Modified basic projection methods for a class of equilibrium problems (Q2413497)

The equilibrium problem considered in the paper is formulated as follows: \[ \text{ Find }x^* \in C \subset \mathbb R^s\text{ such that }f(x^*,y) \geq 0\quad \forall y \in C, \] where \(C\) is a closed convex set, \(f: \mathbb R^s \times \mathbb R^s ~\longrightarrow~ \mathbb R \cup \{ +\infty \},~ f(x,x) = 0 ~\forall x \in\mathbb R^s\). The purpose of the paper is to extend the projection and subgradient methods published in the literature. The extension consists in ``computing an approximate subgradient of a subdifferentiable convex function and an orthogonal projection onto separating hyperplanes''. Convergence of the iteration sequence obtained by combining projection and subgradient methods to the solution of the equilibrium problem is established. Some illustrating numerical examples are presented in the last section of the paper.