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Mixing convex-optimization bounds for maximum-entropy sampling

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Abstract

The maximum-entropy sampling problem is a fundamental and challenging combinatorial-optimization problem, with application in spatial statistics. It asks to find a maximum-determinant order-s principal submatrix of an order-n covariance matrix. Exact solution methods for this NP-hard problem are based on a branch-and-bound framework. Many of the known upper bounds for the optimal value are based on convex optimization. We present a methodology for “mixing” these bounds to achieve better bounds.

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Notes

  1. We note that even in the nondifferentiable case, quasi-Newton methods have very good convergence properties (see [23]).

  2. Helmberg suggested (essentially) the BQP bound in 1995 (see [14, 21]) to Anstreicher and Lee, but no one developed it at all until [5] did so extensively, drawing in and significantly extending some techniques from [2].

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Acknowledgements

J. Lee was supported in part by ONR Grant N00014-17-1-2296, AFOSR Grant FA9550-19-1-0175, and Conservatoire National des Arts et Métiers. M. Fampa was supported in part by CNPq Grants 303898/2016-0 and 434683/2018-3. J. Lee and M. Fampa were supported in part by funding from the Simons Foundation and the Centre de Recherches Mathématiques, through the Simons-CRM scholar-in-residence program. The authors thank Kurt Anstreicher for supplying them with his (and Jon’s) Matlab codes and advice in running them. Moreover, Kurt’s talk at the 2019 Oberwolfach workshop on Mixed-Integer Nonlinear Programming catalyzed the renewed interest of Fampa and Lee in the topic of maximum-entropy sampling.

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Authors and Affiliations

  1. University of Michigan, Ann Arbor, MI, USA

    Zhongzhu Chen & Jon Lee

  2. Universidade Federal do Rio de Janeiro, Rio de Janeiro, Brazil

    Marcia Fampa

  3. Conservatoire National des Arts et Métiers, Paris, France

    Amélie Lambert

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  1. Zhongzhu Chen
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  2. Marcia Fampa
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  3. Amélie Lambert
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  4. Jon Lee
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Correspondence to Jon Lee.

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Chen, Z., Fampa, M., Lambert, A. et al. Mixing convex-optimization bounds for maximum-entropy sampling. Math. Program. 188, 539–568 (2021). https://doi.org/10.1007/s10107-020-01588-w

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