# Mirror Prox Algorithm for Multi-Term Composite Minimization and Semi-Separable Problems

arXiv: https://arxiv.org/abs/1311.1098 · HTML: https://mlsanity.com/paper/1311.1098/

## Key facts

- Cited by 7 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 5 Nov 2013.

## Built on by (most cited first, 7 of 7)

- [A Conditional Gradient-based Method for Simple Bilevel Optimization with Convex Lower-level Problem](https://mlsanity.com/paper/2206.08868/) (2022)
- [Generalized Optimistic Methods for Convex-Concave Saddle Point Problems](https://mlsanity.com/paper/2202.09674/) (2022)
- [Complementary Composite Minimization, Small Gradients in General Norms, and Applications](https://mlsanity.com/paper/2101.11041/) (2021)
- [Nonconvex-Nonconcave Min-Max Optimization with a Small Maximization Domain](https://mlsanity.com/paper/2110.03950/) (2021)
- [Data-Driven Minimax Optimization with Expectation Constraints](https://mlsanity.com/paper/2202.07868/) (2022)
- [Faster Accelerated First-order Methods for Convex Optimization with Strongly Convex Function Constraints](https://mlsanity.com/paper/2212.11143/) (2022)
- [Federated Composite Saddle Point Optimization](https://mlsanity.com/paper/2305.15643/) (2023)

## Abstract (arXiv)

In the paper, we develop a composite version of Mirror Prox algorithm for solving convex-concave saddle point problems and monotone variational inequalities of special structure, allowing to cover saddle point/variational analogies of what is usually called "composite minimization" (minimizing a sum of an easy-to-handle nonsmooth and a general-type smooth convex functions "as if" there were no nonsmooth component at all). We demonstrate that the composite Mirror Prox inherits the favourable (and unimprovable already in the large-scale bilinear saddle point case) O(1/ε) efficiency estimate of its prototype. We demonstrate that the proposed approach can be naturally applied to Lasso-type problems with several penalizing terms (e.g. acting together ℓ1 and nuclear norm regularization) and to problems of the structure considered in the alternating directions methods, implying in both cases methods with the O(ε-1) complexity bounds.