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

TL;DR

This paper introduces a scalable continuous relaxation method using multilinear extensions for optimal sensor selection in nonlinear networks, improving over greedy algorithms with performance guarantees.

Contribution

It presents a novel variational approach that proves observability metrics are submodular and applies multilinear extensions for efficient sensor scheduling in nonlinear systems.

Findings

The method outperforms greedy algorithms in sensor selection tasks.

The approach provides a scalable solution with theoretical performance guarantees.

Validated on nonlinear natural gas combustion networks.

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.