We consider the problem of node localization in sensor networks, and we focus on networks in which the ranging measurements are subject to errors and anchor positions are subject to uncertainty. We consider a statistical model for the uncertainty in the anchor positions and formulate the robust localization problem that finds a maximum likelihood estimation of the node positions. To overcome the non-convexity of the resulting optimization problem, we obtain a convex relaxation that is based on the second order cone programming (SOCP). We also propose a possible distributed implementation using the SOCP convex relaxation. We present numerical studies that compare the presented approach to other existing convex relaxations for the robust localization problem in terms of positioning error and computational complexity.
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