A study of iterative processes for the solution of ill-posed convex variational problems

The authors study a general approach to the construction of stable methods of solving semi-infinite convex optimization problems and variational inequalities of elliptic type in a situation where the problem is ill-posed. The approach is based on the use of iterative prox-regularization in the usual (for well-posed problems) schemes of successive approximation to the original problem. The authors establish conditions which guarantee the strong convergence of the corresponding minimizing sequences, and they analyze the possibility of their algorithmic realization with regard to estimates of the smoothness of the solution.