# Mirror Descent and Convex Optimization Problems With Non-Smooth Inequality Constraints

arXiv: https://arxiv.org/abs/1710.06612 · HTML: https://mlsanity.com/paper/1710.06612/

## Key facts

- Cited by 10 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 18 Oct 2017.

## Built on by (most cited first, 10 of 10)

- [Stochastic Mirror Descent: Convergence Analysis and Adaptive Variants via the Mirror Stochastic Polyak Stepsize](https://mlsanity.com/paper/2110.15412/) (2021)
- [Low-rank Optimal Transport: Approximation, Statistics and Debiasing](https://mlsanity.com/paper/2205.12365/) (2022)
- [Oracle Complexity of Single-Loop Switching Subgradient Methods for Non-Smooth Weakly Convex Functional Constrained Optimization](https://mlsanity.com/paper/2301.13314/) (2023)
- [A single-loop SPIDER-type stochastic subgradient method for expectation-constrained nonconvex nonsmooth optimization](https://mlsanity.com/paper/2501.19214/) (2025)
- [First-Order Methods for Convex Optimization](https://mlsanity.com/paper/2101.00935/) (2021)
- [Primal-Dual Stochastic Mirror Descent for MDPs](https://mlsanity.com/paper/2103.00299/) (2021)
- [Damped Proximal Augmented Lagrangian Method for weakly-Convex Problems with Convex Constraints](https://mlsanity.com/paper/2311.09065/) (2023)
- [GBM-based Bregman Proximal Algorithms for Constrained Learning](https://mlsanity.com/paper/2308.10767/) (2023)
- [Some Primal-Dual Theory for Subgradient Methods for Strongly Convex Optimization](https://mlsanity.com/paper/2305.17323/) (2023)
- [Mirror Descent-Type Algorithms for the Variational Inequality Problem with Functional Constraints](https://mlsanity.com/paper/2605.16262/) (2026)

## Abstract (arXiv)

We consider the problem of minimization of a convex function on a simple set with convex non-smooth inequality constraint and describe first-order methods to solve such problems in different situations: smooth or non-smooth objective function; convex or strongly convex objective and constraint; deterministic or randomized information about the objective and constraint. We hope that it is convenient for a reader to have all the methods for different settings in one place. Described methods are based on Mirror Descent algorithm and switching subgradient scheme. One of our focus is to propose, for the listed different settings, a Mirror Descent with adaptive stepsizes and adaptive stopping rule. This means that neither stepsize nor stopping rule require to know the Lipschitz constant of the objective or constraint. We also construct Mirror Descent for problems with objective function, which is not Lipschitz continuous, e.g. is a quadratic function. Besides that, we address the problem of recovering the solution of the dual problem.