The Augmented Factorization Bound for Maximum-Entropy Sampling

TL;DR

This paper introduces an augmented factorization bound for the maximum-entropy sampling problem that outperforms existing bounds, improves approximation algorithms, and enhances variable fixing strategies, especially for matrices with small condition numbers.

Contribution

It proposes a new concave relaxation-based bound for MESP that dominates previous bounds and provides theoretical and practical improvements in solution quality.

Findings

The new bound outperforms classic and recent bounds in numerical experiments.

It reduces integrality gaps and fixes more variables in benchmark instances.

The improvement is significant for covariance matrices with small condition numbers.

Abstract

The maximum-entropy sampling problem (MESP) aims to select the most informative principal submatrix of a prespecified size from a given covariance matrix. This paper proposes an augmented factorization bound for MESP based on concave relaxation. By leveraging majorization and Schur-concavity theory, we demonstrate that this new bound dominates the classic factorization bound of Nikolov (2015) and a recent upper bound proposed by Li et al. (2024). Furthermore, we provide theoretical guarantees that quantify how much our proposed bound improves the two existing ones and establish sufficient conditions for when the improvement is strictly attained. These results allow us to refine the celebrated approximation bounds for the two approximation algorithms of MESP. Besides, motivated by the strength of this new bound, we develop a variable fixing logic for MESP from a primal perspective. Finally, our numerical experiments demonstrate that our proposed bound achieves smaller integrality gaps and fixes more variables than the tightest bounds in the MESP literature on most benchmark instances, with the improvement being particularly significant when the condition number of the covariance matrix is small.