Finding saddle points on convex polyhedrons in problems of minimax optimization with linear constraints

The paper is concerned with finding the maximin of the convex — concave function $\langle\mathbf{p,x}\rangle+\langle\mathbf{x},D\mathbf{y}\rangle+\langle\mathbf{q,y}\rangle$ on a direct product of convex polyhedrons specified as simultaneous systems of linear inequalities in a finite dimensional space; the problem arises in minimax planning optimization in systems described by linear algebraical models. For the polyhedral game associated with the problem finite methods are suggested for finding the saddle points of its payoff functions. The method reduce solution of the original problem to that of linear programming problems with special constraint sets.