Iterative Schemes for Uniformly Nonconvex Equilibrium Problems
Oday Hazaimah
TL;DR
The paper addresses uniformly prox-regular equilibrium problems posed on prox-regular nonconvex sets and develops new implicit iterative schemes based on the auxiliary principle and inertial proximal ideas. It introduces a gap/merit function framework, enabling descent-type algorithms and associating the equilibrium problem with an auxiliary optimization problem, under mild pseudomonotonicity assumptions on the operator $F$; convergence of the proximal/inertial schemes is established, including in finite-dimensional spaces. A differentiable gap function $g(u)=\max_{w\in\bar{K}}\{F(u,w)-\frac{\alpha}{2}\|u-w\|^2\}$ with $\alpha=\frac{k}{r}$ is constructed, and a descent method using $w(u)=\arg\min_w\{F(u,w)+\frac{\alpha}{2}\|u-w\|^2\}$ and $d(u)=w(u)-u$ is analyzed, with convergence via Zangwill’s theorem. The work unifies equilibrium problems, variational inequalities, and complementarity problems in a nonconvex setting and points to extensions to nonsmooth, Minty-type, and generalized equilibrium frameworks.
Abstract
Uniformly regular equilibrium problems are natural generalizations of abstract equilibrium prob lems and they are defined over the uniformly prox-regular nonconvex sets. Some new efficient implicit methods for solving uniformly regular equilibrium problems are analyzed by the aux iliary principle and inertial proximal methods. The convergence analysis of the new proposed methods is considered under some mild conditions. Gap functions are constructed to suggest some descent-type scheme for uniformly regular equilibrium problems. Our results can be viewed as significant refinements and improvements of the previously known results and they continue to hold for equilibrium problems, variational inequalities and complementarity problems as well.