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Cardinality Constrained Multilinear Sets

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Combinatorial Optimization (ISCO 2020)

Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 12176))

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Abstract

The problem of minimizing a multilinear function of binary variables is a well-studied NP-hard problem. The set of solutions of the standard linearization of this problem is called the multilinear set, and many valid inequalities for its convex hull are available in the literature. Motivated by a machine learning application, we study a cardinality constrained version of this problem with upper and lower bounds on the number of nonzero variables. We call the set of solutions of the standard linearization of this problem a cardinality constrained multilinear set, and give a complete polyhedral description of its convex hull when the multilinear terms in the problem have a nested structure.

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References

  1. Buchheim, C., Klein, L.: Combinatorial optimization with one quadratic term: spanning trees and forests. Discrete Appl. Math. 177, 34–52 (2014)

    Article  MathSciNet  Google Scholar 

  2. Crama, Y., Rodríguez-Heck, E.: A class of valid inequalities for multilinear 0–1 optimization problems. Discrete Optim. 25, 28–47 (2017)

    Article  MathSciNet  Google Scholar 

  3. Dash, S., Gunluk, O., Wei, D.: Boolean decision rules via column generation. In: Advances in Neural Information Processing Systems, pp. 4655–4665 (2018)

    Google Scholar 

  4. Del Pia, A., Khajavirad, A.: A polyhedral study of binary polynomial programs. Math. Oper. Res. 42(2), 389–410 (2016)

    MathSciNet  MATH  Google Scholar 

  5. Del Pia, A., Khajavirad, A.: The multilinear polytope for acyclic hypergraphs. SIAM J. Optim. 28(2), 1049–1076 (2018)

    Article  MathSciNet  Google Scholar 

  6. Del Pia, A., Khajavirad, A.: On decomposability of multilinear sets. Math. Program. 170(2), 387–415 (2018)

    Article  MathSciNet  Google Scholar 

  7. Pia, D., Alberto, K., Aida, S., Nikolaos, V.: On the impact of running intersection inequalities for globally solving polynomial optimization problems. Math. Program. Comput. 1–27 (2019). https://doi.org/10.1007/s12532-019-00169-z

  8. Demiriz, A., Bennett, K.P., Shawe-Taylor, J.: Linear programming boosting via column generation. Mach. Learn. 46, 225–254 (2002)

    Article  Google Scholar 

  9. Dobkin, D.P., Gunopulos, D., Maass, W.: Computing the maximum bichromatic discrepancy, with applications to computer graphics and machine learning. J. Comput. Syst. Sci. 52, 453–470 (1996)

    Article  MathSciNet  Google Scholar 

  10. Eckstein, J., Goldberg, N.: An improved branch-and-bound method for maximum monomial agreement. INFORMS J. Comput. 24(2), 328–341 (2012)

    Article  MathSciNet  Google Scholar 

  11. Eckstein, J., Kagawa, A., Goldberg, N.: REPR: rule-enhanced penalized regression. INFORMS J. Optim. 1(2), 143–163 (2019)

    Article  MathSciNet  Google Scholar 

  12. Fischer, A., Fischer, F.: Complete description for the spanning tree problem with one linearised quadratic term. Oper. Res. Lett. 41, 701–705 (2013)

    Article  MathSciNet  Google Scholar 

  13. Fischer, A., Fischer, F., McCormick, S.T.: Matroid optimisation problems with monotone monomials in the objective (2017, preprint)

    Google Scholar 

  14. Fischer, A., Fischer, F., McCormick, S.T.: Matroid optimisation problems with nested non-linear monomials in the objective function. Math. Program. 169(2), 417–446 (2017). https://doi.org/10.1007/s10107-017-1140-9

    Article  MathSciNet  MATH  Google Scholar 

  15. Goldberg, N., Eckstein, J.: Boosting classifiers with tightened \(l_0\)-relaxation penalties. In: 27th International Conference on Machine Learning, Haifa, Israel (2010)

    Google Scholar 

  16. Golderg, N.: Optimization for sparse and accurate classifiers. Ph.D. thesis, Rutgers University, New Brunswick, NJ (2012)

    Google Scholar 

  17. Günlük, O., Pochet, Y.: Mixing mixed-integer inequalities. Math. Program. 90(3), 429–457 (2001)

    Article  MathSciNet  Google Scholar 

  18. Mehrotra, A.: Cardinality constrained boolean quadratic polytope. Discrete Appl. Math. 79, 137–154 (1997)

    Article  MathSciNet  Google Scholar 

  19. Padberg, M.: The boolean quadric polytope: some characteristics, facets and relatives. Math. Program. 45(1–3), 139–172 (1989)

    Article  MathSciNet  Google Scholar 

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Author information

Authors and Affiliations

  1. University of Wisconsin-Madison, Madison, WI, 53706, USA

    Rui Chen

  2. IBM T. J. Watson Research Center, Yorktown Heights, NY, 10598, USA

    Sanjeeb Dash

  3. School of ORIE, Cornell University, Ithaca, NY, 14853, USA

    Oktay Günlük

Authors
  1. Rui Chen
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  2. Sanjeeb Dash
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  3. Oktay Günlük
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Corresponding author

Correspondence to Rui Chen .

Editor information

Editors and Affiliations

  1. CNRS and Université Clermont Auvergne, Aubière, France

    Mourad Baïou

  2. Département d'informatique et de recherche opérationnelle & CIRRELT, University of Montreal, Montréal, QC, Canada

    Bernard Gendron

  3. School of Operations Research and Information Engineering, Cornell University, Ithaca, NY, USA

    Oktay Günlük

  4. Laboratoire LAMSADE, Université Paris-Dauphine, Paris, France

    A. Ridha Mahjoub

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Chen, R., Dash, S., Günlük, O. (2020). Cardinality Constrained Multilinear Sets. In: Baïou, M., Gendron, B., Günlük, O., Mahjoub, A.R. (eds) Combinatorial Optimization. ISCO 2020. Lecture Notes in Computer Science(), vol 12176. Springer, Cham. https://doi.org/10.1007/978-3-030-53262-8_5

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  • DOI: https://doi.org/10.1007/978-3-030-53262-8_5

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  • Publisher Name: Springer, Cham

  • Print ISBN: 978-3-030-53261-1

  • Online ISBN: 978-3-030-53262-8

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