Low-cost modification of Korpelevich’s methods for monotone equilibrium problems

Abstract

A modification of Korpelevich's method with one metric projection onto the feasible set at an iteration step is proposed to solve monotone equilibrium problems. The weak convergence of the modified method is proved.

Convergence of the Modified Extragradient Method for Variational Inequalities with Non-Lipschitz Operators

“…Algorithm 1 is a modification of the subgradient extragradient algorithms considered in [27,28]. The dynamic step size adjustment (4) is described in [22,23].…”

“…These methods were analyzed in many studies [22][23][24][25][26][27][28][29][30][31][32][33][34]. For variational inequalities and equilibrium programming problems, modifications of the Korpelevich algorithm with one metric projection onto feasible set were proposed [27,28]. In these so-called subgradient extragradient algorithms and in the Korpelevich algorithm, the first stages of the iteration coincide, and then, to obtain the next approximation, projection onto some half-space being the support for the feasible set is carried out instead of projection onto the feasible set.…”

“…In these so-called subgradient extragradient algorithms and in the Korpelevich algorithm, the first stages of the iteration coincide, and then, to obtain the next approximation, projection onto some half-space being the support for the feasible set is carried out instead of projection onto the feasible set. In [27,28], the weak convergence of sequences generated by the subgradient extragradient algorithm to some solution of variational inequality is proved. An obvious disadvantage of the algorithm, which impedes its wide use, is the assumption that the Lipschitz constant of the operator is known or admits a simple estimate.…”

An outer approximation algorithm for equilibrium problems in Hilbert spaces

“…Numerical algorithms for solving EP(f, C) have been proposed based on the auxiliary problem principle, the proximal point technique and projections onto the original set or onto approximations; see for instance [2,4,7,9,12,14,15,17] and the references therein.…”

A new hybrid method for solving variational inequalities

“…At present, there exist many efficient modifications of the extragradient method [4][5][6][7][8][9]. The natural question that arises in the case of an infinite-dimensional Hilbert space is how to construct a modified Korpelevich's extragradient algorithm, which will provide the strong convergence.…”