Set-optimization meets variational inequalities

TL;DR

This paper establishes conditions linking set-optimization solutions to variational inequalities, extending classical optimality conditions to set-valued functions using directional derivatives and scalarizations.

Contribution

It introduces a set-valued directional derivative and relates it to scalarized Dini derivatives, providing new variational inequality characterizations for set-valued optimization.

Findings

Established necessary and sufficient conditions for set-optimization solutions.

Linked set-valued variational inequalities with scalarized derivatives.

Unified variational characterizations for vector-valued optimization problems.

Abstract

We study necessary and sufficient conditions to attain solutions of set-optimization problems in therms of variational inequalities of Stampacchia and Minty type. The notion of a solution we deal with has been introduced Heyde and Loehne, for convex set-valued objective functions. To define the set-valued variational inequality, we introduce a set-valued directional derivative and we relate it to the Dini derivatives of a family of linearly scalarized problems. The optimality conditions are given by Stampacchia and Minty type Variational inequalities, defined both by the set valued directional derivative and by the Dini derivatives of the scalarizations. The main results allow to obtain known variational characterizations for vector valued optimization problems.