Skip to main content
SpringerLink
Log in
Book cover

Advanced Computational Methods for Knowledge Engineering pp 3–17Cite as

Solution Methods for General Quadratic Programming Problem with Continuous and Binary Variables: Overview

  • Conference paper

Part of the book series: Studies in Computational Intelligence ((SCI,volume 479))

Abstract

The nonconvex quadratic programming problem with continuous and/or binary variables is a typical NP-hard optimization problem, which has a wide range of applications. This article presents an overview of actual solution methods for solving this interesting and important class of programming problems. Solution methods are discussed in the sense of global optimization.

This is a preview of subscription content, log in via an institution.

Chapter
USD   29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD   169.00
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD   219.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info
Hardcover Book
USD   219.99
Price excludes VAT (USA)
  • Durable hardcover edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Learn about institutional subscriptions

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. Al-Khayyal, F.A., Falk, J.E.: Jointly constrained biconvex programming. Math. Oper. Res. 8, 273–286 (1983)

    Article  MathSciNet  MATH  Google Scholar 

  2. Al-Khayyal, F.A., Larsen, C., van Voorhis, T.: A relaxation method for nonconvex quadratically constrained quadratic programs. Journal of Global Optimization 6, 215–230 (1995)

    Article  MATH  Google Scholar 

  3. Audet, C., Hansen, P., Jaumard, B., Savard, G.: A branch and cut algorithm for nonconvex quadratically constrained quadratic programming. Math. Program. Ser. A 87, 131–152 (2000)

    MathSciNet  MATH  Google Scholar 

  4. Bazaraa, M.S., Sherali, H.D., Shetty, C.M.: Nonlinear Programming: Theory and Algorithms, 2nd edn. John Wiley & Sons, Inc. (1993)

    Google Scholar 

  5. Ben-Tal, A., Eiger, G., Gershovitz, V.: Global Minimization by Reducing the Duality Gap. Mathematical Programming 63, 193–212 (1994)

    Article  MathSciNet  MATH  Google Scholar 

  6. Berman, A., Shaked-Monderer, N.: Completely positive matrices. World Scientific (2003)

    Google Scholar 

  7. Bomze, I.M.: On standard quadratic optimization problems. Journal of Global Optimization 13, 369–387 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  8. Bomze, I.M., Danninger, G.: A global optimization algorithm for concave quadratic problems. SIAM J. Optimization 3, 826–842 (1993)

    Article  MathSciNet  MATH  Google Scholar 

  9. Bomze, I.M., Duer, M., de Klerk, E., Roos, C., Quist, A.J., Terlaky, T.: On copositive programming and standard quadratic optimization problems. Journal of Global Optimization 18, 301–320 (2000)

    Article  MATH  Google Scholar 

  10. Bomze, I.M.: Branch and bound approaches to standard quadratic optimization problems. Journal of Global Optimization 22, 17–37 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  11. Bomze, I.M., de Klerk, E.: Solving standard quadratic optimization problems via linear, semidefinite and copositive programming. Journal of Global Optimization 24, 163–185 (2002)

    Article  MATH  Google Scholar 

  12. Bomze, I.M., Jarre, F., Rendl, F.: Quadratic factorization heuristics for copositive programming. Math. Prog. Comp. 3, 37–57 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  13. Brimberg, J., Hansen, P., Mladenović, N.: A note on reduction of quadratic and bilinear programs with equality constraints. Journal of Global Optimization 22, 39–47 (2002)

    Article  MATH  Google Scholar 

  14. Bundfuss, S., Dür, M.: An adaptive linear approximation algorithm for copositive programs. SIAM Journal on Optimization 20, 30–53 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  15. Bundfuss, S., Dür, M.: Algorithmic copositivity detection by simplicial partition. Linear Algebra and its Applications 428, 1511–1523 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  16. Burer, S.: On the copositive representation of binary and continuous nonconvex quadratic programs. Mathematical Programming 120, 479–495 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  17. Dür, M.: Copositive programming – a survey. In: Diehl, M., Glineur, F., Jarlebring, E., Michiels, W. (eds.) Recent Advances in Optimization and its Applications in Engineering, pp. 3–20. Springer (2010)

