Global Optimal Solutions to General Sensor Network Localization Problem

TL;DR

This paper introduces a canonical duality approach to efficiently solve the sensor network localization problem, transforming a challenging nonconvex minimization into a solvable dual problem, and demonstrating its effectiveness on large-scale instances.

Contribution

It develops a novel canonical duality framework that reformulates the nonconvex localization problem as a concave dual problem, enabling efficient solutions and insights into problem complexity.

Findings

The dual problem is concave and solvable over a convex set.

Large-scale problems can be solved efficiently using the proposed method.

The problem is not NP-hard if the dual problem has a solution.

Abstract

Sensor network localization problem is to determine the position of the sensor nodes in a network given pairwise distance measurements. Such problem can be formulated as a polynomial minimization via the least squares method. This paper presents a canonical duality theory for solving this challenging problem. It is shown that the nonconvex minimization problem can be reformulated as a concave maximization dual problem over a convex set in a symmetrical matrix space, and hence can be solved efficiently by combining a general (linear or quadratic) perturbation technique with existing optimization techniques. Applications are illustrated by solving some relatively large-scale problems. Our results show that the general sensor network localization problem is not NP-hard unless its canonical dual problem has no solution.