Solution of network localization problem with noisy distances and its convergence

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. Keyword Network localization technique, Localization with non-convex distances constraints, Localization with noisy distances, Applications of Lagrange optimization in localization, Mini-max optimization problem, Non-convex optimization.