An inertial proximal splitting algorithm for hierarchical bilevel equilibria in Hilbert spaces

TL;DR

This paper introduces an inertial proximal splitting algorithm to solve hierarchical bilevel equilibrium problems in Hilbert spaces, proving convergence and demonstrating its effectiveness through a numerical example.

Contribution

It develops a novel inertial proximal splitting method for bilevel equilibria in Hilbert spaces, extending previous algorithms with convergence guarantees.

Findings

The algorithm converges weakly and strongly under certain conditions.

Numerical results validate the theoretical convergence.

The method effectively solves hierarchical bilevel equilibrium problems.

Abstract

In this article, we aim to approximate a solution to the bilevel equilibrium problem $\mathbf{(BEP})$ for short: find $\bar{x} \in \mathbf{S}_f$ such that $ g(\bar{x}, y) \geq 0, \,\, \forall y \in \mathbf{S}_f, $ where $ \mathbf{S}_f = \{ u \in \mathbf{K} : f(u, z) \geq 0, \forall z \in \mathbf{K} \}. $ Here, $\mathbf{K}$ is a closed convex subset of a real Hilbert space $\mathcal{H}$, and $f$ and $g$ are two real-valued bifunctions defined on $\mathbf{K} \times \mathbf{K}$. We propose an inertial version of the proximal splitting algorithm introduced by Z. Chbani and H. Riahi: \textit{Weak and strong convergence of prox-penalization and splitting algorithms for bilevel equilibrium problems}. \textit{Numer. Algebra Control Optim.}, 3 (2013), pp. 353-366. Under suitable conditions, we establish the weak and strong convergence of the sequence generated by the proposed iterative method. We also report a numerical example illustrating our theoretical result.