On extragradient-viscosity methods for solving equilibrium and fixed point problems in a Hilbert space

Research output: Contribution to journalArticle

Abstract

In this paper, new numerical algorithms are introduced for finding the solution of a variational inequality problem whose constraint set is the common elements of the set of fixed points of a demicontractive mapping and the set of solutions of an equilibrium problem for a monotone mapping in a real Hilbert space. The strong convergence of the iterates generated by these algorithms is obtained by combining a viscosity approximation method with an extragradient method. First, this is done when the basic iteration comes directly from the extragradient method, under a Lipschitz-type condition on the equilibrium function. Then, it is shown that this rather strong condition can be omitted when an Armijo-backtracking linesearch is incorporated into the extragradient iteration. The particular case of variational inequality problems is also examined.

Original languageEnglish
Pages (from-to)429-451
Number of pages23
JournalOptimization
Volume64
Issue number2
DOIs
Publication statusPublished - 1 Jan 2015

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Extragradient Method
Viscosity Method
Fixed Point Problem
Hilbert spaces
Equilibrium Point
Hilbert space
Variational Inequality Problem
Viscosity
Viscosity Approximation Method
Iteration
Monotone Mapping
Backtracking
Line Search
Equilibrium Problem
Strong Convergence
Iterate
Numerical Algorithms
Lipschitz
Fixed point
Equilibrium point

Keywords

  • Armijo-backtracking linesearch
  • demicontractive mapping
  • equilibrium problem
  • extragradient method
  • fixed point problem
  • Lipschitz continuity
  • viscosity approximation method

Cite this

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On extragradient-viscosity methods for solving equilibrium and fixed point problems in a Hilbert space. / Vuong, Phan Tu; Strodiot, Jean Jacques; Nguyen, Van Hien.

In: Optimization, Vol. 64, No. 2, 01.01.2015, p. 429-451.

Research output: Contribution to journalArticle

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