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Doubly nonnegative relaxations for quadratic and polynomial optimization problems with binary and box constraints

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Abstract

We propose a doubly nonnegative (DNN) relaxation for polynomial optimization problems (POPs) with binary and box constraints. This work is an extension of the work by Kim, Kojima and Toh in 2016 from quadratic optimization problems to POPs. The dense and sparse DNN relaxations are reduced to a simple conic optimization problem (COP) to which an accelerated bisection and projection (BP) algorithm is applied. The COP involves a single equality constraint in a matrix variable which is restricted to the intersection of the positive semidefinite cone and a polyhedral cone representing the algebraic properties of binary and box constraints as well as the sparsity structure of the objective polynomial. Our DNN relaxation serves as a variant of the Lasserre moment-SOS hierarchy of relaxations for binary and box constrained POPs when the size of the cones is gradually increased to compute tighter lower bounds for their optimal values. A significant part of the paper is devoted to deriving and analyzing a class of polyhedral cones to strengthen the DNN relaxation, and the design of efficient computation of the metric projections onto these cones in the accelerated BP algorithm. Numerical results on various POPs are available in Ito et al. (ACM Trans Math Softw 45, Article 34, 2019).

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Notes

  1. Linear inequality constraints are converted to linear ones by adding slacks.

References

  1. Arima, N., Kim, S., Kojima, M.: A quadratically constrained quadratic optimization model for completely positive cone programming. SIAM J. Optim. 23(4), 2320–2340 (2013)

    Article  MathSciNet  Google Scholar 

  2. Arima, N., Kim, S., Kojima, M., Toh, K.C.: Lagrangian-conic relaxations, part I: a unified framework and its applications to quadratic optimization problems. Pac. J. Optim. 14(1), 161–192 (2018)

    MathSciNet  MATH  Google Scholar 

  3. Arima, N., Kim, S., Kojima, M., Toh, K.C.: A robust Lagrangian-DNN method for a class of quadratic optimization problems. Comput. Optim. Appl. 66(3), 453–479 (2017)

    Article  MathSciNet  Google Scholar 

  4. Beck, A., Teboulle, M.: A fast iterative shrinkage-thresholding algorithm for linear inverse problems. SIAM J. Imaging Sci. 2, 183–202 (2009)

    Article  MathSciNet  Google Scholar 

  5. Best, M.J., Chakravarti, N.: Active set algorithms for isotonic regression; a unifying framework active set algorithms for isotonic regression; a unifying framework. Math. Prog. 47, 425–439 (1990)

    Article  Google Scholar 

  6. Blair, J.R.S., Peyton, B.: An introduction to chordal graphs and clique trees. In: Liu, J.W.H., George, A., Gilbert, J.R. (eds.) Graph Theory and Sparse Matrix Computation. Springer-Verlag, New York (1993)

    Google Scholar 

  7. Burer, S.: Optimizing a polyhedral-semidefinite relaxation of completely positive programs. Math. Program. Comput. 2, 1–19 (2010)

    Article  MathSciNet  Google Scholar 

  8. Dong, H., Anstreicher, K.: Separating doubly nonnegative and completely positive matrices. Math. Program. 137, 131–153 (2013)

    Article  MathSciNet  Google Scholar 

  9. Fu, T., Ge, D., Ye, Y.: On doubly positive semidefinite programming relaxations. J. Comput. Math. 36(3), 391–403 (2018)

    Article  MathSciNet  Google Scholar 

  10. Fukuda, M., Kojima, M., Murota, K., Nakata, K.: Exploiting sparsity in semidefinite programming via matrix completion. I: general framework. SIAM J. Optim. 11, 647–674 (2000)

    Article  MathSciNet  Google Scholar 

  11. Hahn, P., Anjos, M.: QAPLIB—a quadratic assignment problem library. http://www.seas.upenn.edu/qaplib

  12. Ito, N.: Applications of accelerated proximal gradient methods to binary classification and polynomial optimization. Ph.D. thesis, The University of Tokyo (2018)

  13. Ito, N., Kim, S., Kojima, M., Takeda, A., Toh, K.C.: BBCPOP: a sparse doubly nonnegative relaxation of polynomial optimization problems with binary, box and complementarity constraints. ACM Trans. Math. Softw. 45, Article 34 (2019)

  14. Kim, S., Kojima, M., Toh, K.C.: Doubly nonnegative relaxations for quadratic and polynomial optimization problems with binary and box constraints. Research Rport B-483, Tokyo Institute of Technology, Department of Mathematical and Computing Sciences, Oh-Okayama, Meguro-ku, Tokyo 152-8552, July 2016

  15. Kim, S., Kojima, M., Toh, K.C.: A Lagrangian-DNN relaxation: a fast method for computing tight lower bounds for a class of quadratic optimization problems. Math. Program. 156, 161–187 (2016)

