  The problem of sensor network localization (SNL) can be formulated as a
semidefinite programming problem with a rank constraint. We propose a new
method for solving such SNL problems. We factorize a semidefinite matrix with
the rank constraint into a product of two matrices via the Burer--Monteiro
factorization. Then, we add the difference of the two matrices, with a penalty
parameter, to the objective function, thereby reformulating SNL as an
unconstrained multiconvex optimization problem, to which we apply the block
coordinate descent method. In this paper, we also provide theoretical analyses
of the proposed method and show that each subproblem that is solved
sequentially by the block coordinate descent method can also be solved
analytically, with the sequence generated by our proposed algorithm converging
to a stationary point of the objective function. We also give a range of the
penalty parameter for which the two matrices used in the factorization agree at
any accumulation point. Numerical experiments confirm that the proposed method
does inherit the rank constraint and that it estimates sensor positions faster
than other methods without sacrificing the estimation accuracy, especially when
the measured distances contain errors.
