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

Vuong, Phan Tu, Strodiot, Jean Jacques and Nguyen, Van Hien (2015) On extragradient-viscosity methods for solving equilibrium and fixed point problems in a Hilbert space. Optimization, 64 (2), 429-451. (doi:10.1080/02331934.2012.759327).

Record type: Article

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.

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