An asymptotically superlinearly convergent semismooth Newton augmented Lagrangian method for Linear Programming

TL;DR

This paper introduces SNIPAL, a semismooth Newton augmented Lagrangian method that achieves superlinear convergence for large-scale linear programming, especially effective with dense data matrices where traditional interior-point methods struggle.

Contribution

The paper proposes SNIPAL, a novel semismooth Newton-based inexact proximal augmented Lagrangian method that converges superlinearly and handles dense LP problems more efficiently than existing solvers.

Findings

SNIPAL demonstrates superlinear convergence in large-scale LPs.

Numerical results show SNIPAL outperforms Gurobi on dense LP problems.

SNIPAL effectively manages dense data matrices with less computational cost.

Abstract

Powerful interior-point methods (IPM) based commercial solvers, such as Gurobi and Mosek, have been hugely successful in solving large-scale linear programming (LP) problems. The high efficiency of these solvers depends critically on the sparsity of the problem data and advanced matrix factorization techniques. For a large scale LP problem with data matrix $A$ that is dense (possibly structured) or whose corresponding normal matrix $AA^T$ has a dense Cholesky factor (even with re-ordering), these solvers may require excessive computational cost and/or extremely heavy memory usage in each interior-point iteration. Unfortunately, the natural remedy, i.e., the use of iterative methods based IPM solvers, although can avoid the explicit computation of the coefficient matrix and its factorization, is not practically viable due to the inherent extreme ill-conditioning of the large scale normal equation arising in each interior-point iteration. To provide a better alternative choice for solving large scale LPs with dense data or requiring expensive factorization of its normal equation, we propose a semismooth Newton based inexact proximal augmented Lagrangian ({\sc Snipal}) method. Different from classical IPMs, in each iteration of {\sc Snipal}, iterative methods can efficiently be used to solve simpler yet better conditioned semismooth Newton linear systems. Moreover, {\sc Snipal} not only enjoys a fast asymptotic superlinear convergence but is also proven to enjoy a finite termination property. Numerical comparisons with Gurobi have demonstrated encouraging potential of {\sc Snipal} for handling large-scale LP problems where the constraint matrix $A$ has a dense representation or $AA^T$ has a dense factorization even with an appropriate re-ordering.