A Block Coordinate Descent Method for Sensor Network Localization

TL;DR

This paper introduces a novel block coordinate descent approach for sensor network localization that reformulates the problem into an unconstrained multiconvex optimization, providing theoretical guarantees and demonstrating faster, accurate position estimation.

Contribution

It proposes a new Burer--Monteiro factorization-based method for SNL, with convergence analysis and practical efficiency improvements over existing techniques.

Findings

Method converges to a stationary point.

Inherits the rank constraint effectively.

Faster sensor position estimation with maintained accuracy.

Abstract

The problem of sensor network localization (SNL) can be formulated as a semidefinite programming problem with a rank constraint. We propose a new method for solving such SNL problems. We factorize a semidefinite matrix with the rank constraint into a product of two matrices via the Burer--Monteiro factorization. Then, we add the difference of the two matrices, with a penalty parameter, to the objective function, thereby reformulating SNL as an unconstrained multiconvex optimization problem, to which we apply the block coordinate descent method. In this paper, we also provide theoretical analyses of the proposed method and show that each subproblem that is solved sequentially by the block coordinate descent method can also be solved analytically, with the sequence generated by our proposed algorithm converging to a stationary point of the objective function. We also give a range of the penalty parameter for which the two matrices used in the factorization agree at any accumulation point. Numerical experiments confirm that the proposed method does inherit the rank constraint and that it estimates sensor positions faster than other methods without sacrificing the estimation accuracy, especially when the measured distances contain errors.