TY - JOUR

T1 - On the Convergence of Inexact Predictor-Corrector Methods for Linear Programming

AU - Dexter, Gregory

AU - Chowdhury, Agniva

AU - Avron, Haim

AU - Drineas, Petros

N1 - Publisher Copyright:
Copyright © 2022 by the author(s)

PY - 2022

Y1 - 2022

N2 - 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.

AB - 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.

UR - http://www.scopus.com/inward/record.url?scp=85163121829&partnerID=8YFLogxK

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AN - SCOPUS:85163121829

SN - 2640-3498

VL - 162

SP - 5007

EP - 5038

JO - Proceedings of Machine Learning Research

JF - Proceedings of Machine Learning Research

T2 - 39th International Conference on Machine Learning, ICML 2022

Y2 - 17 July 2022 through 23 July 2022

ER -