D-optimal Data Fusion: Exact and Approximation Algorithms

TL;DR

This paper addresses the NP-hard D-optimal Data Fusion problem by proposing exact and approximation algorithms, including convex formulations, randomized sampling, and local search, validated on power grid data.

Contribution

It introduces novel convex integer-programming formulations and scalable algorithms with performance guarantees for the DDF problem, along with enhancements using submodular inequalities.

Findings

Exact algorithm efficiently solves the DDF problem.

Approximation algorithms are scalable and produce high-quality solutions.

Algorithms tested on real power grid data demonstrate effectiveness.

Abstract

We study the D-optimal Data Fusion (DDF) problem, which aims to select new data points, given an existing Fisher information matrix, so as to maximize the logarithm of the determinant of the overall Fisher information matrix. We show that the DDF problem is NP-hard and has no constant-factor polynomial-time approximation algorithm unless P $=$ NP. Therefore, to solve the DDF problem effectively, we propose two convex integer-programming formulations and investigate their corresponding complementary and Lagrangian-dual problems. We also develop scalable randomized-sampling and local-search algorithms with provable performance guarantees. Leveraging the concavity of the objective functions in the two proposed formulations, we design an exact algorithm, aimed at solving the DDF problem to optimality. We further derive a family of submodular valid inequalities and optimality cuts, which can significantly enhance the algorithm performance. Finally, we test our algorithms using real-world data on the new phasor-measurement-units placement problem for modern power grids, considering the existing conventional sensors. Our numerical study demonstrates the efficiency of our exact algorithm and the scalability and high-quality outputs of our approximation algorithms.