Pseudomonotone Variational Inequalities: Convergence Of Proximal Point Algorithm

In this paper, we study the convergence of proximal point algo-rithm(PPA) for solving pseudomonotone variational inequalities. The sub-problems of classical PPA use the square of norm to be the auxiliary func-tion.In part one, we add a differetialble and strongly convex function to substitute for the square norm, then we study the general convergence result based on pseudomonotonicity variational inequalities in a finite-dimensinal space and an infinite-dimensinal Hilbert space. In part two, we propose a modified Bregman-function-based PPA for solving variational inequalities problems. The aglorthm adopts a similar constructive approximate criterion as the one developed by Solodov and Svater for solving the classical vari-ational inequalities problems. Under some suitable conditions, we can get an approximate solution satisfying the accuracy criterion via single Newton-type step.