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A finite branch-and-bound algorithm for nonconvex quadratic programming via semidefinite relaxations

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Abstract

Existing global optimization techniques for nonconvex quadratic programming (QP) branch by recursively partitioning the convex feasible set and thus generate an infinite number of branch-and-bound nodes. An open question of theoretical interest is how to develop a finite branch-and-bound algorithm for nonconvex QP. One idea, which guarantees a finite number of branching decisions, is to enforce the first-order Karush-Kuhn-Tucker (KKT) conditions through branching. In addition, such an approach naturally yields linear programming (LP) relaxations at each node. However, the LP relaxations are unbounded, a fact that precludes their use. In this paper, we propose and study semidefinite programming relaxations, which are bounded and hence suitable for use with finite KKT-branching. Computational results demonstrate the practical effectiveness of the method, with a particular highlight being that only a small number of nodes are required.

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References

  1. Absil, P.-A., Tits, A.: Newton-KKT interior-point methods for indefinite quadratic programming. Manuscript, Department of Electrical and Computer Engineering, University of Maryland, College Park, MD, USA (2006). To appear in Computational Optimization and Applications

  2. Anstreicher, K.M.: Combining RLT and SDP for nonconvex QCQP. Talk given at the Workshop on Integer Programming and Continuous Optimization, Chemnitz University of Technology, November 7–9 (2004)

  3. Bomze, I.M., de Klerk, E.: Solving standard quadratic optimization problems via linear, semidefinite and copositive programming. J. Global Optim. 24(2), 163–185 (2002). Dedicated to Professor Naum Z. Shor on his 65th birthday

    Google Scholar 

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

    Article  MATH  MathSciNet  Google Scholar 

  5. Floudas, C., Visweswaran, V.: Quadratic optimization. In: Horst, R., Pardalos, P. (eds.) Handbook of Global Optimization, pp. 217–269. Kluwer Academic Publishers (1995)

  6. Giannessi, F., Tomasin, E.: Nonconvex quadratic programs, linear complementarity problems, and integer linear programs. In: Fifth Conference on Optimization Techniques (Rome, 1973), Part I, pp. 437–449. Lecture Notes in Computer Science, vol. 3. Springer, Berlin Heidelberg New York (1973)

  7. Globallib: http://www.gamsworld.org/global/globallib.htm

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

    Article  MATH  MathSciNet  Google Scholar 

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

  10. Hansen P., Jaumard B., Ruiz M. and Xiong J. (1993). Global minimization of indefinite quadratic functions subject to box constraints. Naval Res. Logist. 40(3): 373–392

    Article  MATH  MathSciNet  Google Scholar 

  11. ILOG, Inc.: ILOG CPLEX 9.0, User Manual (2003).

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

    Article  MATH  MathSciNet  Google Scholar 

  13. Kojima M. and Tunçel L. (2002). Some fundamental properties of successive convex relaxation methods on LCP and related problems. J. Global Optim. 24(3): 333–348

    Article  MATH  MathSciNet  Google Scholar 

  14. Kozlov M.K., Tarasov S.P. and Khachiyan L.G. (1979). Polynomial solvability of convex quadratic programming. Dokl. Akad. Nauk SSSR 248(5): 1049–1051

    MathSciNet  Google Scholar 

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

    MATH  MathSciNet  Google Scholar 

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

    Article  MATH  MathSciNet  Google Scholar 

  17. Nesterov Y. (1998). Semidefinite relaxation and nonconvex quadratic optimization. Optim. Methods Softw. 9: 141–160

    Article  MATH  MathSciNet  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

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

    Article  MATH  MathSciNet  Google Scholar 

  20. Sahinidis N.V. (1996). BARON a general purpose global optimization software package. J. Glob. Optim. 8: 201–205

    Article  MATH  MathSciNet  Google Scholar 

  21. Sherali H.D. and Fraticelli B.M.P. (2002). Enhancing RLTrelaxations via a new class of semidefinite cuts. J. Global Optim. 22: 233–261

    Article  MATH  MathSciNet  Google Scholar 

  22. Sherali, H.D., Tuncbilek, C.H.: A global optimization algorithm for polynomial programming problems using a reformulation-linearization technique. J. Global Optim. 2(1), 101–112 (1992) Conference on Computational Methods in Global Optimization, I (Princeton, NJ, 1991).

  23. Sherali H.D. and Tuncbilek C.H. (1995). A reformulation-convexification approach for solving nonconvex quadratic programming problems. J. Global Optim. 7: 1–31

    Article  MATH  MathSciNet  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  25. Vandenbussche D. and Nemhauser G. (2005). A polyhedral study of nonconvex quadratic programs with box constraints. Math. Program. 102(3): 531–557

    Article  MATH  MathSciNet  Google Scholar 

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

    Article  MATH  MathSciNet  Google Scholar 

  27. Ye Y. (1999). Approximating quadratic programming with bound and quadratic constraints. Math. Program. 84(2, Ser. A): 219–226

    MATH  MathSciNet  Google Scholar 

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Authors and Affiliations

  1. Department of Management Sciences, University of Iowa, Iowa City, IA, 52242-1000, USA

    Samuel Burer

  2. Axioma, Inc., Marietta, GA, 30068, USA

    Dieter Vandenbussche

Authors
  1. Samuel Burer
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  2. Dieter Vandenbussche
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Correspondence to Samuel Burer.

Additional information

This author was supported in part by NSF Grants CCR-0203426 and CCF-0545514.

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Burer, S., Vandenbussche, D. A finite branch-and-bound algorithm for nonconvex quadratic programming via semidefinite relaxations. Math. Program. 113, 259–282 (2008). https://doi.org/10.1007/s10107-006-0080-6

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  • DOI: https://doi.org/10.1007/s10107-006-0080-6

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