Cole, Richard, Hertrich, Christoph 
ORCID: 0000-0001-5646-8567, Tao, Yixin and Végh, László A. ORCID: 0000-0003-1152-200X
(2024)
A first order method for linear programming parameterized by circuit imbalance.
In: Vygen, Jens and Byrka, Jarosław, (eds.)
Integer Programming and Combinatorial Optimization - 25th International Conference, IPCO 2024, Proceedings.
Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics).
Springer Science and Business Media Deutschland GmbH, pp. 57-70.
ISBN 9783031598340
Abstract
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).
| Item Type: | Book Section |
|---|---|
| Additional Information: | Publisher Copyright: © The Author(s), under exclusive license to Springer Nature Switzerland AG 2024. |
| Divisions: | Mathematics |
| Date Deposited: | 17 Jun 2024 16:00 |
| Last Modified: | 24 Jun 2024 23:14 |
| URI: | http://eprints.lse.ac.uk/id/eprint/123897 |
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