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Non-convex potential games for finding global solutions to sensor network localization

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

TL;DR

The paper tackles the challenge of globally solving sensor network localization (SNL) in the presence of non-convexity by casting SNL as a non-convex multi-player potential game. It then applies canonical duality theory to transform the non-convex problem into a complementary dual problem and develops a conjugation-based algorithm to compute stationary points; a sufficient duality condition identifies when a stationary point corresponds to the global Nash equilibrium (global NE). The authors prove existence and uniqueness of the global NE and provide proofs linking stationary points of the dual to NE of the original game, supported by numerical experiments. Results demonstrate that the proposed framework can recover global SNL solutions and validate the approach on 2D datasets across varying network sizes, with implications for scalable, globally optimal localization in sensor networks.

Abstract

Sensor network localization (SNL) problems require determining the physical coordinates of all sensors in a network. This process relies on the global coordinates of anchors and the available measurements between non-anchor and anchor nodes. Attributed to the intrinsic non-convexity, obtaining a globally optimal solution to SNL is challenging, as well as implementing corresponding algorithms. In this paper, we formulate a non-convex multi-player potential game for a generic SNL problem to investigate the identification condition of the global Nash equilibrium (NE) therein, where the global NE represents the global solution of SNL. We employ canonical duality theory to transform the non-convex game into a complementary dual problem. Then we develop a conjugation-based algorithm to compute the stationary points of the complementary dual problem. On this basis, we show an identification condition of the global NE: the stationary point of the proposed algorithm satisfies a duality relation. Finally, simulation results are provided to validate the effectiveness of the theoretical results.

Non-convex potential games for finding global solutions to sensor network localization

TL;DR

The paper tackles the challenge of globally solving sensor network localization (SNL) in the presence of non-convexity by casting SNL as a non-convex multi-player potential game. It then applies canonical duality theory to transform the non-convex problem into a complementary dual problem and develops a conjugation-based algorithm to compute stationary points; a sufficient duality condition identifies when a stationary point corresponds to the global Nash equilibrium (global NE). The authors prove existence and uniqueness of the global NE and provide proofs linking stationary points of the dual to NE of the original game, supported by numerical experiments. Results demonstrate that the proposed framework can recover global SNL solutions and validate the approach on 2D datasets across varying network sizes, with implications for scalable, globally optimal localization in sensor networks.

Abstract

Sensor network localization (SNL) problems require determining the physical coordinates of all sensors in a network. This process relies on the global coordinates of anchors and the available measurements between non-anchor and anchor nodes. Attributed to the intrinsic non-convexity, obtaining a globally optimal solution to SNL is challenging, as well as implementing corresponding algorithms. In this paper, we formulate a non-convex multi-player potential game for a generic SNL problem to investigate the identification condition of the global Nash equilibrium (NE) therein, where the global NE represents the global solution of SNL. We employ canonical duality theory to transform the non-convex game into a complementary dual problem. Then we develop a conjugation-based algorithm to compute the stationary points of the complementary dual problem. On this basis, we show an identification condition of the global NE: the stationary point of the proposed algorithm satisfies a duality relation. Finally, simulation results are provided to validate the effectiveness of the theoretical results.
Paper Structure (7 sections, 4 theorems, 21 equations, 1 figure, 1 algorithm)

This paper contains 7 sections, 4 theorems, 21 equations, 1 figure, 1 algorithm.

Table of Contents

  1. INTRODUCTION
  2. Problem formulation
  3. Derivation of the global Nash equilibrium
  4. Numerical Experiments
  5. Conclusion
  6. Proof of Theorem \ref{['t1']}
  7. Proof of Theorem \ref{['t2']}

Key Result

Proposition 1

With function $P$ in (potential-fun) and payoffs $J_{i}$ for $i\in\mathcal{N}_s$ in f1, the game ${{G}}=\{\mathcal{N}_{s}, \{\Omega_i\}_{i\in\mathcal{N}_{s}}, \{J_i\}_{i\in\mathcal{N}_{s}}\}$ is a potential game.

Figures (1)

  • Figure 1: Computed sensor location results with different configurations.

Theorems & Definitions (7)

  • Definition 1: potential game monderer1996potential
  • Proposition 1
  • Definition 2: Nash equilibrium nash1951non
  • Definition 3: Nash stationary point pang2011nonconvex
  • Lemma 1
  • Theorem 1
  • Theorem 2