The Alternating Direction Method of Multipliers for Finding the Distance between Ellipsoids

TL;DR

This paper develops and tests ADMM-based algorithms for efficiently computing the distance between ellipsoids, outperforming existing methods in both convex and nonconvex cases through numerical experiments.

Contribution

It introduces novel ADMM variants with heuristic updates and restarting procedures for accurate distance computation between ellipsoids, including solutions for nonconvex cases.

Findings

ADMM methods outperform existing algorithms in convex cases.

Heuristic restarting ensures global optimality in nonconvex cases.

Significant efficiency gains on high-dimensional problems.

Abstract

We study several versions of the alternating direction method of multipliers (ADMM) for solving the convex problem of finding the distance between two ellipsoids and the nonconvex problem of finding the distance between the boundaries of two ellipsoids. In the convex case we present the ADMM with and without automatic penalty updates and demonstrate via numerical experiments on problems of various dimensions that our methods significantly outperform all other existing methods for finding the distance between ellipsoids. In the nonconvex case we propose a heuristic rule for updating the penalty parameter and a heuristic restarting procedure (a heuristic choice of a new starting for point for the second run of the algorithm). The restarting procedure was verified numerically with the use of a global method based on KKT optimality conditions. The results of numerical experiments on various test problems showed that this procedure always allows one to find a globally optimal solution in the nonconvex case. Furthermore, the numerical experiments also demonstrated that our version of the ADMM significantly outperforms existing methods for finding the distance between the boundaries of ellipsoids on problems of moderate and high dimensions.