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

2025/02/28 by Balhag, Aicha, Mazgouri, Zakaria, Riahi, Hassan +1 · 1 citation
#46N10 #49J40 #65K10 #65K15 #90C33 #FOS: Mathematics #Optimization and Control (math.OC)

paper · doi:10.48550/arxiv.2502.20999

Abstract

In this article, we aim to approximate a solution to the bilevel equilibrium problem (BEP) for short: find x ∈ S_f such that g(x, y) ≥ 0, ∀ y ∈ S_f, where S_f = \ u ∈ K : f(u, z) ≥ 0, ∀ z ∈ K \. Here, K is a closed convex subset of a real Hilbert space H, and f and g are two real-valued bifunctions defined on K × K. We propose an inertial version of the proximal splitting algorithm introduced by Z. Chbani and H. Riahi: Weak and strong convergence of prox-penalization and splitting algorithms for bilevel equilibrium problems. 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.

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