  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.
