{"about":{"site":"https://codewithpapers.app","non_affiliation":"Code with Papers and Syntology are not affiliated with, endorsed by, or sponsored by Papers with Code, Meta, or the pwc-archive mirror.","licence":"CC BY-SA 4.0","licence_url":"https://creativecommons.org/licenses/by-sa/4.0/legalcode","attribution":"https://codewithpapers.app/attribution","modified":"archive material modified by Syntology; see the attribution page"},"url":"/paper/faster-first-order-primal-dual-methods-for","title":"Faster First-Order Primal-Dual Methods for Linear Programming using Restarts and Sharpness","arxiv_id":"2105.12715","date":"2021-05-26","proceeding":null,"authors":["David Applegate","Oliver Hinder","Haihao Lu","Miles Lubin"],"abstract":"First-order primal-dual methods are appealing for their low memory overhead, fast iterations, and effective parallelization. However, they are often slow at finding high accuracy solutions, which creates a barrier to their use in traditional linear programming (LP) applications. This paper exploits the sharpness of primal-dual formulations of LP instances to achieve linear convergence using restarts in a general setting that applies to ADMM (alternating direction method of multipliers), PDHG (primal-dual hybrid gradient method) and EGM (extragradient method). In the special case of PDHG, without restarts we show an iteration count lower bound of $\\Omega(\\kappa^2 \\log(1/\\epsilon))$, while with restarts we show an iteration count upper bound of $O(\\kappa \\log(1/\\epsilon))$, where $\\kappa$ is a condition number and $\\epsilon$ is the desired accuracy. Moreover, the upper bound is optimal for a wide class of primal-dual methods, and applies to the strictly more general class of sharp primal-dual problems. We develop an adaptive restart scheme and verify that restarts significantly improve the ability of PDHG, EGM, and ADMM to find high accuracy solutions to LP problems.","url_abs":"https://arxiv.org/abs/2105.12715v4","url_pdf":"https://arxiv.org/pdf/2105.12715v4.pdf","source":{"archive":"pwc-archive (Hugging Face), CC BY-SA 4.0","snapshot":"2025-07-28","licence_url":"https://creativecommons.org/licenses/by-sa/4.0/legalcode","row_kind":"links_only","authors_date_abstract":"arXiv metadata, CC0 1.0 (https://info.arxiv.org/help/license), from the Kaggle arXiv metadata snapshot of 2026-09-12"},"code_links":[{"paper_slug":"faster-first-order-primal-dual-methods-for","repo_url":"https://github.com/google-research/FirstOrderLp.jl","is_official":1,"mentioned_in_paper":1,"mentioned_in_github":1,"framework":"none","reach":{"status":"ok","spdx":"Apache-2.0"}}],"tasks":[],"methods":[],"datasets_introduced":[],"methods_introduced":[],"results":[],"syntology":{"atlas_url":"https://app.syntology.ai/?focus=2105.12715","mcp":null,"developers":"https://syntology.ai/developers"},"arxiv_metadata":null,"syntology_extracted_results":null}