Solving Saddle Point Formulations of Linear Programs with Frank-Wolfe

Abstract

The problem of solving a linear program (LP) is ubiquitous in industry, yet in recent years the size of linear programming problems has grown and continues to do so. State-of-the-art LP solvers make use of the Simplex method and primal-dual interior-point methods which are able to provide accurate solutions in a reasonable amount of time for most problems. However, both the Simplex method and interior-point methods require solving a system of linear equations at each iteration, an operation that does not scale well with the size of the problem.

In response to the growing size of linear programs and poor scalability of existing algorithms, researchers have started to consider first-order methods for solving large scale linear programs. The best known first-order method for general linear programming problems is PDLP. First-order methods for linear programming are characterized by having a matrix-vector product as their primary computational cost.

We present a first-order primal-dual algorithm for solving saddle point formulations of linear programs, named FWLP (Frank-Wolfe Linear Programming). We provide some theoretical results regarding the behavior of our algorithm, however no convergence guarantees are provided. Numerical investigations suggest that our algorithm has error O(1/sqrt(k)) after k iterations, worse than that of PDLP, however we show that our algorithm has advantages for solving very large LPs in practice such as only needing part of the matrix A at each iteration.