    Google Scholar 

  18. Dür, M., Horst, R.: Lagrange Duality and Partioning Techniques in Nonconvex Global Optimization. Journal of Optimization Theory and Applications 95, 347–369 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  19. Falk, J.E., Hoffman, K.L.: A Successive Underestimation Method for Concave Minimization Problems. Mathematics of Operations Research 1, 251–259 (1976)

    Article  MATH  Google Scholar 

  20. Floudas, C.A., Aggarwal, A., Ciric, A.R.: Global optimum search for nonconvex NLP and MINLP problems. Computers and Chemical Engineering 13, 1117–1132 (1989)

    Article  Google Scholar 

  21. Floudas, C.A., Visweswaran, V.: A global optimization algorithm (GOP) for certain classes of nonconvex NLPs- I. Theory. Computers and Chemical Engineering 14, 1397–1417 (1990)

    Article  Google Scholar 

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

    Google Scholar 

  23. Fujie, T., Kojima, M.: Semidefinite programming relaxation for nonconvex quadratic programs. Journal of Global Optimization 10, 367–380 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  24. Geoffrion, A.M.: Duality in Nonlinear Programming: A Simplified Application-Oriented Development. SIAM Review 13, 1–37 (1971)

    Article  MathSciNet  MATH  Google Scholar 

  25. Han, C.G., Pardalos, P.M., Ye, Y.: On the solution of indefinite quadratic problems using an interior point algorithm. Informatica 3, 474–496 (1992)

    MathSciNet  MATH  Google Scholar 

  26. Hansen, P., Jaumard, B.: Reduction of indefinite quadratic programs to bilinear programs. Journal of Global Optimization 2, 41–60 (1992)

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  28. Heinkenschloss, M.: On the solution of a two ball trust region subproblem. Math. Programming 64, 249–276 (1994)

    Article  MathSciNet  MATH  Google Scholar 

  29. Hiriart-Urruty, J.B., Seeger, A.: A variational approach to copositive matrices. SIAM Rev. 52, 593–629 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  30. Horst, R., Pardalos, P.M. (eds.): Handbook of global optimization. Kluwer Academic Publihers (1995)

    Google Scholar 

  31. Horst, R., Pardalos, P.M., Thoai, N.V.: Introduction to Global Optimization, 2nd edn. Kluwer, Dordrecht (2000)

    MATH  Google Scholar 

  32. Horst, R., Thoai, N.V.: A new algorithm for solving the general quadratic programming problem. Computational Optimization and Applications 5, 39–49 (1996)

    Article  MathSciNet  MATH  Google Scholar 

  33. Horst, R., Thoai, N.V.: DC programming: Overview. Journal of Optimization Theory and Applications 103, 1–43 (1999)

    Article  MathSciNet  Google Scholar 

  34. Horst, R., Thoai, N.V., Benson, H.P.: Concave minimization via conical partitions and polyhedral outer approximation. Mathematical Programming 50, 259–274 (1991)

    Article  MathSciNet  MATH  Google Scholar 

  35. Horst, R., Tuy, H.: Global Optimization: Deterministic Approaches, 3rd edn. Springer, Berlin (1996)

    MATH  Google Scholar 

  36. Kojima, M., Tunçel, L.: Cones of matrix and successive convex ralaxations of nonconvex sets. SIAM J. Optimization 10, 750–778 (2000)

    Article  MATH  Google Scholar 

  37. Kojima, M., Tunçel, L.: Discretization and localization in successive convex relaxation methods for nonconvex quadratic optimization. Math. Program. Ser. A 89, 79–111 (2000)

    Article  MATH  Google Scholar 

  38. Kono, H.: Maximizing a convex quadratic function subject to linear constraints. Mathematical Programming 11, 117–127 (1976)

    Article  MathSciNet  Google Scholar 

  39. Le Thi, H.A., Pham, D.T.: Solving a class of linearly constrained indefinite quadratic problems by d.c. algorithm. Journal of Global Optimization 11, 253–285 (1997)

