Parametric nondifferentiable multiobjective fractional programming under (b;;; )-univexity

Tadeusz ANTCZAK,Ram VERMA

doi:10.3906/mat-1705-65

Abstract

In this paper, we are concerned with optimality conditions and duality results for nondifferentiable multiobjective fractional programming problems. Parametric necessary optimality conditions are established for such vector optimization problems in which each component of the involved functions is locally Lipschitz. Further, under the introduced concept of nondifferentiable $(b,\Psi ,\Phi ,\rho )$-univexity, the parametric sufficient optimality conditions are established for a new class of nonconvex multiobjective fractional programming problems. Furthermore, for the considered multiobjective fractional programming problem, its parametric vector dual problem in the sense of Schaible is defined. Then several duality theorems are also established under $(b,\Psi ,\Phi ,\rho )$% -univexity hypotheses.

Highlights

In recent years, multiobjective fractional programming problems have received much attention by many authors due to the fact that in many operations research problems the objective functions are quotients of two functions
Many authors established necessary optimality conditions and employed the conditions to search for optimal solutions as well as duality theorems for such vector optimization problems in the recent past
Kim [18] introduced the concept of generalized invexity for a fractional function and he proved the sufficient optimality conditions and several duality results for the considered nonsmooth multiobjective fractional programming problems involving locally Lipschitz functions

Summary

Introduction

Multiobjective fractional programming problems have received much attention by many authors due to the fact that in many operations research problems the objective functions are quotients of two functions. Kim [18] introduced the concept of generalized invexity for a fractional function and he proved the sufficient optimality conditions and several duality results for the considered nonsmooth multiobjective fractional programming problems involving locally Lipschitz functions. We shall establish parametric necessary optimality conditions for the considered nonsmooth multiobjective fractional programming problem with both inequality and equality constraints in which each component of the involved functions is a locally Lipschitz function. We denote by J(x) the set of active constraints at x ∈ D , that is, J(x) = {j ∈ J : gj(x) = 0} For such multicriterion optimization problems as the considered multiobjective fractional programming problem (MFP), the optimal solution is defined in terms of a (weak) Pareto solution ((weakly) efficient solution) in the following sense: Definition 9 A feasible point x is said to be a weak Pareto solution (weakly efficient solution, weak minimum) for (MFP) if and only if there exists no x ∈ D such that φ(x) < φ(x). Definition 10 A feasible point x is said to be a Pareto solution (efficient solution) for (MFP) if and only if there exists no x ∈ D such that φ(x) ≤ φ(x)

Optimality

Conclusion