Optimality conditions and duality for minimax fractional programming problems with data uncertainty

Xiao-Bing Li,Zhi Lin,Qi-Lin Wang

doi:10.3934/jimo.2018089

Abstract

In this paper, we consider minimax nondifferentiable fractional programming problems with data uncertainty in both the objective and constraints. Via robust optimization, we establish the necessary and sufficient optimality conditions for an uncertain minimax convex-concave fractional programming problem under the robust subdifferentiable constraint qualification. Making use of these optimality conditions, we further obtain strong duality results between the robust counterpart of this programming problem and the optimistic counterpart of its conventional Wolf type and Mond-Weir type dual problems. We also show that the optimistic counterpart of the Wolf type dual of an uncertain minimax linear fractional programming problem with scenario uncertainty (or interval uncertainty) in objective function and constraints is a simple linear programming, and show that the robust strong duality results in sense of Wolf type always hold for this linear minimax fractional programming problem.

Highlights

Consider the following minimax fractional programming with uncertain data (UP)min max fi(x, ui) x∈Rn 1≤i≤p gi(x, vi) s.t. x ∈ C, h(x, w) ∈ −K, where C is a nonempty, closed and convex set of Rn, K is a nonempty, closed and convex cone of Rq, for each i = 1, · · ·, p, fi, gi : Rn × Rq0 → R, h : Rn × Rq0 → Rq, ui, vi and w are uncertain data which belong to the convex, compact and uncertain sets Ui, Vi and W .Robust optimization [2] is a specific and powerful methodology for handing mathematical programming with uncertain data
We show that the optimistic dual of a robust minimax linear fractional with scenario uncertainty in objective function and the scenario uncertainty in the constraints is a simple linear programming
We further verify that the robust strong duality results in sense of Wolf type always hold for this linear minimax fractional programming problem

Summary

Introduction

We further verify that the robust strong duality results in sense of Wolf type always hold for this linear minimax fractional programming problem. The following statements are equivalent: (i): the robust type subdifferential constraint qualification (RSCQ) is fulfilled at x ∈ F ; (ii ): the point x ∈ F is an optimal solution of (RP ) with optimal value αif and only if there exist (yi) ∈ S+p , ui ∈ Ui, vi ∈ Vi, w ∈ W and λ ∈ K∗ such that (4), (5) and (6) hold.

Objectives

Results

Conclusion