Using incomplete indefinite $LDL^T$ preconditioning for inexact interior point methods for linear programming

TL;DR

This paper introduces an efficient inexact interior point method for linear programming that avoids forming the normal equation matrix and uses iterative solutions with an indefinite preconditioner, reducing computational costs.

Contribution

It proposes a novel approach using incomplete indefinite $LDL^T$ preconditioning and iterative solvers for inexact interior point methods, avoiding explicit formation of the normal matrix.

Findings

Preliminary numerical results are promising.

The method reduces computational cost and memory usage.

Inexact solutions with low accuracy are effective in this context.

Abstract

Most linear algebra kernels in interior point methods for linear programming require the solution of linear systems of equation with the matrix $N = A^TD^{-1}A$ (or $AD^{-1}A^T$), where $A$ denotes the constraint matrix of the linear program. This matrix $N$ arises from the reduced KKT system by block elimination. If the number of non-zeros in $N$ or in its Cholesky factorization $N= LL^T$ is very large, the computational cost and memory requirement to solve the linear systems of equations with $N$ may be prohibitively large. In this work we implement an interior point method described by R. Freund and F. Jarre. Forming the normal equation matrix $N$ is avoided altogether and we work with the reduced KKT system instead. We solve the linear systems for the Newton directions iteratively only to low accuracy using SQMR and an indefinite multilevel preconditioner. Preliminary numerical results are encouraging.