Journal of Global Optimization

, Volume 7, Issue 1, pp 1–31 | Cite as

A reformulation-convexification approach for solving nonconvex quadratic programming problems

  • Hanif D. Sherali
  • Cihan H. Tuncbilek


In this paper, we consider the class of linearly constrained nonconvex quadratic programming problems, and present a new approach based on a novel Reformulation-Linearization/Convexification Technique. In this approach, a tight linear (or convex) programming relaxation, or outer-approximation to the convex envelope of the objective function over the constrained region, is constructed for the problem by generating new constraints through the process of employing suitable products of constraints and using variable redefinitions. Various such relaxations are considered and analyzed, including ones that retain some useful nonlinear relationships. Efficient solution techniques are then explored for solving these relaxations in order to derive lower and upper bounds on the problem, and appropriate branching/partitioning strategies are used in concert with these bounding techniques to derive a convergent algorithm. Computational results are presented on a set of test problems from the literature to demonstrate the efficiency of the approach. (One of these test problems had not previously been solved to optimality). It is shown that for many problems, the initial relaxation itself produces an optimal solution.

Key words

Quadratic programming indefinite quadratic problems reformulation-linearization technique reformulation-convexification approach outer-approximations tight linear programming relaxations 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Al-Khayyal, F. A. and J. E. Falk (1983), Jointly Constrained Biconvex Programming,Math, of Oper. Res. 8, 273–286.Google Scholar
  2. 2.
    Al-Khayyal, F. A. and C. Larson (1990), Global Minimization of a Quadratic Function Subject to a Bounded Mixed Integer Constraint Set,Annals of Operations Research 25, 169–180.Google Scholar
  3. 3.
    Balas, E. (1975), Nonconvex Quadratic Programming via Generalized Polars,SIAM Journal on Applied Math. 28, 335–349.Google Scholar
  4. 4.
    Benacer, R. and PhamDinh, Tao (1986), Global Maximization of a Nondefinite Quadratic Function over a Convex Polyhedron, pp. 65–76, inFermat Days 85: Mathematics for Optimization, J. B. Hiriart-Urruty (ed.), North-Holland, Amsterdam.Google Scholar
  5. 5.
    Bomze, I. M. (1992), Copositivity Conditions for Global Optimality Indefinite Quadratic Programming Problems,Czechoslovak Journal for Operations Research 1, 1–19.Google Scholar
  6. 6.
    Fisher, M. L. (1981), The Lagrangian Relaxation Methods for Solving Integer Programming Problems,Management Science 27, 1–18.Google Scholar
  7. 7.
    Floudas, C. A. and P. M. Parsalos (1990),A Collection of Test Problems for Constrained Global Optimization Algorithms, Springer-Verlag, Berlin.Google Scholar
  8. 8.
    Floudas, C. A. and V. Visweswaran (1990), A Global Optimization Algorithm (GOP) for Certain Classes of Nonconvex NLP's-1. Theory,Computers and Chemical Engineering 14, 1419.Google Scholar
  9. 9.
    Floudas, C. A. and V. Visweswaran (1993), A Primal-Relaxed Dual Global Optimization Approach: Theory,JOTA 78, 187–225.Google Scholar
  10. 10.
    Golub, G. H. and C. F. Van Loan (1989),Matrix Computations, Second Edition, The Johns Hopkins University Press, Baltimore.Google Scholar
  11. 11.
    Guignard, M. and S. Kim (1987), Lagrangian Decomposition: A Model Yielding Stronger Lagrangian Bounds,Mathematical Programming 39, 215–228.Google Scholar
  12. 12.
    Hansen, P., B. Jaumard, and S. Lu (1991), An Analytical Approach to Global Optimization,Math. Programming 52, 227–254.Google Scholar
  13. 13.
    Kough, P. F., The Indefinite Quadratic Programming Problem,Operations Research 27(3), 516–533.Google Scholar
  14. 14.
    Larsson, T. and Z. Liu (1989), A Primal Convergence Result for Dual Subgradient Optimization with Applications to Multicommunity Network Flows, Research Report, Department of Mathematics, Linkoping Institute of Technology, S-581 83, Linkoping, Sweden.Google Scholar
