Implementation of Interior-point Methods for LP based on Krylov Subspace Iterative Solvers with Inner-iteration Preconditioning

TL;DR

This paper introduces a robust interior-point method for linear programming that leverages inner-iteration preconditioned Krylov subspace solvers, effectively handling ill-conditioned problems without preprocessing and outperforming standard solvers on large instances.

Contribution

The paper proposes a novel interior-point algorithm using inner-iteration preconditioned Krylov methods, eliminating the need for preprocessing and improving robustness on large, ill-conditioned LP problems.

Findings

Successfully solved LP problems with up to 434,580 unknowns.

Outperformed standard solvers like SeDuMi, SDPT3, and PDCO in robustness.

Demonstrated effectiveness on diverse benchmark collections.

Abstract

We apply novel inner-iteration preconditioned Krylov subspace methods to the interior-point algorithm for linear programming (LP). Inner-iteration preconditioners recently proposed by Morikuni and Hayami enable us to overcome the severe ill-conditioning of linear equations solved in the final phase of interior-point iterations. The Krylov subspace methods do not suffer from rank-deficiency and therefore no preprocessing is necessary even if rows of the constraint matrix are not linearly independent. By means of these methods, a new interior-point recurrence is proposed in order to omit one matrix-vector product at each step. Extensive numerical experiments are conducted over diverse instances of 138 LP problems including the Netlib, QAPLIB, Mittelmann and Atomizer Basis Pursuit collections. The largest problem has 434,580 unknowns. It turns out that our implementation is more robust than the standard public domain solvers SeDuMi (Self-Dual Minimization), SDPT3 (Semidefinite Programming Toh-Todd-T\"ut\"unc\"u) and the LSMR iterative solver in PDCO (Primal-Dual Barrier Method for Convex Objectives) without increasing CPU time. The proposed interior-point method based on iterative solvers succeeds in solving a fairly large number of LP instances from benchmark libraries under the standard stopping criteria. The work also presents a fairly extensive benchmark test for several renowned solvers including direct and iterative solvers.