Multilinear Extensions in Submodular Optimization for Optimal Sensor Scheduling in Nonlinear Networks

TL;DR

The paper tackles SNS in nonlinear dynamical networks by formulating observability via a variational Gramian, revealing modularity and submodularity properties for key metrics. It then solves the SNS by a multilinear extension with continuous greedy optimization and pipage rounding, providing a $(1-1/e)$ performance guarantee. The authors prove that the variational Gramian induces a modular $\mathrm{trace}$ and submodular $\mathrm{rank}$ and $\log \det$ measures, enabling scalable, provable sensor selection under matroid constraints. Validation on the nonlinear GRI30 combustion network demonstrates near-optimal sensor placements with substantial computational advantages over greedy methods, highlighting practical applicability to large nonlinear systems and flexible constraints beyond simple cardinality.

Abstract

Optimal sensing nodes selection (SNS) in dynamic systems is a combinatorial optimization problem that has been thoroughly studied in the recent literature. This problem can be formulated within the context of set optimization. For high-dimensional nonlinear systems, the problem is extremely difficult to solve. It scales poorly too. Current literature poses combinatorial submodular set optimization problems via maximizing observability performance metrics subject to matroid constraints. Such an approach is typically solved using greedy algorithms that require lower computational effort yet often yield sub-optimal solutions. In this paper, we address the SNS problem for nonlinear dynamical networks using a variational form of the system dynamics, that basically perturb the system physics. As a result, we show that the observability performance metrics under such system representation are indeed submodular. The optimal problem is then solved using the multilinear continuous extension. This extension offers a computationally scalable and approximate continuous relaxation with a performance guarantee. The effectiveness of the extended submodular program is studied and compared to greedy algorithms. We demonstrate the proposed set optimization formulation for SNS on nonlinear natural gas combustion networks.

Figures (1)

• Figure 1: State estimation error based on the optimal selected node obtained from greedy algorithm (left) and continuous greedy algorithm (right).

Theorems & Definitions (17)

• Remark 1
• Theorem 1
• proof
• Definition 1
• Definition 2
• Theorem 2
• Lemma 1
• Theorem 3
• Theorem 4
• proof