Комбинаторно-аналитический метод максимизации негладкой точной нижней границы множества вогнутых гладких функций, зависящих от параметра

We consider systems that operate according a continuous-discrete maximin criterion. The combinatorial-analytic method of maximizing of nonsmooth greatest lower boundary of the finite set of concave smooth functions, depending on the parameter, is proposed. The method is based on necessary conditions for both maxima and intersections of the set of functions. It produces a set of points-applicants that may become the solution of the nonsmooth maximin problem. A combinatorial algorithm for finding solution is constructed. The proposed combinatorial-analytic method for solving discrete-continuous maximin optimization problem reduces the problem of maximizing nondifferentiable lower boundary of a finite set of differentiable concave functions to a finite set of differentiable optimization problems with subsequent solution of the problem of finding lower price of some matrix game. In general, this game has no saddle point (solution in pure strategies), because the lower and upper values of the game do not always coincide. The use of the combinatorial-analytic method is more preferable in comparison with subgradient optimization methods by construction and study of algorithms for solving the maximin problems depending on parameters. An example of solving of the problem of finding of the supremum of the greatest lower boundary of the set of quadratic concave functions is shown.