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

TL;DR

The paper addresses BEP in a real Hilbert space by developing an inertial proximal splitting algorithm (IPSA) that extends prior splitting methods to a bilevel setting. It leverages the Fitzpatrick transform and resolvent concepts to obtain weak convergence under a discrete geometric condition, and strengthens results to strong convergence under monotonicity and standard step-size assumptions; a generalization to class (S_+) bifunctions further broadens applicability. A numerical example in $\mathbb{R}^5$ validates the theoretical findings and shows the impact of inertial and penalization parameters on convergence. The work contributes a practical, convergent scheme for hierarchical equilibrium problems with two levels of bifunctions, broadening the toolbox for nonlinear variational inequalities and optimization with equilibrium constraints. Overall, IPSA provides a robust approach to solving BEP with provable convergence guarantees and demonstrated performance improvements.

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.

Figures (1)

• Figure 1: The rate of convergence of $\|x_n - \bar{x}\|_2$.

Theorems & Definitions (16)

• Definition 2.1
• Lemma 2.2: Minty's Lemma, BO
• Lemma 2.3
• Remark 1
• Lemma 3.1
• Corollary 3.2
• Theorem 3.3
• Theorem 3.4
• Theorem 3.5
• Remark 2