Mixing convex-optimization bounds for maximum-entropy sampling

Chen, Zhongzhu; Fampa, Marcia; Lambert, Amélie; Lee, Jon

doi:10.1007/s10107-020-01588-w

Cited by 11 publications

(14 citation statements)

References 25 publications

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“…, for each i ∈ N . This gives a nice concave decreasing sequence of eigenvalues that is preserved under matrix inversion (see [CFLL21]).…”

Section: Appendix 2: Test Instancesmentioning

confidence: 97%

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Tridiagonal Maximum-Entropy Sampling and Tridiagonal Masks

Al‐Thani¹,

Lee²

2021

Preprint

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The NP-hard maximum-entropy sampling problem (MESP) seeks a maximum (log-)determinant principal submatrix, of a given order, from an input covariance matrix C.We give an efficient dynamic-programming algorithm for MESP when C (or its inverse) is tridiagonal and generalize it to the situation where the support graph of C (or its inverse) is a spider graph (and beyond). We give a class of arrowhead covariance matrices C for which a natural greedy algorithm solves MESP.A mask M for MESP is a correlation matrix with which we pre-process C, by taking the Hadamard product M • C. Upper bounds on MESP with M • C give upper bounds on MESP with C. Most upper-bounding methods are much faster to apply, when the input matrix is tridiagonal, so we consider tridiagonal masks M (which yield tridiagonal M • C). We make a detailed analysis of such tridiagonal masks, and develop a combinatorial local-search based upper-bounding method that takes advantage of fast computations on tridiagonal matrices.

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“…, for each i ∈ N . This gives a nice concave decreasing sequence of eigenvalues that is preserved under matrix inversion (see [CFLL21]).…”

Section: Appendix 2: Test Instancesmentioning

confidence: 97%

“…where γ > 0 is a scaling parameter that must be judiciously selected (see [Ans20,CFLL21]), and e is an all-ones vector.…”

Section: Local Search On Tridiagonal Masksmentioning

confidence: 99%

Tridiagonal Maximum-Entropy Sampling and Tridiagonal Masks

Al‐Thani¹,

Lee²

2021

Preprint

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“…Because the factorization bound shifts by the same amount as z(C, s, A, b), under scaling of C by γ, we cannot improve on the factorization bound by scaling. In contrast, the linx bound is very sensitive to the choice of the scale factor, and while we can compute an optimal scale factor for the linx bound (see Chen et al (2021)), it is a significant computational burden to do so.…”

Section: Dfactmentioning

confidence: 99%

“…In Section 3, we discuss the numerical experiments where, using a commercial nonlinearprogramming solver, we calculate upper bounds for benchmark instances of MESP from the literature with the three relaxations presented, namely DDFact, complementary DDFact and linx, and with the "mixing" strategy described in Chen et al (2021). Generally, we found that a commercial nonlinear-programming solver is quite viable for our relaxations, even for DDFact which has some nondifferentiabilty.…”

Section: Introductionmentioning

confidence: 99%

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On computing with some convex relaxations for the maximum-entropy sampling problem

Chen¹,

Fampa²,

Lee³

2021

Preprint

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Based on a factorization of an input covariance matrix, we define a mild generalization of an upper bound of Nikolov (2015) and Li and Xie (2020) for the NP-Hard constrained maximum-entropy sampling problem (CMESP). We demonstrate that this factorization bound is invariant under scaling and also independent of the particular factorization chosen. We give a variable-fixing methodology that could be used in a branchand-bound scheme based on the factorization bound for exact solution of CMESP. We report on successful experiments with a commercial nonlinear-programming solver. We further demonstrate that the known "mixing" technique can be successfully used to combine the factorization bound with the factorization bound of the complementary CMESP, and also with the "linx bound" of Anstreicher (2020).

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An Outer-Approximation Algorithm for Maximum-Entropy Sampling

Fampa

Lee

2022

Lecture Notes in Computer Science

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Mixing convex-optimization bounds for maximum-entropy sampling

Cited by 11 publications

References 25 publications

Tridiagonal Maximum-Entropy Sampling and Tridiagonal Masks

Tridiagonal Maximum-Entropy Sampling and Tridiagonal Masks

On computing with some convex relaxations for the maximum-entropy sampling problem

An Outer-Approximation Algorithm for Maximum-Entropy Sampling

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