Solution of network localization problem with noisy distances and its convergence

TL;DR

This paper introduces a novel iterative method using nonlinear Lagrangian techniques and smoothing gradient methods to solve the non-convex network localization problem with noisy distances, ensuring convergence and real-time solutions.

Contribution

It presents a new approach that directly handles non-convex constraints without relaxation, improving solution accuracy and convergence in network localization.

Findings

Method converges to the actual solution.

Achieves real-time localization with desired accuracy.

Handles noisy distance measurements effectively.

Abstract

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.