Proximal point algorithm extended to equilibrium problems

Abstract : The author proposes a generalized proximal method for solving an equilibrium problem which consists in finding x 2 K such that F(x, y) 0, for all y 2 K, where K is a nonempty, convex and closed set of a real Hilbert space X, and F:K ×K !Ris a given bifunction with F(x, x) = 0 for all x 2 K. The weak convergence of the sequence generated by the method is proved under the assumptions of monotonicity and convexity, with respect to the second argument y (for every fixed x 2 K), of the bifunction F. Replacing the assumption of monotonicity with the one of strong monotonicity on F, a strong convergence result is obtained. A second strong convergence theorem is proved under the hypotheses of monotonicity and a new assumption of "co-Lipschitz continuity at 0" of the operator F. Applications to convex optimization, to the problem of finding a zero of a maximal monotone operator and to Nash equilibria problems are provided.
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https://hal.univ-antilles.fr/hal-00773194
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Submitted on : Friday, January 11, 2013 - 7:48:27 PM
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  • HAL Id : hal-00773194, version 1

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Abdellatif Moudafi. Proximal point algorithm extended to equilibrium problems. Journal of Natural Geometry, 1999, 15 (1-2), pp.91-100. ⟨hal-00773194⟩

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