{"title":"Worst-case analysis of restarted primal-dual hybrid gradient on totally unimodular linear programs","authors":"","doi":"10.1016/j.orl.2024.107199","DOIUrl":null,"url":null,"abstract":"<div><div>We analyze restarted PDHG on totally unimodular linear programs. In particular, we show that restarted PDHG finds an <em>ϵ</em>-optimal solution in <span><math><mi>O</mi><mo>(</mo><mi>H</mi><msubsup><mrow><mi>m</mi></mrow><mrow><mn>1</mn></mrow><mrow><mn>2.5</mn></mrow></msubsup><msqrt><mrow><mtext>nnz</mtext><mo>(</mo><mi>A</mi><mo>)</mo></mrow></msqrt><mi>log</mi><mo></mo><mo>(</mo><mi>H</mi><msub><mrow><mi>m</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>/</mo><mi>ϵ</mi><mo>)</mo><mo>)</mo></math></span> matrix-vector multiplies where <span><math><msub><mrow><mi>m</mi></mrow><mrow><mn>1</mn></mrow></msub></math></span> is the number of constraints, <span><math><msub><mrow><mi>m</mi></mrow><mrow><mn>2</mn></mrow></msub></math></span> the number of variables, <span><math><mtext>nnz</mtext><mo>(</mo><mi>A</mi><mo>)</mo></math></span> is the number of nonzeros in the constraint matrix, <em>H</em> is the largest absolute coefficient in the right hand side or objective vector, and <em>ϵ</em> is the distance to optimality of the outputted solution.</div></div>","PeriodicalId":54682,"journal":{"name":"Operations Research Letters","volume":null,"pages":null},"PeriodicalIF":0.8000,"publicationDate":"2024-10-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Operations Research Letters","FirstCategoryId":"91","ListUrlMain":"https://www.sciencedirect.com/science/article/pii/S0167637724001354","RegionNum":4,"RegionCategory":"管理学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q4","JCRName":"OPERATIONS RESEARCH & MANAGEMENT SCIENCE","Score":null,"Total":0}
引用次数: 0
Abstract
We analyze restarted PDHG on totally unimodular linear programs. In particular, we show that restarted PDHG finds an ϵ-optimal solution in matrix-vector multiplies where is the number of constraints, the number of variables, is the number of nonzeros in the constraint matrix, H is the largest absolute coefficient in the right hand side or objective vector, and ϵ is the distance to optimality of the outputted solution.
期刊介绍：
Operations Research Letters is committed to the rapid review and fast publication of short articles on all aspects of operations research and analytics. Apart from a limitation to eight journal pages, quality, originality, relevance and clarity are the only criteria for selecting the papers to be published. ORL covers the broad field of optimization, stochastic models and game theory. Specific areas of interest include networks, routing, location, queueing, scheduling, inventory, reliability, and financial engineering. We wish to explore interfaces with other fields such as life sciences and health care, artificial intelligence and machine learning, energy distribution, and computational social sciences and humanities. Our traditional strength is in methodology, including theory, modelling, algorithms and computational studies. We also welcome novel applications and concise literature reviews.
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