    Article  MATH  Google Scholar 

  40. Le Thi, H.A., Pham, D.T.: A branch and bound method via d.c. optimization algorithms and ellipsoidal technique for box constrained nonconvex quadratic problems. Journal of Global Optimization 13, 171–206 (1998)

    Article  MATH  Google Scholar 

  41. Locatelli, M., Raber, U.: Packing equal circes into a square: a deterministic global optimization approach. Discrete Apllied Mathematics 122, 139–166 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  42. Lovász, L., Schreijver, A.: Cone of matrices and set functions and 0-1 optimization. SIAM J. Optimization 1, 166–190 (1991)

    Article  MATH  Google Scholar 

  43. Mangasarian, O.L.: Nonlinear Programming. Robert E. Krieger Publishing Company, Huntington (1979)

    Google Scholar 

  44. Martinez, J.M.: Local minimizers of quadratic functions on Euclidean balls and spheres. SIAM J. Optimization 4, 159–176 (1994)

    Article  MATH  Google Scholar 

  45. Moré, J.: Generalizations of the trust region problem. Optimization Methods & Software 2, 189–209 (1993)

    Article  Google Scholar 

  46. Novak, I.: A new semidefinite bound for indefinite quadratic forms over a simplex. Journal of Global Optimization 14, 357–364 (1999)

    Article  Google Scholar 

  47. Novak, I.: Dual bounds and optimality cuts for all-quadratic programs with convex constraints. Journal of Global Optimization 18, 337–356 (2000)

    Article  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  49. Pardalos, P.M., Rendl, F., Wolkowicz, H.: The Quadratic Assignment Problem: A Survey and Recent Developments. DIMACS Series in Discrete Mathematics and Theoretical Computer Science, vol. 16, pp. 1–39 (1994)

    Google Scholar 

  50. Pardalos, P.M., Rosen, J.B.: Constrained Global Optimization: Algorithms and Applications. LNCS, vol. 268. Springer, Heidelberg (1987)

    Book  MATH  Google Scholar 

  51. Pardalos, P.M., Vavasis, S.A.: Quadratic programming with one negative eigenvalue is NP-hard. Journal of Global Optimization 1, 843–855 (1991)

    MathSciNet  Google Scholar 

  52. Pardalos, P.M., Ye, Y., Han, C.G.: Algorithms for the solution of quadratic knapsack problems. Linear Algebra & its Applications 152, 69–91 (1991)

    Article  MathSciNet  MATH  Google Scholar 

  53. Pham, D.T., Le Thi, H.A.: D.c. optimization algorithm for solving the trust region subproblem. SIAM J. on Optimization 8, 1–30 (1998)

    Article  Google Scholar 

  54. Phong, T.Q., Pham, D.T., Le Thi, H.A.: A method for solving d.c. programming problems. Application to fuel mixture nonconvex optimization problem. J. of Global Optimization 6, 87–105 (1995)

    Article  MATH  Google Scholar 

  55. Raber, U.: A simplicial branch and bound method for solving nonconvex all-quadratic programs. J. of Global Optimization 13, 417–432 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  56. Rendl, F., Wolkowicz, H.: A semidefinite framwork for trust region subproblems with applications to large scale minimization. Mathematical Programming (Serie B) 77, 273–299 (1997)

    MathSciNet  MATH  Google Scholar 

  57. Sahni, S.: Computationally related problems. SIAM J. Comput. 3, 262–279 (1974)

    Article  MathSciNet  Google Scholar 

  58. Sherali, H.D., Adams, W.P.: A hierarchy of relations between the continuous and convex hull representations of zero-one programming problems. SIAM J. Discrete Mathematics 3, 411–430 (1990)

    Article  MathSciNet  MATH  Google Scholar 

  59. Sherali, H.D., Adams, W.P.: A refomulation-linearization technique for solving discrete and continuous nonconvex problems. Kluwer Academic Publihers (1999)

    Google Scholar 

  60. Sherali, H.D., Tuncbilek, C.H.: A reformulation-convexification approach for solving nonconvex quadratic programming problems. J. of Global Optimization 7, 1–31 (1995)

