Gradient approach of computing fixed points of equilibrium problems
type: Item
instance of: scholarly article
title: Gradient approach of computing fixed points of equilibrium problems
zbMATH Open document ID: 1056.91001
DOI: 10.1023/A:1020321209606
published in: Journal of Global Optimization
publication date: 2003-06-09T00:00:00Z
review text: The author considers the general equilibrium problem: Find \(v^*\in\Omega_0\) such that \[ \Phi(v^*,w)\geq\Phi (v^*, v^*) \quad \forall w \in \Omega _{0}, \] where \(\Phi : \mathbb{R}^{n} \times \mathbb{R}^{n} \rightarrow \mathbb{R}\) is a differentiable bifunction which is convex in the second variable and \( \Omega _{0}\) is a convex closed subset of \(\mathbb{R}^n\). Then it is equivalent to the variational inequality which is to find \(v^* \in \Omega _{0}\) such that \[ \langle F (v^*), w-v^* \rangle \geq 0 \quad \forall w \in \Omega _{0}, \] where \(F(v) = \nabla _{w} \left.\Phi (v,w) \right | _{w=v}\). The author considers several conditions on \(\Phi \) and the Jacobian of \(F\) which provide the integrability of \(F\) and the convergence of the extragradient method. Its convergence is also proved under monotonicity type assumptions on \(\Phi \). Some rates of convergence, which are similar to the previous ones, see e.g. \textit{A. S. Antipin} [Comput. Math. Math. Phys. 35, 539--551 (1995; Zbl 0852.65046)], are also given.
reviewed by: Igor V. Konnov
Mathematics Subject Classification ID: 91A10
Mathematics Subject Classification ID: 90C47
Mathematics Subject Classification ID: 49J35
zbMATH DE Number: 1924945
zbMATH Keywords: extragradient method
zbMATH Keywords: potential mapping
author: Anatoliǐ Sergeevich Antipin
MaRDI profile type: Publication
full work available at URL: a:1020321209606
OpenAlex ID: W1559058103
description: scientific article; zbMATH DE number 1924945