Non-smooth analysis and applications

The talk deals with the nonsmooth analysis of nondifferentiable functions. Methods of approximation of nonsmooth functions and multivalued mappings are studied. The second mixed derivative of the reference function of Lipschitzian multivalued mappings is introduced, through which the kind of cones of multivalued mappings is written. A new first- and second-order subdifferential is introduced, using which the necessary and sufficient conditions for optimality at a point are written down and a continuous extension of the Clark subdifferential is constructed. Using the first and second order subdifferentials, the first order codifferentials are found. Under sufficiently general conditions, the continuity of the codifferential is proved. New methods of optimization of nonsmooth functions are developed, which find applications in game theory, economics, and medicine. Necessary and sufficient conditions of representability of an arbitrary function of two variables as a difference of convex ones are given, which is important for answering the question of quasi-differentiability of a function of two variables.