Some enhanced existence results for strong vector equilibrium problems

TL;DR

This paper establishes new sufficient conditions for the enhanced solvability of strong vector equilibrium problems using a variational approach, nonsmooth analysis, and slope conditions, with implications for error bounds and optimality conditions.

Contribution

It introduces verifiable slope-based conditions involving generalized derivatives for the enhanced solvability of strong vector equilibrium problems, replacing traditional KKM theory methods.

Findings

Derived conditions for enhanced solvability using nonsmooth analysis tools.

Established error bounds and estimates for the distance to the solution set.

Provided a variational framework characterizing solutions as zeros of merit functions.

Abstract

This paper explores some sufficient conditions for the enhanced solvability of strong vector equilibrium problems, which can be established via a variational approach. Enhanced solvability here means existence of solutions, which are strong with respect to the partial ordering, complemented with inequalities estimating the distance from the solution set (namely, error bounds). This kind of estimates plays a crucial role in the tangential (first-order) approximation of the solution set as well as in formulating optimality conditions for mathematical programming with equilibrium constraints (MPEC). The approach here followed characterizes solutions as zeros (or global minimizers) of some merit functions associated to the original problem. Thus, to achieve the main results the traditional employment of the KKM theory is replaced by proper conditions on the slope of the merit functions. In turn, to make such conditions verifiable, some tools of nonsmooth analysis are exploited. As a result, several conditions for the enhanced solvability of strong equilibrium problems are derived, which are expressed in terms of generalized (Bouligand) derivatives, convex normals and various (Fenchel and Mordukhovich) subdifferentials.