Comparative analysis of the extragradient methods for solution of the variational inequalities of some problems

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

“…The convergence without problem modification is provided in iterative extragradient methods first proposed by Korpelevich in [21]. 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].…”

A family of extragradient methods for solving equilibrium problems

“…A convergence theorem is given without proof and computational results are presented to show the efficiency of the two-step extragradient method for solving linear programming problems with filled (without zeroes) matrices of dimension 10 to 50. The proof of convergence appeared in 2012 with some other numerical examples in [20]. In that paper, the authors proved that the distance between the iterate x k and the solution x * is sufficiently decreasing at each iteration.…”

“…Notice that when α k = 0, the vectors xk and x k coincide and the corresponding iteration becomes the classical extragradient iteration. On the other hand, when α k = β k = β > 0, the iteration corresponds to the two-step extragradient iteration [20]. The aim of the paper is twofold.…”

“…When α k = 0 for all k, the corresponding method coincides with the classical extragradient method used for solving an equilibrium problem. When α k = β k = β > 0 for all k, and the variational inequality problem is considered, the method corresponds to the two-step extragradient method of [20]. The convergence of the new methods is deduced following the value of the parameters α k and β k , and under a Lipschitz continuity assumption on f .…”

Finite number of iterations in the two-step extragradient method

“…From theorem in [14] about the convergence with respect to the norm of two-step extragradient method (2) for variational inequalities we obtain…”