A Primal-Dual Gradient Descent Approach to the Connectivity Constrained Sensor Coverage Problem

TL;DR

This work addresses static sensor placement to maximize region-wide coverage while enforcing connectivity, formulating a constrained optimization using a smooth connectivity measure derived from the graph Laplacian. A primal-dual gradient descent approach, PPALA, is developed to minimize the coverage objective $\mathcal{H}(\bm{x})$ subject to the connectivity constraint $\tau - \det(P^T\mathcal{L}(\bm{x})P) \le 0$ and minimum-distance constraints, with a regularization option to incorporate priors. The authors establish convergence to a KKT-point by verifying Mangasarian-Fromowitz constraint qualification, compactness, and Lipschitz continuity of gradients, and validate the method numerically under unimodal and bimodal spatial densities, showing feasible, well-connected deployments that maintain high coverage. They also discuss how the smooth connectivity formulation approximates a boolean transmission model as $w$ grows and outline future directions toward discrete formulations and scalability.

Abstract

Sensor networks play a critical role in many situational awareness applications. In this paper, we study the problem of determining sensor placements to balance coverage and connectivity objectives over a target region. Leveraging algebraic graph theory, we formulate a novel optimization problem to maximize sensor coverage over a spatial probability density of event likelihoods while adhering to connectivity constraints. To handle the resulting non-convexity under constraints, we develop an augmented Lagrangian-based gradient descent algorithm inspired by recent approaches to efficiently identify points satisfying the Karush-Kuhn-Tucker (KKT) conditions. We establish convergence guarantees by showing necessary assumptions are satisfied in our setup, including employing Mangasarian-Fromowitz constraint qualification to prove the existence of a KKT point. Numerical simulations under different probability densities demonstrate that the optimized sensor networks effectively cover high-priority regions while satisfying desired connectivity constraints.

Figures (3)

• Figure 1: The deployment trajectories obtained when using the PPALA to solve problem \ref{['problem:constrained']} with the density $\phi_1$ in \ref{['eq:unimodal']} for $w = 20$ and $\varepsilon = 0.1$. Subfigures (c) - (d) capture the unconstrained problem with $\tau = -1$, while $\tau = 0.1$ and $\tau = 1$ in subfigures (e) - (f) and (g) - (h), respectively.
• Figure 2: The deployments obtained when using the PPALA to solve problem \ref{['problem:constrained']} with the bimodal density $\phi_2$ in \ref{['eq:bimodal']} for $w = 20$ and $\varepsilon = 0.1$. Subfigures (c) - (d) capture the unconstrained problem with $\tau = -1$, while $\tau = 0.1$ and $\tau = 1$ in subfigures (e) - (f) and (g) - (h), respectively.
• Figure 3: The deployments obtained from the PPALA when solving the regularized problem \ref{['problem:regularized']} with the bimodal density $\phi_2$ in \ref{['eq:bimodal']} for $w = 20$, $\varepsilon = 0.1$ and $\tau = 0.1$. Subfigure (b) captures the unregularized problem, while $\alpha = 0.01$, $\alpha = 0.02$ and $\alpha = 0.03$ in subfigures (c) - (e), respectively.

Theorems & Definitions (17)

• Definition II.1
• Definition II.2
• Lemma II.1: laplacian
• Lemma II.2: connectivity
• Remark II.1
• Lemma II.3: connectivity
• Definition V.1: MFCQ
• Lemma V.1: horn2012matrix
• Theorem V.1
• proof