Convex Relaxation for the Generalized Maximum-Entropy Sampling Problem

Authors Gabriel Ponte , Marcia Fampa , Jon Lee

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Author Details

Gabriel Ponte
  • University of Michigan, Ann Arbor, MI, USA
Marcia Fampa
  • Universidade Federal do Rio de Janeiro, Brazil
Jon Lee
  • University of Michigan, Ann Arbor, MI, USA

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Gabriel Ponte, Marcia Fampa, and Jon Lee. Convex Relaxation for the Generalized Maximum-Entropy Sampling Problem. In 22nd International Symposium on Experimental Algorithms (SEA 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 301, pp. 25:1-25:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)


The generalized maximum-entropy sampling problem (GMESP) is to select an order-s principal submatrix from an order-n covariance matrix, to maximize the product of its t greatest eigenvalues, 0 < t ≤ s < n. It is a problem that specializes to two fundamental problems in statistical design theory: (i) maximum-entropy sampling problem (MESP); (ii) binary D-optimality (D-Opt). In the general case, it is motivated by a selection problem in the context of PCA (principal component analysis). We introduce the first convex-optimization based relaxation for GMESP, study its behavior, compare it to an earlier spectral bound, and demonstrate its use in a branch-and-bound scheme. We find that such an approach is practical when s-t is very small.

Subject Classification

ACM Subject Classification
  • Theory of computation → Mixed discrete-continuous optimization
  • Mathematics of computing → Mathematical optimization
  • maximum-entropy sampling
  • D-optimality
  • convex relaxation
  • branch-and-bound
  • integer nonlinear optimization
  • principal component analysis


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  1. Kurt M. Anstreicher. Maximum-entropy sampling and the Boolean quadric polytope. Journal of Global Optimization, 72(4):603-618, 2018. URL:
  2. Kurt M. Anstreicher. Efficient solution of maximum-entropy sampling problems. Operations Research, 68(6):1826-1835, 2020. URL:
  3. Kurt M. Anstreicher, Marcia Fampa, Jon Lee, and Joy Williams. Continuous relaxations for constrained maximum-entropy sampling. In Integer programming and Combinatorial Optimization (Vancouver, BC, 1996), volume 1084 of Lecture Notes in Comput. Sci., pages 234-248. Springer, Berlin, 1996. URL:
  4. Kurt M. Anstreicher, Marcia Fampa, Jon Lee, and Joy Williams. Using continuous nonlinear relaxations to solve constrained maximum-entropy sampling problems. Mathematical Programming, Series A, 85(2):221-240, 1999. URL:
  5. Kurt M. Anstreicher and Jon Lee. A masked spectral bound for maximum-entropy sampling. In mODa 7 - Advances in Model-Oriented Design and Analysis, Contrib. Statist., pages 1-12. Physica, Heidelberg, 2004. URL:
  6. Samuel Burer and Jon Lee. Solving maximum-entropy sampling problems using factored masks. Mathematical Programming, 109(2-3, Ser. B):263-281, 2007. URL:
  7. Zhongzhu Chen, Marcia Fampa, Amélie Lambert, and Jon Lee. Mixing convex-optimization bounds for maximum-entropy sampling. Mathematical Programming, Series B, 188:539-568, 2021. URL:
  8. Zhongzhu Chen, Marcia Fampa, and Jon Lee. On computing with some convex relaxations for the maximum-entropy sampling problem. INFORMS Journal on Computing, 35(2):368-385, 2023. URL:
  9. Marcia Fampa and Jon Lee. Maximum-Entropy Sampling: Algorithms and Application. Springer, 2022. URL:
  10. Peter Guttorp, Nhu D. Le, Paul D. Sampson, and James V. Zidek. Using entropy in the redesign of an environmental monitoring network. In G.P. Patil, C.R. Rao, and N.P. Ross, editors, Multivariate Environmental Statistics, volume 6, pages 175-202. North-Holland, 1993. Google Scholar
  11. Alan Hoffman, Jon Lee, and Joy Williams. New upper bounds for maximum-entropy sampling. In mODa 6 - Advances in Model-Oriented Design and Analysis (Puchberg/Schneeberg, 2001), Contrib. Statist., pages 143-153. Physica, Heidelberg, 2001. URL:
  12. Ian Jolliffe and Jorge Cadima. Principal component analysis: A review and recent developments. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences, 374:20150202, April 2016. URL:
  13. Chun-Wa Ko, Jon Lee, and Maurice Queyranne. An exact algorithm for maximum entropy sampling. Oper. Res., 43(4):684-691, 1995. URL:|.
  14. Ole Kröger, Carleton Coffrin, Hassan Hijazi, and Harsha Nagarajan. Juniper: An open-source nonlinear branch-and-bound solver in Julia, Proc. of CPAIOR 2018, 2018. URL:
  15. Jon Lee. Constrained maximum-entropy sampling. Operations Research, 46(5):655-664, 1998. URL:
  16. Jon Lee and Joy Lind. Generalized maximum-entropy sampling. INFOR: Information Systems and Oper. Res., 58(2):168-181, 2020. URL:
  17. Jon Lee and Joy Williams. A linear integer programming bound for maximum-entropy sampling. Mathematical Programming, Series B, 94(2-3):247-256, 2003. URL:
  18. Yongchun Li, Marcia Fampa, Jon Lee, Feng Qiu, Weijun Xie, and Rui Yao. D-optimal data fusion: Exact and approximation algorithms. INFORMS Journal on Computing, 36(1):97-120, 2024. URL:
  19. Yongchun Li and Weijun Xie. Best principal submatrix selection for the maximum entropy sampling problem: scalable algorithms and performance guarantees. Operations Research, 2023. URL:
  20. Aleksandar Nikolov. Randomized rounding for the largest simplex problem. In STOC 2015, pages 861-870, 2015. URL:
  21. Gabriel Ponte, Marcia Fampa, and Jon Lee. Branch-and-bound for D-optimality with fast local search and variable-bound tightening, 2023. Preprint: URL:
  22. Gabriel Ponte, Marcia Fampa, and Jon Lee. Convex relaxation for the generalized maximum-entropy sampling problem, 2024. Preprint: URL:
  23. Claude E. Shannon. A mathematical theory of communication. The Bell System Technical Journal, 27(3):379-423, 1948. URL:
  24. George W. Snedecor and William G. Cochran. Statistical Methods. Iowa State University Press, Ames, IA, sixth edition, 1967. Google Scholar
  25. Joy Denise Williams. Spectral Bounds for Entropy Models. Ph.D. thesis, University of Kentucky, April 1998. URL: