Solving strongly convex-concave composite saddle point problems with a small dimension of one of the variables.

The article is devoted to the development of algorithmic methods for strongly convex-concave saddle-point problems in the case when one of the groups of variables has a large dimension, and the other is sufficiently small (up to a hundred). The proposed technique is based on reducing problems of this type to a problem of minimizing a convex (maximizing a concave) functional in one of the variables, for which it is possible to find an approximate gradient at an arbitrary point with the required accuracy using an auxiliary optimization subproblem with another variable. In this case for low-dimensional problems the ellipsoid method is used (if necessary, with an inexact $\delta$-subgradient), and for multidimensional problems accelerated gradient methods are used. For the case of a very small dimension of one of the groups of variables (up to 5) there is proposed an approach based on a new version of the multidimensional analogue of the Yu. E. Nesterov method on the square (multidimensional dichotomy) with the possibility of using inexact values of the gradient of the objective functional.