Abstract

The variational inequality problem provides a unified framework for a large family of problems such as complementarity problems, minimization problems, Nash games, vector minimization problems, to name a few. Although a large number of solution algorithms have been developed for this problem, there is still a wide scope for improvement and a need for extensive additional research in this realm. In particular, efficient and convergent algorithms for solving such problems are still being sought. With above motivations, we introduce two algorithms for solving the variational inequality problem with pseudomonotone operators in Banach spaces which consist of two principal steps and admit an interesting geometric interpretation. In the first algorithm, at a certain iterate, an auxiliary point is first computed through a line search. Using this auxiliary point and the previous iterate, we construct an appropriate hyperplane separating the solution set of the given problem from the current iterate. Considering this information, the next iterate is then defined as the Bregman projection of a certain point onto the feasible set of the problem. Assuming standard hypotheses, we show that the generated sequence by the algorithm weakly and globally converges to some solution of the problem. In the second algorithm, we study the issue of upgrading from weak to strong convergence, for which the Bregman projection step in the above algorithm is replaced by the Bregman projection of the initial point onto the intersection of the feasible set of the problem and two proper halfspaces, built at the current iterate. Besides the above weak convergence result versus the strong one, the latter algorithm has a distinctive feature meaning that the generated sequence strongly and globally converges to the closest solution of the problem to the initial point which is of interest even in finite dimensional spaces.



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