We consider the problem of locating a user's position from a set of noisy pseudoranges to a group of satellites. Two different formulations are studied: the nonlinear least squares formulation in which the objective function is nonconvex and nonsmooth, and the nonlinear squared least squares variant in which the objective function is smooth, but still nonconvex. We show that the squared least squares problem can be solved efficiently, despite is nonconvexity. Conditions for attainment of the optimal solutions of both problems are derived. The problem is shown to have tight connections to the well known circle fitting and orthogonal regression problems. Finally, a fixed point method for the nonlinear least squares problems is derived and analyzed. This is a joint work with Dror Pan (Technion).
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