In applying methods of operations research to economics, planning, control, optimization of complex technical systems, optimal design, and to other areas it frequently becomes necessary to solve minimax problems of the following form: \[ (1)\quad \min_{u\in U}\max_{x\in X}f(x,u),\quad U\subset E^ m,\quad X\subset E^ n. \] Effective computational methods for solving (1) have only been developed for the case when one can solve the inner problem comparatively easily, for example, the set of X is convex and the function f(x,u) is convex in x. Now if the inner problem max f(x,u) for fixed u has many local extrema, the existence of numerical methods encounters serious difficulties which cannot always be overcome. At the same time minimax problems with nonconvex inner problems occur rather frequently. In the present paper we offer a stochastic numerical method of solution of (1) in the case of a multiextremal inner problem of maximization, and we prove its convergence almost surely. The method has proved its effectiveness in solving a series of practical problems.