About the estimation of the convergence rate of projection-iteration processes of conditional minimization of a functional

Abstract

We study projection-iterative processes based on the conditional gradient method to solve the problem of minimizing a functional in a real separable Hilbert space. To solve extremal problems, methods of approximate (projection) type are often used, which make it possible to replace the initial problem by a sequence of auxiliary approximating extremal problems. The work of many authors is devoted to the problems of approximating various classes of extremal problems. Investigations of projection and projection-iteration methods for solving extremal problems with constraints in Hilbert and reflexive Banach spaces were carried out, in particular, in the works of S.D. Balashova, in which the general conditions for approximation and convergence of sequences of exact and approximate solutions of approximating extremal problems considered both in subspaces of the original space and in certain spaces isomorphic to them were proposed. The projection-iterative approach to the approximate solution of an extremal problem is based on the possibility of applying iterative methods to the solution of approximating problems. Moreover, for each of the "approximate" extremal problems, only a few approximations are obtained with the help of a certain iteration method and the last of them as the initial approximation for the next "approximate" problem is used. This paper, in continuation of the author's past work to solve the problem of minimizing a functional on a convex set of Hilbert space, is devoted to obtaining theoretical estimates of the rate of convergence of the projection-iteration method based on the conditional gradient method (for different ways of specifying a step multiplier) of minimization of approximating functionals in certain spaces isomorphic to subspaces of the original space. We prove theorems on the convergence of a projection-iteration method and obtain estimates of error and convergence degree