Iterative minimization schemes for solving the single source localization problem

We consider the problem of locating a single radiating source from several noisy measurements using a maximum likelihood (ML) criteria. The resulting optimization problem is nonconvex and nonsmooth, and thus finding its global solution is in principle a hard task. Exploiting the special structure of the objective function, we introduce and analyze two iterative schemes for solving this problem. The first algorithm is a very simple explicit fixed-point-based formula, and the second is based on solving at each iteration a nonlinear least squares problem, which can be solved globally and efficiently after transforming it into an equivalent quadratic minimization problem with a single quadratic constraint. We show that the nonsmoothness of the problem can be avoided by choosing a specific "good" starting point for both algorithms, and we prove the convergence of the two schemes to stationary points. We present empirical results that support the underlying theoretical analysis and suggest that, despite of its nonconvexity, the ML problem can effectively be solved globally using the devised schemes.