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

## TL;DR

This paper develops a new interpolation technique for a class of non-smooth quasi-convex functionals and applies it to analyze the convergence of an adaptive mirror descent method for constrained optimization.

## Contribution

It introduces an analogue of quadratic interpolation for non-smooth quasi-convex functionals with specific non-smoothness conditions.

## Key findings

- Derived convergence rate estimates for the adaptive mirror descent method.
- Extended interpolation techniques to locally Lipschitz quasi-convex functionals.
- Provided theoretical foundations for optimization methods under weaker smoothness assumptions.

## Abstract

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.