# Mirror Descent for Constrained Optimization Problems with Large Subgradient Values

## TL;DR

This paper introduces novel Mirror Descent strategies with adaptive step-sizes and stopping rules for constrained optimization problems involving non-smooth, Holder-continuous functionals, achieving optimal convergence rates across various smoothness levels.

## Contribution

It proposes new adaptive Mirror Descent algorithms with optimal convergence guarantees for non-smooth and strongly convex functionals under constraints.

## Key findings

- Algorithms are applicable to various smoothness levels of the objective.
- The methods achieve optimal convergence rates according to lower oracle bounds.
- An optimal restart technique is developed for strongly convex objectives.

## Abstract

Based on the ideas of arXiv:1710.06612, we consider the problem of minimization of the Holder-continuous non-smooth functional $f$ with non-positive convex (generally, non-smooth) Lipschitz-continuous functional constraint. We propose some novel strategies of step-sizes and adaptive stopping rules in Mirror Descent algorithms for the considered class of problems. It is shown that the methods are applicable to the objective functionals of various levels of smoothness. Applying the restart technique to the Mirror Descent Algorithm there was proposed an optimal method to solve optimization problems with strongly convex objective functionals. Estimates of the rate of convergence of the considered algorithms are obtained depending on the level of smoothness of the objective functional. These estimates indicate the optimality of considered methods from the point of view of the theory of lower oracle bounds. In addition, the case of a quasi-convex objective functional and constraint was considered.