A Regularized Alternating Least-Squares Method for Minimizing a Sum of Squared Euclidean Norms with Rank Constraint

Abstract

Minimizing a sum of Euclidean norms (MSEN) is a classic minimization problem widely used in several applications, including the determination of single and multifacility locations. The objective of the MSEN problem is to find a vector $x$ such that it minimizes a sum of Euclidean norms of systems of equations. In this paper, we propose a modification of the MSEN problem, which we call the problem of minimizing a sum of squared Euclidean norms with rank constraint, or simply the MSSEN-RC problem. The objective of the MSSEN-RC problem is to obtain a vector $x$ and rank-constrained matrices $A_1,\cdots ,A_p$ such that they minimize a sum of squared Euclidean norms of systems of equations. Additionally, we present an algorithm based on the regularized alternating least-squares (RALS) method for solving the MSSEN-RC problem. We show that given the existence of critical points of the alternating least-squares method, the limit points of the converging sequences of the RALS are the critical points of the objective function. Finally, we show numerical experiments that demonstrate the efficiency of the RALS method.