First-order approximation of strong vector equilibria with application to nondifferentiable constrained optimization

TL;DR

This paper develops a first-order approximation framework for strong vector equilibria using Bouligand derivatives, enabling new optimality conditions in nondifferentiable constrained optimization.

Contribution

It introduces conical approximations of the strong solution set to vector equilibrium problems and derives optimality conditions for related constrained optimization problems.

Findings

Contingent cone formulas for strong vector equilibria

Necessary and sufficient optimality conditions derived

Application to mathematical programming with equilibrium constraints

Abstract

Vector equilibrium problems are a natural generalization to the context of partially ordered spaces of the Ky Fan inequality, where scalar bifunctions are replaced with vector bifunctions. In the present paper, the local geometry of the strong solution set to these problems is investigated through its inner/outer conical approximations. Formulae for approximating the contingent cone to the set of strong vector equilibria are established, which are expressed via Bouligand derivatives of the bifunctions. These results are subsequently employed for deriving both necessary and sufficient optimality conditions for problems, whose feasible region is the strong solution set to a vector equilibrium problem, so they can be cast in mathematical programming with equilibrium constraints.