    Article  MathSciNet  Google Scholar 

  16. Lasserre, J.B.: An explicit exact SDP relaxation for nonlinear 0-1 programs. In: Integer Programming and Combinatorial Optimization, pp. 293–303. Springer (2001)

  17. Lasserre, J.B.: Global optimization with polynomials and the problems of moments. SIAM J. Optim. 11, 796–817 (2001)

    Article  MathSciNet  Google Scholar 

  18. Lasserre, J.B., Toh, K.C., Yang, S.: A bounded degree SOS hierarchy for polynomial optimization. EURO J. Comput. Optim. 5(1–2), 87–117 (2017)

    Article  MathSciNet  Google Scholar 

  19. Laurent, M.: Sums of squares, moment matrices and optimization over polynomials sums of squares, moment matrices and optimization over polynomials. In: Putinar, M., Sullivant, S. (eds.) Emerging Applications of Algebraic Geometry. The IMA Volumes in Mathematics and Its Applications, vol. 149, pp. 157–270. Springer, New York, NY (2008)

    Google Scholar 

  20. Moreau, J.J.: Décomposition orthogonale d’un espace hilbertien selon deux cones mutuellement polaires. C. R. Acad. Sci. 255, 238–240 (1962)

    MATH  Google Scholar 

  21. Nie, J., Wang, L.: Regularization methods for SDP relaxations in large scale polynomial optimization. SIAM J. Optim. 22(2), 408–428 (2012)

    Article  MathSciNet  Google Scholar 

  22. Pardalos, P.M., Xue, G.: Algorithms for a class of isotonic regression problems. Algorithmica 23, 211–222 (1999)

    Article  MathSciNet  Google Scholar 

  23. Sloane, N.: Challenge problems: independent sets in graphs

  24. Waki, H., Kim, S., Kojima, M., Muramatsu, M.: Sums of squares and semidefinite programming relaxations for polynomial optimization problems with structured sparsity. SIAM J. Optim. 17, 218–242 (2006)

    Article  MathSciNet  Google Scholar 

  25. Weisser, T., Lasserre, J.B., Toh, K.C.: Sparse-BSOS: a bounded degree SOS hierarchy for large scale polynomial optimization with sparsity. Math. Program. Comput. 10(1), 1–32 (2018)

    Article  MathSciNet  Google Scholar 

  26. Wiegele, A.: Biq Mac library. http://www.biqmac.uni-klu.ac.at/biqmaclib.html (2007)

  27. Yang, L.Q., Sun, D.F., Toh, K.C.: SDPNAL+: a majorized semismooth Newton-CG augmented Lagrangian method for semidefinite programming with nonnegative constraints. Math. Program. Comput. 7, 331–366 (2015)

    Article  MathSciNet  Google Scholar 

  28. Yoshise, A., Matsukawa, Y.: On optimization over the doubly nonnegative cone. In: IEEE Multi-conference on Systems and Control (2010)

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Acknowledgements

The authors would like to thank Prof. Satoru Iwata for the information on a closed relation of the class of polyhedral cone discussed in Sect. 3 to the isotonic regression problem, and Dr. Naoki Ito for improving the Matlab codes for Algorithm 2.1.

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Authors and Affiliations

  1. Department of Mathematics, Ewha W. University, 52 Ewhayeodae-gil, Seodaemun-gu, Seoul, 120-750, Korea

    Sunyoung Kim

  2. Department of Industrial and Systems Engineering, Chuo University, Tokyo, 192-0393, Japan

    Masakazu Kojima

  3. Department of Mathematics, and Institute of Operations Research and Analytics, National University of Singapore, 10 Lower Kent Ridge Road, Singapore, 119076, Singapore

    Kim-Chuan Toh

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  1. Sunyoung Kim
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  2. Masakazu Kojima
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  3. Kim-Chuan Toh
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Correspondence to Kim-Chuan Toh.

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S.K.: The research was supported by National Research Foundation of Korea (Grant Nos. 2017-R1A2B2005119, 2014-R1A2A1A11049618). M.K.: This research was supported by Grant-in-Aid for Scientific Research (A) 26242027 and the Japan Science and Technology Agency (JST), the Core Research of Evolutionary Science and Technology (CREST) research project. K.-C.T.: This research was supported by Ministry of Education - Singapore (Grant No. R-146-000-257-112).

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Kim, S., Kojima, M. & Toh, KC. Doubly nonnegative relaxations for quadratic and polynomial optimization problems with binary and box constraints. Math. Program. 193, 761–787 (2022). https://doi.org/10.1007/s10107-020-01594-y

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  • DOI: https://doi.org/10.1007/s10107-020-01594-y

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