Outer approximation methods for solving variational inequalities in Hilbert space

In this paper, we study variational inequalities in a real Hilbert space, which are governed by a strongly monotone and Lipschitz continuous operator F over a closed and convex set C. We assume that the set C can be outerly approximated by the fixed point sets of a sequence of certain quasi-nonexpan...

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Veröffentlicht in:Optimization Jg. 66; H. 3; S. 417 - 437
Hauptverfasser: Gibali, Aviv, Reich, Simeon, Zalas, Rafał
Format: Journal Article
Sprache:Englisch
Veröffentlicht: Philadelphia Taylor & Francis 04.03.2017
Taylor & Francis LLC
Schlagworte:
Approximation
Common fixed point
Convergence
Cutters
Hilbert space
Inequalities
iterative method
Iterative methods
Mathematical analysis
Operators
Optimization
Quantum theory
quasi-nonexpansive operator
subgradient projection
variational inequality
ISSN:0233-1934, 1029-4945
Online-Zugang:Volltext
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Zusammenfassung:In this paper, we study variational inequalities in a real Hilbert space, which are governed by a strongly monotone and Lipschitz continuous operator F over a closed and convex set C. We assume that the set C can be outerly approximated by the fixed point sets of a sequence of certain quasi-nonexpansive operators called cutters. We propose an iterative method, the main idea of which is to project at each step onto a particular half-space constructed using the input data. Our approach is based on a method presented by Fukushima in 1986, which has recently been extended by several authors. In the present paper, we establish strong convergence in Hilbert space. We emphasize that to the best of our knowledge, Fukushima's method has so far been considered only in the Euclidean setting with different conditions on F. We provide several examples for the case where C is the common fixed point set of a finite number of cutters with numerical illustrations of our theoretical results.
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ISSN:0233-1934
1029-4945
DOI:10.1080/02331934.2016.1271800