Fast, Accurate Second Order Methods for Network Optimization

TL;DR

This paper introduces a distributed second order method for network flow optimization that leverages Hessian sparsity and a novel solver to achieve superlinear convergence, improving speed and accuracy over traditional dual descent methods.

Contribution

It presents an efficient distributed Newton-like method utilizing Hessian sparsity and a new solver, enhancing convergence speed and accuracy in network optimization.

Findings

Achieves superlinear convergence near the optimum.

Utilizes a novel distributed solver for symmetric diagonally dominant systems.

Demonstrates improved performance on randomly generated networks.

Abstract

Dual descent methods are commonly used to solve network flow optimization problems, since their implementation can be distributed over the network. These algorithms, however, often exhibit slow convergence rates. Approximate Newton methods which compute descent directions locally have been proposed as alternatives to accelerate the convergence rates of conventional dual descent. The effectiveness of these methods, is limited by the accuracy of such approximations. In this paper, we propose an efficient and accurate distributed second order method for network flow problems. The proposed approach utilizes the sparsity pattern of the dual Hessian to approximate the the Newton direction using a novel distributed solver for symmetric diagonally dominant linear equations. Our solver is based on a distributed implementation of a recent parallel solver of Spielman and Peng (2014). We analyze the properties of the proposed algorithm and show that, similar to conventional Newton methods, superlinear convergence within a neighbor- hood of the optimal value is attained. We finally demonstrate the effectiveness of the approach in a set of experiments on randomly generated networks.