  The sensor network localization, SNL, problem in embedding dimension r,
consists of locating the positions of wireless sensors, given only the
distances between sensors that are within radio range and the positions of a
subset of the sensors (called anchors). Current solution techniques relax this
problem to a weighted, nearest, (positive) semidefinite programming, SDP,
completion problem, by using the linear mapping between Euclidean distance
matrices, EDM, and semidefinite matrices. The resulting SDP is solved using
primal-dual interior point solvers, yielding an expensive and inexact solution.
  This relaxation is highly degenerate in the sense that the feasible set is
restricted to a low dimensional face of the SDP cone, implying that the Slater
constraint qualification fails. Cliques in the graph of the SNL problem give
rise to this degeneracy in the SDP relaxation. In this paper, we take advantage
of the absence of the Slater constraint qualification and derive a technique
for the SNL problem, with exact data, that explicitly solves the corresponding
rank restricted SDP problem. No SDP solvers are used. For randomly generated
instances, we are able to efficiently solve many huge instances of this NP-hard
problem to high accuracy, by finding a representation of the minimal face of
the SDP cone that contains the SDP matrix representation of the EDM. The main
work of our algorithm consists in repeatedly finding the intersection of
subspaces that represent the faces of the SDP cone that correspond to cliques
of the SNL problem.
