A First Order Method for Linear Programming Parameterized by Circuit Imbalance

Richard Cole, Christoph Hertrich, Yixin Tao, László A. Végh

Research output: Chapter in Book/Report/Conference proceedingConference contribution


Various first order approaches have been proposed in the literature to solve Linear Programming (LP) problems, recently leading to practically efficient solvers for large-scale LPs. From a theoretical perspective, linear convergence rates have been established for first order LP algorithms, despite the fact that the underlying formulations are not strongly convex. However, the convergence rate typically depends on the Hoffman constant of a large matrix that contains the constraint matrix, as well as the right hand side, cost, and capacity vectors. We introduce a first order approach for LP optimization with a convergence rate depending polynomially on the circuit imbalance measure, which is a geometric parameter of the constraint matrix, and depending logarithmically on the right hand side, capacity, and cost vectors. This provides much stronger convergence guarantees. For example, if the constraint matrix is totally unimodular, we obtain polynomial-time algorithms, whereas the convergence guarantees for approaches based on primal-dual formulations may have arbitrarily slow convergence rates for this class. Our approach is based on a fast gradient method due to Necoara, Nesterov, and Glineur (Math. Prog. 2019); this algorithm is called repeatedly in a framework that gradually fixes variables to the boundary. This technique is based on a new approximate version of Tardos’s method, that was used to obtain a strongly polynomial algorithm for combinatorial LPs (Oper. Res. 1986).

Original languageEnglish (US)
Title of host publicationInteger Programming and Combinatorial Optimization - 25th International Conference, IPCO 2024, Proceedings
EditorsJens Vygen, Jarosław Byrka
PublisherSpringer Science and Business Media Deutschland GmbH
Number of pages14
ISBN (Print)9783031598340
StatePublished - 2024
Event25th International Conference on Integer Programming and Combinatorial Optimization, IPCO 2024 - Wroclaw, Poland
Duration: Jul 3 2024Jul 5 2024

Publication series

NameLecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics)
Volume14679 LNCS
ISSN (Print)0302-9743
ISSN (Electronic)1611-3349


Conference25th International Conference on Integer Programming and Combinatorial Optimization, IPCO 2024


  • Circuit Imbalances
  • First Order Methods
  • Hoffman Proximity
  • Linear Programming

ASJC Scopus subject areas

  • Theoretical Computer Science
  • General Computer Science


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