# On Some Adaptive Mirror Descent Algorithms for Convex and Strongly Convex Optimization Problems with Functional Constraints

## TL;DR

This paper introduces adaptive mirror descent algorithms for convex optimization problems with multiple convex constraints, capable of handling various levels of smoothness, and proposes restart techniques for strongly convex cases, with convergence analysis and numerical validation.

## Contribution

The paper develops adaptive mirror descent algorithms applicable to non-smooth and smooth convex problems with constraints, including restart methods for strongly convex cases, and provides convergence estimates.

## Key findings

- Algorithms effectively handle diverse smoothness levels.
- Proposed methods demonstrate improved convergence rates.
- Numerical experiments confirm practical advantages.

## Abstract

In this paper some adaptive mirror descent algorithms for problems of minimization convex objective functional with several convex Lipschitz (generally, non-smooth) functional constraints are considered. It is shown that the methods are applicable to the objective functionals of various level of smoothness: the Lipschitz condition is valid either for the objective functional itself or for its gradient or Hessian (and the functional may not satisfy the Lipschitz condition). By using the restart technique methods for strongly convex minimization problems are proposed. Estimates of the rate of convergence of the considered algorithms are obtained depending on the level of smoothness of the objective functional. Numerical experiments illustrating the advantages of the proposed methods for some examples are presented.