Journal of Global Optimization

, Volume 13, Issue 2, pp 151–170

A Polyhedral Approach for Nonconvex Quadratic Programming Problems with Box Constraints

  • Yasutoshi Yajima
  • Tetsuya Fujie
Article

Abstract

We apply a linearization technique for nonconvex quadratic problems with box constraints. We show that cutting plane algorithms can be designed to solve the equivalent problems which minimize a linear function over a convex region. We propose several classes of valid inequalities of the convex region which are closely related to the Boolean quadric polytope. We also describe heuristic procedures for generating cutting planes. Results of preliminary computational experiments show that our inequalities generate a polytope which is a fairly tight approximation of the convex region.

Cutting plane method Linearization technique Nonconvex quadratic programs Valid inequalities 

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References

  1. 1.
    Alizadeh, W.F. (1995), Interior point methods in semidefinite programming with application to combinatorial optimization, SIAM Journal on Optimization 5: 13–51.Google Scholar
  2. 2.
    Barahona, F., Jünger, M. and Reinelt, G. (1989), Experiments in quadratic 0–1 programming, Mathematical Programming 44: 127–137.Google Scholar
  3. 3.
    Barahona, F. and Mahjoub, A.R. (1986), On the cut polytope, Mathematical Programming 36: 157–173.Google Scholar
  4. 4.
    Bazaraa, M., Sherali, H.D. and Shetty, C.M. (1993), Nonlinear Programming: Theory and Algorithms, John Wiley & Sons, New York.Google Scholar
  5. 5.
    Boros, E. and Hammer, P.L. (1991), The max–cut problem and quadratic 0–1 optimization; polyhedral aspect, relaxations and bounds, Annals of Operations Research 33: 151–180.Google Scholar
  6. 6.
    Boros, E. and Hammer, P.L. (1993), Cut polytopes, Boolean quadric polytopes and nonnegative pseudo–Boolean functions, Mathematics of Operations Research 18: 245–253.Google Scholar
  7. 7.
    Coleman, T.F. and Hulbert, L.A. (1989), A direct active set algorithm for large sparse quadratic programs with simple bounds, Mathematical Programming, 45: 373–406.Google Scholar
  8. 8.
    De Angelis, P.L., Pardalos, P.M. and Toraldo, G. (1997), Quadratic Programming with box constraints, in I.M. Bomze et al. (eds.), Developments in Global Optimization (pp. 73–93), Kluwer Academic Publishers, Dordrecht, Boston, London.Google Scholar
  9. 9.
    Delorme, C. and Poljak, S. (1993), Laplacian eigenvalues and the maximum cut problem, Mathematical Programming 62: 557–574.Google Scholar
  10. 10.
    Fujie, T. and Kojima, M. (1997), Semidefinite programming relaxation for nonconvex quadratic programs, Journal of Global Optimization 10: 367–380.Google Scholar
  11. 11.
    Hansen, P., Jaumard, B., Ruiz, M. and Xiong, J. (1993), Global minimization of indefinite quadratic functions subject to box constraints, Naval Research Logistics Quarterly 40: 373–392.Google Scholar
  12. 12.
    Helmberg, C., Rendl, F., Vanderbei, R.J. and Wolkowicz, H. (1996), An interior-point method for semidefinite programming, SIAM Journal on Optimization 6: 342–361.Google Scholar
  13. 13.
    Helmberg, C. and Rendl, F. (1995), Solving quadratic (0,1)-problems by semidefinite programs and cutting planes, ZIB Preprint SC–95-35.Google Scholar
  14. 14.
    Kalantari, B. and Bagchi, A. (1990), An algorithm for quadratic zero–one programs, Naval Research Logistics Quarterly 37: 527–538.Google Scholar
  15. 15.
    Kojima, M., Shindoh, S. and Hara, S. (1997), Interior-point methods for the monotone linear complementarity problem in symmetric matrices, SIAM Journal on Optimization 7: 86–125.Google Scholar
  16. 16.
    Padberg, M. (1989), The Boolean quadric polytope: Some characteristics, facets and relatives, Mathematical Programming 45: 139–172.Google Scholar
  17. 17.
    Pardalos, P.M. and Rodgers, G.P. (1990), Computational aspects of a branch and bound algorithm for quadratic zero–one programming, Computing 45: 131–144.Google Scholar
  18. 18.
    Pardalos, P.M. and Vavasis, S.A. (1991), Quadratic programming with one negative eigenvalue is NP–hard, Journal of Global Optimization 1: 15–22.Google Scholar
  19. 19.
    Poljak, S. and Rendl, F. (1995), Solving the max–cut problem using eigenvalues, Discrete Applied Mathematics 62: 249–278.Google Scholar
  20. 20.
    Ramana, M. (1993), An Algorithmic Analysis of Multiquadratic and Semidefinite Programming Problems, PhD thesis, Johns Hopkins University, Baltimore, MD.Google Scholar
  21. 21.
    Sherali, H. and Alameddine, A. (1990), An explicit characterization of the convex envelope of a bivariate bilinear function over special polytopes, Annals of Operations Research 25: 197–214.Google Scholar
  22. 22.
    Sherali, H. and Alameddine, A. (1992), A new reformulation–linearization for solving bilinear programming problems, Journal of Global Optimization 2: 397–410.Google Scholar
  23. 23.
    Sherali, H., Lee, Y. and Adams, W.P. (1995), A simultaneous lifting strategy for identifying new class of facets for Boolean quadric polytope, Operations Research Letters 17: 19–26.Google Scholar
  24. 24.
    Sherali, H. and Tuncbilek, C.H. (1995), A reformulation–convexification approach for solving nonconvex quadratic programming problems, Journal of Global Optimization 7: 1–31.Google Scholar
  25. 25.
    Simone, C.D. (1989), The cut polytope and the Boolean quadric polytope, Discrete Mathematics 79: 71–75.Google Scholar
  26. 26.
    Vavasis, S.A. (1992), Approximate algorithms for indefinite quadratic programming, Mathematical Programming 57: 279–311.Google Scholar
  27. 27.
    Ye, Y. (1992), On the affine scaling algorithm for nonconvex quadratic programming, Mathematical Programming 56: 285–300.Google Scholar

Copyright information

© Kluwer Academic Publishers 1998

Authors and Affiliations

  • Yasutoshi Yajima
    • 1
  • Tetsuya Fujie
    • 2
  1. 1.Department of Industrial Engineering and ManagementTokyo Institute of TechnologyOh-Okayama, Meguro-ku, TokyoJapan (e-mail: Email
  2. 2.Department of Mathematical and Computing SciencesTokyo Institute of TechnologyOh-Okayama, Meguro-ku, TokyoJapan

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