Solving a linear program via a single unconstrained minimization

TL;DR

This paper introduces a new method for solving linear programs by reformulating them as unconstrained minimization problems and applying a regularized Newton method, demonstrating faster convergence in practice.

Contribution

The paper presents a novel reformulation of primal-dual linear programs as unconstrained convex minimization problems and develops a regularized Newton method with proven convergence.

Findings

Global convergence within $O( ext{epsilon}^{-3/2})$ iterations.

Faster convergence observed in numerical experiments compared to worst-case bounds.

A heuristic with a parameter $ u$ improves convergence speed, especially in high dimensions.

Abstract

This paper proposes a novel approach for solving linear programs. We reformulate a primal-dual linear program as an unconstrained minimization of a convex and twice continuously differentiable merit function. When the optimal set of the primal-dual pair is nonempty, its optimal set is equal to the optimal set of the proposed merit function. Minimizing this merit function poses some challenges due to its Hessian being singular at some points in the domain, including the optimal solutions. We handle singular Hessians using the Newton method with Levenberg-Marquardt regularization. We show that the Newton method with Levenberg-Marquardt regularization yields global convergence to a solution of the primal-dual linear program in at most $O(\epsilon^{-3/2})$ iterations requiring only the assumption that the optimal set of the primal-dual linear program is bounded. Testing on random synthetic problems demonstrates convergence to optimal solutions to very high accuracy significantly faster than the derived worst-case bound. We further introduce a modified merit function that depends on a scalar parameter $\nu > 0$, whose Hessian is nonsingular for all $\nu > 0$ and which reduces exactly to the original merit function when $\nu = 0$. Based on this formulation, we propose a heuristic scheme that performs Newton steps while gradually decreasing $\nu$ toward zero. Numerical experiments indicate that this approach achieves faster convergence, particularly on higher-dimensional problems.