Second Order Cone Programming for Sensor Network Localization with Anchor Position Uncertainty

Node localization is a difficult task in sensor networks in which the ranging measurements are subject to errors and anchor positions are subject to uncertainty. In this paper, the robust localization problem is formulated using the maximum likelihood criterion under an unbounded uncertainty model for the anchor positions. To overcome the non-convexity of the resulting optimization problem, a convex relaxation leading to second order cone programming (SOCP) is devised. Furthermore, an analysis is performed in order to identify the set of nodes which are accurately positioned using robust SOCP, and to establish a relation between the solution of the proposed robust SOCP optimization and the existing robust optimization using semidefinite programming (SDP). Based on this analysis, a mixed robust SDP-SOCP localization framework is proposed which benefits from the better accuracy of SDP and the lower complexity of SOCP. Since the centralized optimization involves a high computational complexity in large networks, we also derive the distributed implementation of the proposed robust SOCP convex relaxation. Finally, we propose an iterative optimization based on the expectation maximization (EM) algorithm for the cases where anchor uncertainty parameters are unavailable. Simulations confirm that the robust SOCP and mixed robust SDP-SOCP provide tradeoffs between localization accuracy and computational complexity that render them attractive solutions, especially in networks with a large number of nodes.