Accelerated Dual Descent for Network Optimization

TL;DR

This paper presents a family of distributed dual descent algorithms that leverage approximate Newton directions to significantly accelerate convergence in network optimization problems, maintaining distributed implementation benefits.

Contribution

Introduces a novel family of dual descent algorithms using approximate Newton directions computed via local information exchanges, enhancing convergence speed while preserving distributed nature.

Findings

Convergence times are 10 to 100 times faster than existing methods.

Algorithms exhibit superlinear convergence near the optimum.

Numerical results confirm significant acceleration in distributed network optimization.

Abstract

Dual descent methods are commonly used to solve network optimization problems because their implementation can be distributed through the network. However, their convergence rates are typically very slow. This paper introduces a family of dual descent algorithms that use approximate Newton directions to accelerate the convergence rate of conventional dual descent. These approximate directions can be computed using local information exchanges thereby retaining the benefits of distributed implementations. The approximate Newton directions are obtained through matrix splitting techniques and sparse Taylor approximations of the inverse Hessian.We show that, similarly to conventional Newton methods, the proposed algorithm exhibits superlinear convergence within a neighborhood of the optimal value. Numerical analysis corroborates that convergence times are between one to two orders of magnitude faster than existing distributed optimization methods. A connection with recent developments that use consensus iterations to compute approximate Newton directions is also presented.