjective optimization problems have been of much interest in the recent past and explored the extent of optimality conditions and duality applicability in mathematical programming problems
We introduce a new generalized class of (φ, α, ρ, σ)-dI-V-type I univex functions which generalizes the class of functions introduced by Kharbanda et al [18], Ahmad [17], Slimani and Radjef [12], Mishra and Noor [10], Mishra et al [16], Antczak [9], Suneja and Srivastava [19], and Ye [8]
(f, g) is pseudo-quasi (φ, α, ρ, σ)-dI-V-type I univex function at xo

Summary

Introduction

Generalizations of convexity related to optimality conditions and duality for nonlinear single objective or multiobjective optimization problems have been of much interest in the recent past and explored the extent of optimality conditions and duality applicability in mathematical programming problems. Instead of Clarke generalized subgradient, Ye [8] used the concept of directional derivative to define the class of d invex functions He derived necessary and sufficient optimality conditions taking functions f󸀠(xo; y) and gJ󸀠(xo; y) to be convex. Mishra and Noor [10] extended the class of functions to d-V-type I functions and obtained sufficient optimality and duality results for Mond-Weir type multiobjective dual program. Slimani and Radjef [12] introduced a far wider class of nondifferentiable functions called dI-V-type I functions in which each component is directionally differentiable in its own direction instead of the same direction and established sufficient optimality and duality results. Rueda et al [14] obtained optimality and duality results for several mathematical programs by combining the concepts of type I and univex functions. We establish weak, strong, converse, and strict converse duality results for Mond-Weir type multiobjective dual program

Preliminaries and Definitions

Illustration

Sufficient Optimality Conditions

Mond-Weir Type Duality