Optimality and duality results for a nondifferentiable multiobjective fractional programming problem

Ramu Dubey,Meraj Ali Khan,Shiv K Gupta

doi:10.1186/s13660-015-0876-0

Abstract

The purpose of this paper is to study a class of nondifferentiable multiobjective fractional programming problems in which every component of objective functions contains a term involving the support function of a compact convex set. For a differentiable function, we introduce the definition of higher-order $(C,\alpha,\gamma,\rho,d)$ -convex function. A nontrivial example is also constructed which is in this class but not $(F,\alpha,\gamma,\rho,d)$ -convex. Based on the $(C,\alpha,\gamma,\rho,d)$ -convexity, sufficient optimality conditions for an efficient solution of the nondifferentiable multiobjective fractional programming problem are established. Further, a higher-order Mond-Weir type dual is formulated for this problem and appropriate duality results are proved under higher-order $(C,\alpha,\gamma,\rho,d)$ -assumptions.

Highlights

Higher-order duality is significant due to its computational importance as it provides higher bounds whenever an approximation is used
A higher-order vector optimization problem and its dual have been studied by Kassem [ ]
We prove that the ratio of higher-order (C, α, γ, ρ, d)-convex functions is higher-order (C, α, γ, ρ, d )-convex

Summary

Introduction

Higher-order duality is significant due to its computational importance as it provides higher bounds whenever an approximation is used. By introducing two different functions, h : Rn × Rn → R and k : Rn × Rn → Rm, Mangasarian [ ] formulated a higher-order dual for a nonlinear optimization problem, {min f (x), subject to g(x) ≤ }. Inspired by this concept, many researchers have worked in this direction. Chen [ ] has formulated higherorder multiobjective symmetric dual programs and established duality relations under higher-order F-convexity assumptions. Ying [ ] has studied higher-order multiobjective symmetric fractional problem and formulated its Mond-Weir type dual. Duality results are obtained under higher-order (F, α, ρ, d)-convexity

Objectives

Conclusion