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The Augmented Factorization Bound for Maximum-Entropy Sampling

Yongchun Li

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.

The Augmented Factorization Bound for Maximum-Entropy Sampling

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.

Paper Structure

This paper contains 15 sections, 14 theorems, 34 equations, 7 figures.

Table of Contents

  1. Introduction
  2. Related work
  3. Contributions and Outline
  4. Reformulating \ref{['mesp']} as a concave integer program
  5. A naive reformulation of \ref{['mesp']}
  6. The concave envelope
  7. The augmented factorization bound and its properties
  8. The monotonicity of \ref{['eq:upper']} and its dominance over \ref{['eq:fact']}
  9. Theoretical guarantees for the improvement of \ref{['eq:upper']} over \ref{['eq:fact']}
  10. Theoretical guarantees for the improvement of \ref{['eq:upper']} over \ref{['eq:ddf']}
  11. A primal certificate for variable fixing using \ref{['eq:upper']}
  12. Numerical experiments
  13. MESP: Three benchmark data sets
  14. DDF: IEEE 118- and 300-bus data sets
  15. Conclusions

Key Result

Proposition 1

For any $t$ with $0\le t\le \lambda_{\min}(\bm C)$, mesp can be reduced to

Figures (7)

  • Figure 1: $n=63$ with the condition number $\lambda_{\max}(\bm C)/ \lambda_{\min}(\bm C) = 48.42$
  • Figure 2: $n=90$ with with the condition number $\lambda_{\max}(\bm C)/\lambda_{\min}(\bm C) = 200.45$
  • Figure 3: $n=124$ with the condition number $\lambda_{\max}(\bm C)/\lambda_{\min}(\bm C) = 78340.48$
  • Figure 4: IEEE $118$-bus instance and large PMU standard deviations with $\lambda_{\max}(\bm C)/\lambda_{\min}(\bm C)=313.27$
  • Figure 5: IEEE $118$-bus instance and small PMU standard deviations with $\lambda_{\max}(\bm C)/\lambda_{\min}(\bm C)=2690744.66$
  • ...and 2 more figures

Theorems & Definitions (33)

  • Remark 1
  • Definition 1
  • Definition 2
  • Proposition 1
  • proof
  • Definition 3
  • Remark 2: nikolov2015randomized
  • Proposition 2
  • proof
  • Corollary 1
  • ...and 23 more