On mirror descent methods for some types of composite optimization problems with functional constraints

The paper is devoted to some methods of mirror descent and theoretical estimates of their rate of convergence for problems of convex composite optimization, where the functionals $f$ and $g$ are convex, and $r$ and $\nu$ have a simple structure. On the class of Lipschitz functionals, we propose a modified mirror descent algorithm with adaptively chosen steps and a stopping criterion. The mirror descent operator is defined as standart. In this case, both problems with Lipschitz functionalities and problems with functionalities that satisfy substantially more general Lipschitz «relative Lipschitz property» to some not necessarily strongly convex prox function, which was recently proposed by Y.\;E.\;Nesterov and H.\;Lu. For example, this condition is applicable to such applied problems as Support Vector Machine, Truss Topology Design, and to the problem of finding a common point for a system of ellipsoids. For a more general class of relatively Lipschitz problems, a method with constant steps but with an adaptive stopping criterion is proposed. This made it possible to obtain estimates for the rate of convergence of the mirror descent method that are optimal for the class of problems with relatively Lipschitz objective functionals and constraint functionals. In paper was also considered a generalization of the last result to the case of the assumption that $\delta$-subgradients of functionals are available instead of ordinary subgradients, and an estimate for the corresponding mirror descent algorithm is obtained.