Weak and strong convergence of prox-penalization and splitting algorithms  for   bilevel equilibrium problems

Zaki Chbani; Hassan Riahi

doi:10.3934/naco.2013.3.353

Article Contents

2013, Volume 3, Issue 2: 353-366. Doi: 10.3934/naco.2013.3.353

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Weak and strong convergence of prox-penalization and splitting algorithms for bilevel equilibrium problems

Zaki Chbani^1, and
Hassan Riahi^1,

1.
Cadi Ayyad University, Faculty of Sciences Semlalia, Mathematics, 40000 Marrakech

Received: March 31, 2012

Revised: October 31, 2012

Published: March 2013

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Abstract

The aim of this paper is to obtain, in a Hilbert space $H$, the weak and strong convergence of a penalty proximal algorithm and a splitting one for a bilevel equilibrium problem: find $ x\in S_F $ such that $\ G(x,y)\geq 0\ $ for all $ \ y\in S_F$, where $S_F :=\lbrace y\in K\; :\; F(y,u)\geq 0\;\; \forall u\in K \rbrace$, and $F,G:K\times K\longrightarrow \mathbb{R}$ are two bifunctions with $K$ a nonempty closed convex subset of $H$. In our framework, results of convergence generalize those recently obtained by Attouch et al. (SIAM Journal on Optimization 21, 149-173 (2011)). We show in particular that for the strong convergence of the penalty algorithm, the geometrical condition they impose is not required. We also give applications of the iterative schemes to fixed point problems and variational inequalities.

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Mathematics Subject Classification: Primary: 46N10, 65K15, 90C33; Secondary: 47H05, 65K10.

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