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Faster, but weaker, relaxations for quadratically constrained quadratic programs

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Abstract

We introduce a new relaxation framework for nonconvex quadratically constrained quadratic programs (QCQPs). In contrast to existing relaxations based on semidefinite programming (SDP), our relaxations incorporate features of both SDP and second order cone programming (SOCP) and, as a result, solve more quickly than SDP. A downside is that the calculated bounds are weaker than those gotten by SDP. The framework allows one to choose a block-diagonal structure for the mixed SOCP-SDP, which in turn allows one to control the speed and bound quality. For a fixed block-diagonal structure, we also introduce a procedure to improve the bound quality without increasing computation time significantly. The effectiveness of our framework is illustrated on a large sample of QCQPs from various sources.

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References

  1. Burer, S., Monteiro, R.: A nonlinear programming algorithm for solving semidefinite programs via low-rank factorization. Math. Program., Ser. B 95, 329–357 (2003)

    Article  MATH  MathSciNet  Google Scholar 

  2. de Klerk, E., Sotirov, R.: Exploiting group symmetry in semidefinite programming relaxations of the quadratic assignment problem. Math. Program., Ser. A 122(2), 225–246 (2010)

    Article  MATH  Google Scholar 

  3. Fujie, T., Kojima, M.: Semidefinite programming relaxation for nonconvex quadratic programs. J. Glob. Optim. 10(4), 367–380 (1997)

    Article  MATH  MathSciNet  Google Scholar 

  4. Fukuda, M., Kojima, M., Murota, K., Nakata, K.: Exploiting sparsity in semidefinite programming via matrix completion. I. General framework. SIAM J. Optim. 11(3), 647–674 (2000)

    Article  MATH  MathSciNet  Google Scholar 

  5. See the website: www.gamsworld.org/global/globallib/globalstat.htm

  6. Goemans, M., Williamson, D.: Improved approximation algorithms for maximum cut and satisfiability problems using semidefinite programming. J. ACM 42, 1115–1145 (1995)

    Article  MATH  MathSciNet  Google Scholar 

  7. Grone, R., Johnson, C., Sá, E., Wolkowicz, H.: Positive definite completions of partial Hermitian matrices. Linear Algebra Appl. 58, 109–124 (1984)

    Article  MATH  MathSciNet  Google Scholar 

  8. Helmberg, C., Rendl, F.: A spectral bundle method for semidefinite programming. SIAM J. Optim. 10, 673–696 (2000)

    Article  MATH  MathSciNet  Google Scholar 

  9. Kim, S., Kojima, M.: Second order cone programming relaxation of nonconvex quadratic optimization problems. Optim. Methods Softw. 15(3–4), 201–224 (2001)

    Article  MATH  MathSciNet  Google Scholar 

  10. Kojima, M., Tunçel, L.: Cones of matrices and successive convex relaxations of nonconvex sets. SIAM J. Optim. 10(3), 750–778 (2000)

    Article  MATH  MathSciNet  Google Scholar 

  11. Lasserre, J.B.: Global optimization with polynomials and the problem of moments. SIAM J. Optim. 11(3), 796–817 (2001)

    Article  MATH  MathSciNet  Google Scholar 

  12. Laurent, M., Rendl, F.: Semidefinite programming and integer programming. In: Karen Aardal, R.W., Nemhauser, G. (eds.) Handbook on Discrete Optimization, pp. 393–514. Elsevier, Amsterdam (2005)

    Chapter  Google Scholar 

  13. Lovász, L., Schrijver, A.: Cones of matrices and set-functions and 0-1 optimization. SIAM J. Optim. 1, 166–190 (1991)

    Article  MATH  MathSciNet  Google Scholar 

  14. Monteiro, R.D.C.: First- and second-order methods for semidefinite programming. Math. Program. 97, 209–244 (2003)

    MATH  MathSciNet  Google Scholar 

  15. Nakata, K., Fujisawa, K., Fukuda, M., Kojima, M., Murota, K.: Exploiting sparsity in semidefinite programming via matrix completion. II. Implementation and numerical results. Math. Program., Ser. B 95(2), 303–327 (2003). Computational semidefinite and second order cone programming: the state of the art

    Article  MATH  MathSciNet  Google Scholar 

  16. Rendl, F., Rinaldi, G., Wiegele, A.: Solving Max-Cut to optimality by intersecting semidefinite and polyhedral relaxations. Math. Program. 121(2), 307 (2010)

    Article  MATH  MathSciNet  Google Scholar 

  17. Saxena, A., Bonami, P., Lee, J.: Convex relaxations of non-convex mixed integer quadratically constrained programs: projected formulations. Math. Program. (2010). doi:10.1007/s10107-010-0340-3

    MathSciNet  Google Scholar 

  18. Sherali, H.D., Adams, W.P.: A Reformulation-Linearization Technique for Solving Discrete and Continuous Nonconvex Problems. Kluwer, Dordrecht (1997)

    Google Scholar 

  19. Zhao, X., Sun, D., Toh, K.: A Newton-CG augmented Lagrangian method for semidefinite programming. SIAM J. Optim. 20, 1737–1765 (2010)

    Article  MATH  MathSciNet  Google Scholar 

  20. Zheng, X., Sun, X., Li, D.: Nonconvex quadratically constrained quadratic programming: best d.c. decompositions and their sdp representations. J. Glob. Optim. 50, 695–712 (2011)

    Article  MATH  MathSciNet  Google Scholar 

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Authors and Affiliations

  1. Department of Management Sciences, University of Iowa, Iowa City, IA, 52242-1994, USA

    Samuel Burer

  2. Department of Mathematics, Ewha W. University, 11-1 DaHyun-Dong, SuhDaeMoon-Gu, Seoul, Korea, 120-750

    Sunyoung Kim

  3. Research and Development Initiative & JST CREST, Chuo University, 1-13-27, Kasuga, Bunkyo-ku, Tokyo, 112-8551, Japan

    Masakazu Kojima

Authors
  1. Samuel Burer
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  2. Sunyoung Kim
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  3. Masakazu Kojima
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Corresponding author

Correspondence to Samuel Burer.

Additional information

The research of S. Burer was supported in part by NSF Grant CCF-0545514. The research of S. Kim was supported by NRF 2012-R1A1A2-038982 and NRF 2010-000-8784. The research of M. Kojima was partially supported by the Japan Science and Technology Agency (JST), the Core Research of Evolutionary Science and Technology (CREST) Research Project.

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Burer, S., Kim, S. & Kojima, M. Faster, but weaker, relaxations for quadratically constrained quadratic programs. Comput Optim Appl 59, 27–45 (2014). https://doi.org/10.1007/s10589-013-9618-8

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  • DOI: https://doi.org/10.1007/s10589-013-9618-8

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