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Joint Cooperative and Non-Cooperative Localization in WSNs with Distributed Scaled Proximal ADMM Algorithms

Qiaojia Zhu, Xiaojing Shen, Haiqi Liu, Pramod K. Varshney

Abstract

The integration of cooperative and non-cooperative localization is fundamentally important, as these two modes frequently coexist in wireless sensor networks, especially when sensor positions are uncertain and targets are unable to communicate with the network. This paper presents a joint modeling approach that formulates cooperative and non-cooperative localization as a single optimization problem. By processing both tasks jointly, the proposed method eliminates the latency inherent in sequential approaches that perform cooperative localization first, followed by non-cooperative localization. However, this joint formulation introduces complex variable coupling, posing challenges in both modeling and optimization. To address this coupling, we introduce auxiliary variables that enable structural decoupling and facilitate distributed computation. Building on this formulation, we develop the Scaled Proximal Alternating Direction Method of Multipliers for Joint Cooperative and Non-Cooperative Localization (SP-ADMM-JCNL). Leveraging the structured design of the problem, we provide theoretical guarantees that the algorithm generates a sequence converging globally to a KKT point of the reformulated problem, and further to a critical point of the original non-convex objective function, with the convergence rate of O(1/T). Experiments demonstrate that SP-ADMM-JCNL achieves accurate and reliable localization performance.

Joint Cooperative and Non-Cooperative Localization in WSNs with Distributed Scaled Proximal ADMM Algorithms

Abstract

The integration of cooperative and non-cooperative localization is fundamentally important, as these two modes frequently coexist in wireless sensor networks, especially when sensor positions are uncertain and targets are unable to communicate with the network. This paper presents a joint modeling approach that formulates cooperative and non-cooperative localization as a single optimization problem. By processing both tasks jointly, the proposed method eliminates the latency inherent in sequential approaches that perform cooperative localization first, followed by non-cooperative localization. However, this joint formulation introduces complex variable coupling, posing challenges in both modeling and optimization. To address this coupling, we introduce auxiliary variables that enable structural decoupling and facilitate distributed computation. Building on this formulation, we develop the Scaled Proximal Alternating Direction Method of Multipliers for Joint Cooperative and Non-Cooperative Localization (SP-ADMM-JCNL). Leveraging the structured design of the problem, we provide theoretical guarantees that the algorithm generates a sequence converging globally to a KKT point of the reformulated problem, and further to a critical point of the original non-convex objective function, with the convergence rate of O(1/T). Experiments demonstrate that SP-ADMM-JCNL achieves accurate and reliable localization performance.

Paper Structure

This paper contains 20 sections, 9 theorems, 217 equations, 8 figures, 2 tables, 2 algorithms.

Table of Contents

  1. Introduction
  2. Problem Formulation
  3. Notation
  4. Problem Statement
  5. Problem Reformulation
  6. The Proposed SP-ADMM-JCNL Algorithm
  7. Convergence Analysis
  8. Numerical Experiments
  9. Synthetic Network Evaluation
  10. Parameter Selection and Sensitivity Analysis
  11. Parameter Selection for $\rho$ and $c$
  12. Localization Performance With $u_0$ Selection:
  13. Conclusion
  14. Solution to Remark \ref{['remark1']}
  15. (Proof of Lemma\ref{['lem12']})
  16. ...and 5 more sections

Key Result

Lemma 1

Suppose that the sequence $\{ (\mathbf{z}_i^t, \mathbf{w}_i^t, \boldsymbol{\lambda}_i^t) \}_{t \geq 1}$ is generated by Algorithm al1 and that $c \mathbf{B}_i^T \mathbf{B}_i$ follows the form given in Equation(23). Then, the following results hold: where $L_1 = N_{\max} + M_{\max} + 1, L_2 = 2N_{\max} + 1$, with $N_{\max} := \max \{ N_i : i \in \mathcal{N} \}$, $M_{\max} := \max \{ M_i : i \in \m

Figures (8)

  • Figure 1: Exemplary illustration of the considered scenario.
  • Figure 2: Synthetic network layout.
  • Figure 3: Performance with synthetic network. Measurement noise: AWGN (first row) and range-dependent Gaussian noise $\sigma_{i,j}^2 = \sigma_{\text{add}}^2 \| \mathbf{p}_i - \mathbf{p}_j \|^2$ (second row).RMSE value(left column); feasibility gap and stationarity gap(right column).
  • Figure 4: Localization results with synthetic network. Measurement noise: AWGN and range-dependent Gaussian noise. Anchors: ; true agent positions: +; true target positions: +; estimated agent positions: $\circ$; estimated target positions: $\circ$.
  • Figure 5: Comparison of RMSE between JCNL and SCNL.
  • ...and 3 more figures

Theorems & Definitions (30)

  • Remark 1
  • Remark 2
  • Lemma 1
  • proof
  • Remark 3
  • Theorem 1
  • proof
  • Lemma 2
  • proof
  • Remark 4
  • ...and 20 more