Combinatorial-analyticalmethod for maximizing of nonsmooth greatest lower boundary of the set of smooth concavefunctions depending on the parameter

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

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