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Globally solving nonconvex quadratic programming problems via completely positive programming

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Abstract

Nonconvex quadratic programming (QP) is an NP-hard problem that optimizes a general quadratic function over linear constraints. This paper introduces a new global optimization algorithm for this problem, which combines two ideas from the literature—finite branching based on the first-order KKT conditions and polyhedral-semidefinite relaxations of completely positive (or copositive) programs. Through a series of computational experiments comparing the new algorithm with existing codes on a diverse set of test instances, we demonstrate that the new algorithm is an attractive method for globally solving nonconvex QP.

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References

  1. Belotti, P.: Couenne: a user’s manual. Technical report, Clemson University. http://projects.coin-or.org/Couenne/browser/trunk/Couenne/doc/couenne-user-manual.pdf (2010)

  2. Belotti, P., Lee, J., Liberti, L., Margot, F., Wächter, A.: Branching and bounds tightening techniques for non-convex MINLP. Optim. Methods Softw. 24(4–5), 597–634 (2009). http://www.optimization-online.org/DB_HTML/2008/08/2059.html

  3. Burer S.: On the copositive representation of binary and continuous nonconvex quadratic programs. Math. Program. 120(2), 479–495 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  4. Burer S.: Optimizing a polyhedral-semidefinite relaxation of completely positive programs. Math. Prog. Comp. 2(1), 1–19 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  5. Burer S., Vandenbussche D.: Solving lift-and-project relaxations of binary integer programs. SIAM J. Optim. 16(3), 726–750 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  6. Burer S., Vandenbussche D.: A finite branch-and-bound algorithm for nonconvex quadratic programming via semidefinite relaxations. Math. Program. 113(2, Ser. A), 259–282 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  7. Burer S., Vandenbussche D.: Globally solving box-constrained nonconvex quadratic programs with semidefinite-based finite branch-and-bound. Comput. Optim. Appl. 43(2), 181–195 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  8. Celis, M.R., Dennis, J.E., Tapia, R.A.: A trust region strategy for nonlinear equality constrained optimization. In: Numerical Optimization, 1984 (Boulder, Colo., 1984), pp. 71–82. SIAM, Philadelphia (1985)

  9. Dolan E.D., Moré J.J.: Benchmarking optimization software with performance profiles. Math. Program. 91(2, Ser. A), 201–213 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  10. Globallib: Gamsworld global optimization library. http://www.gamsworld.org/global/globallib.htm

  11. Gould N., Orban D., Toint P.L.: CUTEr (and SifDec): a constrained and unconstrained testing environment, revisited. ACM Trans. Math. Softw. 29(4), 373–394 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  12. Gould, N.I.M., Toint, P.L.: Numerical methods for large-scale non-convex quadratic programming. In: Trends in Industrial and Applied Mathematics (Amritsar, 2001). Appl. Optim., vol. 72, pp. 149–179. Kluwer, Dordrecht (2002)

  13. Lin Y., Cryer C.W.: An alternating direction implicit algorithm for the solution of linear complementarity problems arising from free boundary problems. Appl. Math. Optim. 13(1), 1–17 (1985)

    Article  MathSciNet  MATH  Google Scholar 

  14. Lootsma F.A., Pearson J.D.: An indefinite-quadratic-programming model for a continuous-production problem. Philips Res. Rep. 25, 244–254 (1970)

    MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  16. MathWorks: MATLAB Reference Guide. The MathWorks Inc., Natick (2010)

  17. Moré J.J., Toraldo G.: Algorithms for bound constrained quadratic programming problems. Numer. Math. 55(4), 377–400 (1989)

    Article  MathSciNet  MATH  Google Scholar 

  18. Nguyen T., Welsch R.: Outlier detection and least trimmed squares approximation using semi-definite programming. Comput. Stat. Data Anal. 54, 3212–3226 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  19. Nocedal J., Wright S.: Numerical Optimization. Springer, New York (1999)

    Book  MATH  Google Scholar 

  20. Pardalos P.: Global optimization algorithms for linearly constrained indefinite quadratic problems. Comput. Math. Appl. 21, 87–97 (1991)

    Article  MathSciNet  MATH  Google Scholar 

  21. Pardalos P.M., Vavasis S.A.: Quadratic programming with one negative eigenvalue is NP-hard. J. Global Optim. 1(1), 15–22 (1991)

    Article  MathSciNet  MATH  Google Scholar 

  22. Sahinidis N.V.: BARON: a general purpose global optimization software package. J. Global Optim. 8(2), 201–205 (1996)

    Article  MathSciNet  MATH  Google Scholar 

  23. Sherali H.D., Adams W.P.: A Reformulation-Linearization Technique for Solving Discrete and Continuous Nonconvex Problems. Nonconvex Optimization and its Applications, vol. 31. Kluwer, Dordrecht (1999)

    Book  Google Scholar 

  24. Skutella M.: Convex quadratic and semidefinite programming relaxations in scheduling. J. ACM 48(2), 206–242 (2001)

    Article  MathSciNet  Google Scholar 

  25. Vandenbussche D., Nemhauser G.: A branch-and-cut algorithm for nonconvex quadratic programs with box constraints. Math. Program. 102(3), 559–575 (2005)

    Article  MathSciNet  MATH  Google Scholar 

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Authors and Affiliations

  1. Mathematics and Computer Science Division, Argonne National Laboratory, Argonne, IL, 60439, USA

    Jieqiu Chen

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

    Samuel Burer

Authors
  1. Jieqiu Chen
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  2. Samuel Burer
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Correspondence to Jieqiu Chen.

Additional information

The submitted manuscript has been created by the UChicago Argonne, LLC, Operator of Argonne National Laboratory (“Argonne”) under Contract No. DE-AC02-06CH11357 with the U.S. Department of Energy. The U.S. Government retains for itself, and others acting on its behalf, a paid-up, nonexclusive, irrevocable worldwide license in said article to reproduce, prepare derivative works, distribute copies to the public, and perform publicly and display publicly, by or on behalf of the Government.

The research of J. Chen and S. Burer was supported in part by NSF Grant CCF-0545514. J. Chen was supported in part by the Office of Advanced Scientific Computing Research, Office of Science, U.S. Department of Energy, under Contract DE-AC02-06CH11357.

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Chen, J., Burer, S. Globally solving nonconvex quadratic programming problems via completely positive programming. Math. Prog. Comp. 4, 33–52 (2012). https://doi.org/10.1007/s12532-011-0033-9

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