Variational inequalities and convex semi-infinite programming problems

The present paper is concerned with a general approach for the construction of stable methods solving convex variational problems. This approach uses the procedure of iterative PROX-regularization in connection with suitable methods of sequential discretizations of convex variational inequalities and semi-infinite programming problems. The presented investigation scheme for such methods allows for the establishment of conditions which control the behaviour of the methods and guarantees the strong convergence of the obtained minimizing sequence. The possibility of the realization of this scheme is described for some concrete elliptical variational inequalities and also for some numerical algorithms, where the parameters of discretization and of convergence controlling are coordinated.