Scalarization approach for approximation of weakly efficient solutions of set-valued optimization problems

M Oveisiha,Ryan Loxton

doi:10.1080/23311835.2016.1243840

Abstract

In this paper, by using the scalarization method and normal subdifferential for set-valued maps, we consider an extension of Minty variational-like inequalities and obtain some relations between their solutions and set-valued optimization problems. An existence result for generalized Minty variational-like inequalities and set-valued optimization problems is also given. Moreover, the concept of approximate efficient solutions due to Kutateladze is investigated and by the Tammer–Weidner nonlinear functional, we characterize them for cone constrained set-valued optimization problems.

Highlights

Giannessi (1980) was the first author who obtained the equivalence between solutions of a Minty variational inequality and efficient solution of differentiable, convex optimization problem
By introducing a scalarized Minty variational-like inequalities (MVLI), we show that any solution of a scalarized set-valued optimization problems (SOP) is a solution of Minty variational-like inequalities under standard assumptions and that the inverse implications hold under the additional generalized K-convexity assumption, where K is an ordering cone of the considered image space
Approximation of weakly efficient solutions we present the concept of approximate efficiency for set-valued maps that is a generalization of the same notion due to Kutateladze (1979)

Summary

Introduction

Variational inequalities are identified either in the form presented by the Minty (1967) or in the form by Stampacchia (1960). Giannessi (1980) was the first author who obtained the equivalence between solutions of a Minty variational inequality and efficient solution of differentiable, convex optimization problem. Variational inequalities are identified either in the form presented by the Minty (1967) or in the form by Stampacchia (1960). Giannessi (1980) was the first author who obtained the equivalence between solutions of a Minty variational inequality and efficient solution of differentiable, convex optimization problem. Some authors focused their works to nonsmooth functions Al-Homidan & Ansari, 2010; Alshahrani, Ansari, & Al-Homidan, 2014; Chen & Huang, 2012; Yang & Yang, 2006). Al-Homidan and Ansari (2010) obtained these results for invex functions with Clarke’s generalized directional derivative. By using the scalarization method, Santos, Rojas-Medar, M.

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Conclusions