# Mirror Descent for Constrained Optimization Problems with Large Subgradient Values

arXiv: https://arxiv.org/abs/1908.00218 · HTML: https://mlsanity.com/paper/1908.00218/

## Key facts

- Cited by 3 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 1 Aug 2019.

## Built on by (most cited first, 3 of 3)

- [Oracle Complexity of Single-Loop Switching Subgradient Methods for Non-Smooth Weakly Convex Functional Constrained Optimization](https://mlsanity.com/paper/2301.13314/) (2023)
- [First-Order Softmax Weighted Switching Gradient Method for Distributed Stochastic Minimax Optimization with Stochastic Constraints](https://mlsanity.com/paper/2603.05774/) (2026)
- [Unified High-Probability Analysis of Stochastic Variance-Reduced Estimation](https://mlsanity.com/paper/2605.15388/) (2026)

## Abstract (arXiv)

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.