    Article  MathSciNet  MATH  Google Scholar 

  61. Shor, N.Z.: Nondifferentiable optimization and polynomial problems. Kluwer Academic Publihers (1998)

    Google Scholar 

  62. Shor, N.Z., Stetsyuk, P.I.: Lagrangian bounds in multiextremal polynomial and discrete optimization problems. J. of Global Optimization 23, 1–41 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  63. Stern, R., Wolkowicz, H.: Indefinite trust region subproblems and nonsymmetric eigenvalue perturbation. SIAM J. Optimization 5, 286–313 (1995)

    Article  MathSciNet  MATH  Google Scholar 

  64. Takeda, A., Dai, Y., Fukuda, M., Kojima, M.: Towards the implementation of successive convex relaxation method for nonconvex quadratic optimization problems. In: Pardalos, P. (ed.) Approximation and Complexity in Numerical Optimization: Continuous and Discrete Problems. Kluwer Academic Publishers (2000)

    Google Scholar 

  65. Takeda, A., Fujisawa, K., Fukaya, Y., Kojima, M.: Parallel implementation of successive convex relation methods for quadratic optimization problems. J. of Global Optimization 24, 237–260 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  66. Thoai, N.V.: Global Optimization Techniques for Solving the General Quadratic Integer Programming Problem. Computational Optimization and Applications 10, 149–163 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  67. Thoai, N.V.: Duality Bound Method for the General Quadratic Programming Problem with Quadratic Constraints. Journal of Optimization Theory and Applications 107, 331–354 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  68. Thoai, N.V.: Convergence and Application of a Decomposition Method Using Duality Bounds for Nonconvex Global Optimization. Journal of Optimization Theory and Applications 113, 165–193 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  69. Vandenberghe, L., Boyd, S.: Semidefinite Programming. SIAM Review 38, 49–95 (1996)

    Article  MathSciNet  MATH  Google Scholar 

  70. Visweswaran, V., Floudas, C.A.: A global optimization algorithm (GOP) for certain classes of nonconvex NLPs- II. Application of theory and test problems. Computers and Chemical Engineering 14, 1419–1434 (1990)

    Article  Google Scholar 

  71. Yajima, Y., Fujie, T.: A polyhedral approach for nonconvex quadratic programming problems with box constraints. Journal of Gloabl Optimization 13, 151–170 (1998)

    Article  MathSciNet  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Mathematics, University of Trier, D-54286, Trier, Germany

    Nguyen Van Thoai

Authors
  1. Nguyen Van Thoai
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Nguyen Van Thoai .

Editor information

Editors and Affiliations

  1. , Division of Knowledge Management Systems, Wroclaw University of Technology, Str. Wyb. Wyspianskiego 27, Wroclaw, 50-370, Poland

    Ngoc Thanh Nguyen

  2. and Economics, Department of Telecommunications, Budapest University of Technology, Pazmany Peter setany 1/D, Budapest, 1111, Hungary

    Tien van Do

  3. LITA - UFR MIM, Université de Lorraine – Metz, Ile du Saulcy, Metz Cedex 01, 57045, France

    Hoai An le Thi

Rights and permissions

Reprints and permissions

Copyright information

© 2013 Springer International Publishing Switzerland

About this paper

Cite this paper

Van Thoai, N. (2013). Solution Methods for General Quadratic Programming Problem with Continuous and Binary Variables: Overview. In: Nguyen, N., van Do, T., le Thi, H. (eds) Advanced Computational Methods for Knowledge Engineering. Studies in Computational Intelligence, vol 479. Springer, Heidelberg. https://doi.org/10.1007/978-3-319-00293-4_1

Download citation

  • DOI: https://doi.org/10.1007/978-3-319-00293-4_1

  • Publisher Name: Springer, Heidelberg

  • Print ISBN: 978-3-319-00292-7

  • Online ISBN: 978-3-319-00293-4

  • eBook Packages: EngineeringEngineering (R0)

Publish with us

Policies and ethics

Search

Navigation