Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Global solution to sensor network localization: A non-convex potential game approach and its distributed implementation

Gehui Xu, Guanpu Chen, Yiguang Hong, Baris Fidan, Thomas Parisini, Karl H. Johansson

TL;DR

The paper tackles the global solution to sensor network localization in a non-convex setting by modeling non-anchor nodes as players in a multi-player potential game, where the NE coincides with the network’s true locations. It introduces canonical duality to transform the non-convex problem into a complementary dual, establishing necessary and sufficient conditions for NE via dual variables, and provides a centralized conjugate-based algorithm with $\mathcal{O}(1/\sqrt{k})$ convergence. A distributed implementation (DSDEG) based on sliding mode control and extragradient methods is developed and shown to globally converge to the NE under the duality criterion. Extensive simulations, including comparisons with SDP-based, ARMA, and RBR methods, demonstrate high localization accuracy and scalability, validating the proposed approach's potential for robust, distributed SNL in large networks.

Abstract

Consider a sensor network consisting of both anchor and non-anchor nodes. We address the following sensor network localization (SNL) problem: given the physical locations of anchor nodes and relative measurements among all nodes, determine the locations of all non-anchor nodes. The solution to the SNL problem is challenging due to its inherent non-convexity. In this paper, the problem takes on the form of a multi-player non-convex potential game in which canonical duality theory is used to define a complementary dual potential function. After showing the Nash equilibrium (NE) correspondent to the SNL solution, we provide a necessary and sufficient condition for a stationary point to coincide with the NE. An algorithm is proposed to reach the NE and shown to have convergence rate $\mathcal{O}(1/\sqrt{k})$. With the aim of reducing the information exchange within a network, a distributed algorithm for NE seeking is implemented and its global convergence analysis is provided. Extensive simulations show the validity and effectiveness of the proposed approach to solve the SNL problem.

Global solution to sensor network localization: A non-convex potential game approach and its distributed implementation

TL;DR

The paper tackles the global solution to sensor network localization in a non-convex setting by modeling non-anchor nodes as players in a multi-player potential game, where the NE coincides with the network’s true locations. It introduces canonical duality to transform the non-convex problem into a complementary dual, establishing necessary and sufficient conditions for NE via dual variables, and provides a centralized conjugate-based algorithm with convergence. A distributed implementation (DSDEG) based on sliding mode control and extragradient methods is developed and shown to globally converge to the NE under the duality criterion. Extensive simulations, including comparisons with SDP-based, ARMA, and RBR methods, demonstrate high localization accuracy and scalability, validating the proposed approach's potential for robust, distributed SNL in large networks.

Abstract

Consider a sensor network consisting of both anchor and non-anchor nodes. We address the following sensor network localization (SNL) problem: given the physical locations of anchor nodes and relative measurements among all nodes, determine the locations of all non-anchor nodes. The solution to the SNL problem is challenging due to its inherent non-convexity. In this paper, the problem takes on the form of a multi-player non-convex potential game in which canonical duality theory is used to define a complementary dual potential function. After showing the Nash equilibrium (NE) correspondent to the SNL solution, we provide a necessary and sufficient condition for a stationary point to coincide with the NE. An algorithm is proposed to reach the NE and shown to have convergence rate . With the aim of reducing the information exchange within a network, a distributed algorithm for NE seeking is implemented and its global convergence analysis is provided. Extensive simulations show the validity and effectiveness of the proposed approach to solve the SNL problem.
Paper Structure (26 sections, 7 theorems, 86 equations, 5 figures, 1 table, 2 algorithms)

This paper contains 26 sections, 7 theorems, 86 equations, 5 figures, 1 table, 2 algorithms.

Table of Contents

  1. Introduction
  2. Motivations and Challenges
  3. Contributions
  4. Related work
  5. Outline
  6. Preliminaries
  7. Problem formulation
  8. Sensor network localization problem
  9. Potential game formulation
  10. Derivation of global Nash equilibrium
  11. Dealing with non-convexity
  12. Algorithm design and analysis
  13. Distributed implementation
  14. Distributed algorithm design
  15. Interpretation of DSDEG
  16. ...and 11 more sections

Key Result

Lemma 1

Under Assumption 1, the global minimum of $\Phi$ in (potential-fun) is unique and is equal to $\boldsymbol{x}^{\star}$ of SNL.

Figures (5)

  • Figure 1: The initial distribution and final localization results.
  • Figure 2: Convergence of all non-anchor nodes' decisions
  • Figure 3: Computed sensor location results with different configurations.
  • Figure 4: Computed sensor location results with different configurations.
  • Figure 5: Computed sensor location results with different configurations.

Theorems & Definitions (14)

  • Definition 1: NE
  • Definition 2: potential game
  • Lemma 1
  • Proposition 1
  • Remark 1
  • Theorem 1
  • Corollary 1
  • Remark 2
  • Theorem 2
  • Remark 3
  • ...and 4 more