  The network localization problem with convex and non-convex distance
constraints may be modeled as a nonlinear optimization problem. The existing
localization techniques are mainly based on convex optimization. In those
techniques, the non-convex distance constraints are either ignored or relaxed
into convex constraints for using the convex optimization methods like SDP,
least square approximation, etc.. We propose a method to solve the nonlinear
non-convex network localization problem with noisy distance measurements
without any modification of constraints in the general model. We use the
nonlinear Lagrangian technique for non-convex optimization to convert the
problem to a root finding problem of a single variable continuous function.
This problem is then solved using an iterative method. However, in each step of
the iteration the computation of the functional value involves a finite
mini-max problem (FMX). We use smoothing gradient method to fix the FMX
problem. We also prove that the solution obtained from the proposed iterative
method converges to the actual solution of the general localization problem.
The proposed method obtains the solutions with a desired label of accuracy in
real time.
