Weak and strong convergence of an inertial proximal method for solving bilevel monotone equilibrium problems

TL;DR

This paper proposes an inertial proximal algorithm for bilevel monotone equilibrium problems in Hilbert spaces, proving its convergence and demonstrating its effectiveness through examples and numerical tests.

Contribution

It introduces a novel inertial proximal method for bilevel equilibrium problems, establishing convergence without restrictive assumptions and extending previous results.

Findings

Proved weak and strong convergence of the proposed method.

Applied the method to hierarchical minimization and saddle point problems.

Provided numerical example demonstrating algorithm implementability.

Abstract

In this paper, we introduce an inertial proximal method for solving a bilevel problem involving two monotone equilibrium bifunctions in Hilbert spaces. Under suitable conditions and without any restrictive assumption on the trajectories, the weak and strong convergence of the sequence generated by the iterative method are established. Two particular cases illustrating the proposed method are thereafter discussed with respect to hierarchical minimization problems and equilibrium problems under saddle point constraint. Furthermore, a numerical example is given to demonstrate the implementability of our algorithm. The algorithm and its convergence results improve and develop previous results in the field.