On the Convergence of Inexact Predictor-Corrector Methods for Linear Programming

Gregory Dexter*, Agniva Chowdhury*, Haim Avron*, Petros Drineas*

*Corresponding author for this work

Research output: Contribution to journalConference articlepeer-review

1 Scopus citations

Abstract

Interior point methods (IPMs) are a common approach for solving linear programs (LPs) with strong theoretical guarantees and solid empirical performance. The time complexity of these methods is dominated by the cost of solving a linear system of equations at each iteration. In common applications of linear programming, particularly in machine learning and scientific computing, the size of this linear system can become prohibitively large, requiring the use of iterative solvers, which provide an approximate solution to the linear system. However, approximately solving the linear system at each iteration of an IPM invalidates the theoretical guarantees of common IPM analyses. To remedy this, we theoretically and empirically analyze (slightly modified) predictor-corrector IPMs when using approximate linear solvers: our approach guarantees that, when certain conditions are satisfied, the number of IPM iterations does not increase and that the final solution remains feasible. We also provide practical instantiations of approximate linear solvers that satisfy these conditions for special classes of constraint matrices using randomized linear algebra.

Original languageEnglish
Pages (from-to)5007-5038
Number of pages32
JournalProceedings of Machine Learning Research
Volume162
StatePublished - 2022
Event39th International Conference on Machine Learning, ICML 2022 - Baltimore, United States
Duration: 17 Jul 202223 Jul 2022

Funding

FundersFunder number
BSF2017698
Department of Statistics
NSF
Office of Advanced Scientific Computing Research
U.S. Department of Energy
UT-Battelle LLCDE-AC05-00OR22725
National Science FoundationAF 1814041, FRG 1760353, DOE-SC0022085
U.S. Department of Energy
Bloom's Syndrome Foundation
Advanced Scientific Computing Research
Purdue University
UT-Battelle

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