# Additive Schwarz Methods for Convex Optimization as Gradient Methods

arXiv: https://arxiv.org/abs/1912.03617 · HTML: https://mlsanity.com/paper/1912.03617/

## Key facts

- Cited by 4 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 8 Dec 2019.

## Built on by (most cited first, 4 of 4)

- [A Nonoverlapping Domain Decomposition Method for Extreme Learning Machines: Elliptic Problems](https://mlsanity.com/paper/2406.15959/) (2024)
- [Recursive Bound-Constrained AdaGrad with Applications to Multilevel and Domain Decomposition Minimization](https://mlsanity.com/paper/2507.11513/) (2025)
- [DualFL: A Duality-based Federated Learning Algorithm with Communication Acceleration in the General Convex Regime](https://mlsanity.com/paper/2305.10294/) (2023)
- [SVD-Preconditioned Gradient Descent Method for Solving Nonlinear Least Squares Problems](https://mlsanity.com/paper/2602.09057/) (2026)

## Abstract (arXiv)

This paper gives a unified convergence analysis of additive Schwarz methods for general convex optimization problems. Resembling to the fact that additive Schwarz methods for linear problems are preconditioned Richardson methods, we prove that additive Schwarz methods for general convex optimization are in fact gradient methods. Then an abstract framework for convergence analysis of additive Schwarz methods is proposed. The proposed framework applied to linear elliptic problems agrees with the classical theory. We present applications of the proposed framework to various interesting convex optimization problems such as nonlinear elliptic problems, nonsmooth problems, and nonsharp problems.