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

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