The augmented weak sharpness of solution sets in equilibrium problems

TL;DR

The paper studies finite termination of feasible solution sequences in equilibrium problems $EP(\\phi,S)$. It introduces augmented weak sharpness, using an augmented mapping $H$ on the solution set $\\bar{S}$, and establishes a necessary and sufficient termination condition that involves an $\\alpha>0$ with $\\alpha B \\subset H(z)+[T_{S}(z)\\cap \\\hat{N}_{\\bar{S}}(z)]^{\\circ}$ for all $z\\in\\bar{S}$ and a limsup criterion on $u^{k}\\in\\partial_{y}\\phi(x^{k},x^{k})$ and $v^{k}\\in H(P_{\\bar{S}}(x^{k}))$. This framework generalizes weak sharpness and strong non-degeneracy, yielding finite-termination results for generic equilibrium problems and their special cases (MP, VIP, SPP, NEP) under broader conditions and with less stringent assumptions. The results provide broad corollaries and illustrate the applicability of augmented weak sharpness to a wide class of optimization and equilibrium algorithms, expanding termination analysis beyond classical sharpness requirements.

Abstract

This study delves into equilibrium problems, focusing on the identification of finite solutions for feasible solution sequences. We introduce an innovative extension of the weak sharp minimum concept from convex programming to equilibrium problems, coining this as weak sharpness for solution sets. Recognizing situations where the solution set may not exhibit weak sharpness, we propose an augmented mapping approach to mitigate this limitation. The core of our research is the formulation of augmented weak sharpness for the solution set, a comprehensive concept that encapsulates both weak sharpness and strong non-degeneracy within feasible solution sequences. Crucially, we identify a necessary and sufficient condition for the finite termination of these sequences under the premise of augmented weak sharpness for the solution set in equilibrium problems. This condition significantly broadens the scope of existing literature, which often assumes the solution set to be weakly sharp or strongly non-degenerate, especially in the context of mathematical programming and variational inequality problems. Our findings not only shed light on the termination conditions in equilibrium problems but also introduce a less stringent sufficient condition for the finite termination of various optimization algorithms. This research, therefore, makes a substantial contribution to the field by enhancing our understanding of termination conditions in equilibrium problems and expanding the applicability of established theories to a wider range of optimization scenarios.