On Constrained Set-Valued Semi-Infinite Programming Problems with ρ-Cone Arcwise Connectedness

Koushik Das,Savin Treanţă

doi:10.3390/axioms10040302

Abstract

In this paper, we establish sufficient Karush–Kuhn–Tucker (for short, KKT) conditions of a set-valued semi-infinite programming problem (SP) via the notion of contingent epiderivative of set-valued maps. We also derive duality results of Mond–Weir (MWD), Wolfe (WD), and mixed (MD) types of the problem (SP) under ρ-cone arcwise connectedness assumptions.

Highlights

Over the last few decades, many authors, such as Hanson [1], Craven [2], Corley [3], Zalmai [4] etc., studied different types of optimization problems
We study the duality theorems of Mond–Weir (MWD), Wolfe (WD), and mixed (MD) types associated with the problem (SP)
We introduce the notion of ρ-cone arcwise connectedness of set-valued maps as a generalization of cone arcwise connected set-valued maps

Summary

Introduction

Over the last few decades, many authors, such as Hanson [1], Craven [2], Corley [3], Zalmai [4] etc., studied different types of optimization problems. 3. ρ-Cone Arcwise Connectedness Das and Nahak [22,23,24,25] and Treantaand Das [26] introduced the notion of ρ-cone convexity of set-valued maps They establish the sufficient KKT optimality conditions of set-valued optimization problems under contingent epiderivative and ρ-cone convexity assumptions. Let A be an arcwise connected subset of a real normed space U, e ∈ int(Ω), and F : U → 2V be a set-valued map, with A ⊆ dom(F ). We establish the sufficient KKT optimality conditions of the set-valued semi-infinite programming problem (SP) under ρ-cone arcwise connectedness assumption.

Mond–Weir Type Dual

Wolfe Type Dual

Mixed Type Dual

Conclusions