Sharp Lagrange multipliers for set-valued optimization problems

Abdessamad Oussarhan,Tijani Amahroq

doi:10.1051/ro/2022034

Abstract

In this paper, we give a comparison among some notions of weak sharp minima introduced in Amahroq et al. [Le matematiche J. 73 (2018) 99–114], Durea and Strugariu [Nonlinear Anal. 73 (2010) 2148–2157] and Zhu et al. [Set-Valued Var. Anal. 20 (2012) 637–666] for set-valued optimization problems. Besides, we establish sharp Lagrange multiplier rules for general constrained set-valued optimization problems involving new scalarization functionals based on the oriented distance function. Moreover, we provide sufficient optimality conditions for the considered problems without any convexity assumptions.

Highlights

The concept of sharp minimizer has been investigated for different types of optimization problems: real-valued, vector-valued as well as set-valued optimization problems
Studniarski [34] comes to extend the results of Auslender [6] for any extended real-valued objective function and the feasible set not necessary closed where the order of sharp minimizer (γ ≥ 2)
For vector-valued optimization problems, Jimenez [19] has introduced the notion of sharp minimizer of order γ, in addition, he has developed with Novo in Jimenez [20] and Jimenez and Novo [21] the theory on minimizer of order (γ ≥ 1 integer) considering different frameworks

Summary

Introduction

The concept of sharp minimizer has been investigated for different types of optimization problems: real-valued, vector-valued as well as set-valued optimization problems. Studniarski [35] introduced the notion of weak ψ-sharp local minima in vector optimization problems He has extended some necessary and sufficient optimality conditions obtained by Jimenez [19]. In connection with the paper of Durea and Strugariu [13], the sharp minimizer was introduced by means of the oriented distance function and its necessary optimality conditions are established with the use of the Mordukhovich generalized differentiation. Amahroq et al [5] introduced this notion in set-valued optimization problems without recourse to the use of distances adopted in Durea and Strugariu [13] and Zhu et al [40] They have established necessary and sufficient optimality conditions involving set-valued derivatives, besides they have provided optimality conditions in terms of Fritz-John multipliers under convexity assumptions on the objective set-valued mapping using the classical separation theorem.

Preliminaries

Necessary conditions for sharp minima of higher order in the weak sense

Necessary conditions for sharp minima in the strong sense

Sufficient conditions

Conclusions