  15. 15.
    Manas, M. (1968), An Algorithm for a Nonconvex Programming Problem,Econ Math Obzor Acad. Nad. Ceskoslov 4(2), 202–212.Google Scholar
  16. 16.
    Meyer, G. G. (1988), Convergence of Relaxation Algorithms by Averaging,Mathematical Programming 40, 205–212.Google Scholar
  17. 17.
    Mueller, R. K. (1970), A Method for Solving the Indefinite Quadratic Programming Problem,Management Science 16(5), 333–339.Google Scholar
  18. 18.
    Murtagh, B. A. and M. A. Saunders (1987), MINOS 5.1 User's Guide, Technical Report Sol 83-20R, Systems Optimization Laboratory, Department of Operations Research, Stanford University, Stanford, California.Google Scholar
  19. 19.
    Muu, L. D. and W. Oettli (1991), An Algorithm for Indefinite Quadratic Programming with Convex Constraints,Operations Research Letters 10, 323–327.Google Scholar
  20. 20.
    Pardalos, P. M. (1991), Global Optimization Algorithms for Linearly Constrained Indefinite Quadratic Programs,Computers Math. Applic. 21, 87–97.Google Scholar
  21. 21.
    Pardalos, P. M., J. H. Glick, and J. B., Rosen (1987), Global Minimization of Indefinite Quadratic Problems,Computing 39, 281–291.Google Scholar
  22. 22.
    Pardalos, P. M. and J. B. Rosen (1987),Constrained Global Optimization: Algorithms and Applications, Springer-Verlag, Berlin.Google Scholar
  23. 23.
    Pardalos, P. M. and S. A. Vavasis (1991), Quadratic Programming with One Negative Eigenvalue Is NP-Hard,Journal of Global Optimization 1, 15–22.Google Scholar
  24. 24.
    Phillips, A. T. and J. B. Rosen (1990), Guaranteedɛ-Approximate Solution for Indefinite Quadratic Global Minimization,Naval Research Logistics 37, 499–514.Google Scholar
  25. 25.
    Ritter, K. (1966), A Method for Solving Maximum Problems with a Nonconcave Quadratic Objective Function,Z. Wahrscheinlichkeitstheorie,4, 340–351.Google Scholar
  26. 26.
    Sherali, H. D. and A. R. Alameddine (1992), A New Reformulation-Linearization Technique for Bilinear Programming Problems,Journal of Global Optimization 2(3), 379–410.Google Scholar
  27. 27.
    Sherali, H. D. and D. C. Myers (1985/6), The Design of Branch and Bound Algorithms for a Class of Nonlinear Integer Programs,Annals of Oper. Res. 5, 463–484.Google Scholar
  28. 28.
    Sherali, H. D. and C. H. Tuncbilek (1992), A Global Optimization Algorithm for Polynomial Programming Problems Using a Reformulation-Linearization Technique,The Journal of Global Optimization 2, 101–112.Google Scholar
  29. 29.
    Sherali, H. D. and O. Ulular (1989), A Primal-Dual Conjugate Subgradient Algorithm for Specially Structured Linear and Convex Programming Problems,Appl. Math. Optim. 20, 193–221.Google Scholar
  30. 30.
    Tuncbilek, C. H. (1994),Polynomial and Indefinite Quadratic Programming Problems: Algorithms and Applications, PhD Dissertation, Industrial and Systems Engineering, Virginia Polytechnic Institute and State University.Google Scholar
  31. 31.
    Tuy, H. (1987), Global Minimization of a Difference of Two Convex Functions,Mathematical Programming Study 30, 150–182.Google Scholar
  32. 32.
    Vavasis, S. A. (1992), Approximation Algorithms for Indefinite Quadratic Programming,Mathematical Programming 57, 279–311.Google Scholar
  33. 33.
    Visweswaran, V. and C. A. Floudas, (1993), New Properties and Computational Improvement of the GOP Algorithm for Problems with Quadratic Objective Function and Constraints,Journal of Global Optimization 3, 439–462.Google Scholar
  34. 34.
    Zwart, P. B. (1973), Nonlinear Programming: Counterexamples to Two Global Optimization Algorithms,Operations Research 21(6), 1260–1266.Google Scholar

Copyright information

© Kluwer Academic Publishers 1995

Authors and Affiliations

  • Hanif D. Sherali
    • 1
  • Cihan H. Tuncbilek
    • 1
  1. 1.Department of Industrial and Systems Engineering Virginia Polytechnic Institute and State University BlacksburgUSA

Personalised recommendations