Smooth Optimization Methods for Minimax Problems

The classical discrete minimax problem is considered. It is transformed into an equivalent problem by a monotone transformation of the initial functions. It was found that the classical Lagrangian of the equivalent problem has a number of important properties both in primal and dual spaces in convex as well as in nonconvex cases. In particular, the classical Lagrangian of the equivalent problem, being as smooth as the initial functions, has the main advantages of augmented Lagrangians. This makes it possible to construct a multiplier method for the minimax problem and a general method for the simultaneous solution of the primal and the dual problems. These methods are based on the theory of methods of smooth optimization and preserve the main advantages of the latter for nonsmooth minimax problems.