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In this paper, the energy-based localization problem in wireless sensor networks is addressed. We focus on the weighted least squares (WLS) estimation of the source location. Due to the nonconvex nature of the WLS formulation, its global solution is hard to obtain without a good initial estimate. We propose a semidefinite relaxation method for this localization problem. To do so, we transform the original WLS formulation into a nonconvex approximate WLS (AWLS) formulation, which is then relaxed as a semidefinite programming (SDP). We show that it is possible for the SDP to be tight, i.e., the SDP solves the original AWLS problem. For the cases where the SDP is not tight, a procedure called Gaussian randomization is applied to further refine the SDP solution. Simulation results show that the proposed method can outperform the existing methods at high noise levels.