  The single source localization problem (SSLP) appears in several fields such
as signal processing and global positioning systems. The optimization problem
of SSLP is nonconvex and difficult to find its globally optima solution. It can
be reformulated as a rank constrained Euclidean distance matrix (EDM)
completion problem with a number of equality constraints. In this paper, we
propose a facial reduction approach to solve such an EDM completion problem.
For the constraints of fixed distances between sensors, we reduce them to a
face of the EDM cone and derive the closed formulation of the face. We prove
constraint nondegeneracy for each feasible point of the resulting EDM
optimization problem without a rank constraint, which guarantees the quadratic
convergence of semismooth Newton's method. To tackle the nonconvex rank
constraint, we apply the majorized penalty approach developed by Zhou et al.
(IEEE Trans Signal Process 66(3):4331-4346, 2018). Numerical results verify the
fast speed of the proposed approach while giving comparable quality of
solutions as other methods.
