Penalized Semidefinite Programming Method for Elliptic Localization

Abstract

Elliptic localization has become a hot research topic and has been widely used in various fields. In this paper, we propose a penalized semidefinite programming (SDP) method for the elliptic localization problem when both the transmitter position and signal propagation speed are unknown. We first formulate a nonconvex constrained weighted least squares (CWLS) problem using the connections between the unknown variables in the measurement model. Since the nonconvex CWLS problem is difficult to solve, we relax it into a convex SDP problem by applying semidefinite relaxation (SDR). Although the SDP problem avoids local convergence or divergence, its solution cannot reach the Cramer-Rao lower bound (CRLB) accuracy owing to the relaxation. In order to address this issue, we propose to add a penalty term to tighten the above SDP problem. Furthermore, we propose an adaptive selection procedure for choosing a proper penalty factor. Simulation results validate the effectiveness and efficiency of the penalized SDP method and show that the proposed method is able to reach the CRLB.