Some Analogue of Quadratic Interpolation for a Special Class of Non-Smooth Functionals and One Application to Adaptive Mirror Descent for Constrained Optimization Problems

Theoretical estimates of the convergence rate of many well-known gradient-type optimization methods are based on quadratic interpolation, provided that the Lipschitz condition for the gradient is satisfied. In this article we obtain a possibility of constructing an analogue of such interpolation in the class of locally Lipschitz quasi-convex functionals with the special conditions of non-smoothness (Lipshitz-continuous subgradient) introduced in this paper. As an application, estimates are obtained for the rate of convergence of the previously proposed adaptive mirror descent method for the problems of minimizing a quasi-convex locally Lipschitz functional with several convex functional constraints.