Polynomial Linear Programming with Gaussian Belief Propagation

TL;DR

This paper introduces a novel approach for solving linear programming problems by integrating Gaussian belief propagation into interior-point methods, reducing computational complexity and enabling distributed, efficient inference.

Contribution

It proposes using Gaussian belief propagation within interior-point methods to efficiently solve LP problems by exploiting Hessian sparsity, avoiding direct matrix inversion, and enabling distributed computation.

Findings

GaBP effectively replaces Hessian inversion in LP solving.

The method exploits sparsity for computational efficiency.

Applicable to general interior-point algorithms, including non-linear solvers.

Abstract

Interior-point methods are state-of-the-art algorithms for solving linear programming (LP) problems with polynomial complexity. Specifically, the Karmarkar algorithm typically solves LP problems in time O(n^{3.5}), where $n$ is the number of unknown variables. Karmarkar's celebrated algorithm is known to be an instance of the log-barrier method using the Newton iteration. The main computational overhead of this method is in inverting the Hessian matrix of the Newton iteration. In this contribution, we propose the application of the Gaussian belief propagation (GaBP) algorithm as part of an efficient and distributed LP solver that exploits the sparse and symmetric structure of the Hessian matrix and avoids the need for direct matrix inversion. This approach shifts the computation from realm of linear algebra to that of probabilistic inference on graphical models, thus applying GaBP as an efficient inference engine. Our construction is general and can be used for any interior-point algorithm which uses the Newton method, including non-linear program solvers.