ADAPTIVE OPERATOR EXTRAPOLATION METHOD FOR VARIATIONAL INEQUALITIES IN BANACH SPACES

Abstract

Many problems of operations research and mathematical physics can be formulated in the form of variational inequalities. The development and research of algorithms for solving variational inequalities is an actively developing area of applied nonlinear analysis. Note that often nonsmooth optimization problems can be effectively solved if they are reformulated in the form of saddle point problems and algorithms for solving variational inequalities are applied. Recently, there has been progress in the study of algorithms for problems in Banach spaces. This is due to the wide involvement of the results and constructions of the geometry of Banach spaces. A new algorithm for solving variational inequalities in a Banach space is proposed and studied. In addition, the Alber generalized projection is used instead of the metric projection onto the feasible set. An attractive feature of the algorithm is only one computation at the iterative step of the projection onto the feasible set. For variational inequalities with monotone Lipschitz operators acting in a 2-uniformly convex and uniformly smooth Banach space, a theorem on the weak convergence of the method is proved.