Perturbation analysis of the euclidean distance matrix optimization problem and its numerical implications

Abstract

Euclidean distance matrices have lately received increasing attention in applications such as multidimensional scaling and molecular conformation from nuclear magnetic resonance data in computational chemistry. In this paper, we focus on the perturbation analysis of the Euclidean distance matrix optimization problem (EDMOP). Under Robinson’s constraint qualification, we establish a number of equivalent characterizations of strong regularity and strong stability at a locally optimal solution of EDMOP. Those results extend the corresponding characterizations in Semidefinite Programming and are tailored to the special structure in EDMOP. As an application, we demonstrate a numerical implication of the established results on an alternating direction method of multipliers (ADMM) to a stress minimization problem, which is an important instance of EDMOP. The implication is that the ADMM method converges to a strongly stable solution under reasonable assumptions.