Advertisement

Journal of Global Optimization

, Volume 46, Issue 2, pp 191–206 | Cite as

Global optimality conditions for quadratic 0-1 optimization problems

  • Wei Chen
  • Liansheng Zhang
Article

Abstract

In the present work, we intend to derive conditions characterizing globally optimal solutions of quadratic 0-1 programming problems. By specializing the problem of maximizing a convex quadratic function under linear constraints, we find explicit global optimality conditions for quadratic 0-1 programming problems, including necessary and sufficient conditions and some necessary conditions. We also present some global optimality conditions for the problem of minimization of half-products.

Keywords

Global optimization Quadratic programming Global optimality condition Quadratic 0-1 programming Half-products 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Alidaee B., Kochenberger G., Ahmadian A.: 0-1 quadratic programming approach for the optimal solution of two scheduling problems. Int. J. Syst. Sci. 25, 401–408 (1994)CrossRefGoogle Scholar
  2. 2.
    Beck A., Teboulle M.: Global optimality conditions for quadratic optimization problems with binary constraints. Siam. J. Optim. 11(1), 179–188 (2000)CrossRefGoogle Scholar
  3. 3.
    Bomze I.M., Danninger G.: A global optimization algorithm for concave quadratic programming problems. Siam. J. Optim. 3(4), 826–842 (1993)CrossRefGoogle Scholar
  4. 4.
    Bomze I.M., Danninger G.: A finite algorithm for solving general quadratic problems. J. Glob. Optim. 4(1), 1–16 (1994)CrossRefGoogle Scholar
  5. 5.
    Boros E., Hammer P.L.: Pseudo-Boolean Optimization. Discret. Appl. Math. 123(1–3), 155–225 (2002)Google Scholar
  6. 6.
    Danninger, G.: Communicated by G. Leitmann, Role of copositivity in optimality criteria for nonconvex optimization problems. J. Optim. Theory Appl. 75(3) (1992)Google Scholar
  7. 7.
    Forrester R., Greenberg H.: Quadratic binary programming models in computational biology. Algorithmic Oper. Res. 3, 110–129 (2008)Google Scholar
  8. 8.
    Fung H.K., Taylor M.S., Floudas C.A.: Novel formulations for the sequence selection problem in de novo protein design with flexible templates. Optim. Meth. Softw. 22, 51–71 (2007)CrossRefGoogle Scholar
  9. 9.
    Helmberg C., Rendl F.: Solving quadratic (01)-problems by semidefinite programs and cutting planes. Math. Program. 82, 291–315 (1998)Google Scholar
  10. 10.
    Hiriart-Urruty J.-B.: Conditions for global optimality. In: Horst, R., Pardalos, P.M. (eds) Handbook of global optimization, pp. 1–26. Kluwer, Dordrecht (1995)Google Scholar
  11. 11.
    Hiriart-Urruty J.-B.: Global optimality conditions in maximizing a convex quadratic function under convex quadratic constraints. J. Glob. Optim. 21, 445–455 (2001)CrossRefGoogle Scholar
  12. 12.
    Horst R., Pardalos P.M., Thoai N.V.: Introduction to global optimization. Kluwer, Dordrecht (1995)Google Scholar
  13. 13.
    Huang H.-Z., Pardalos P.M., Prokopyev O.A.: Lower bound improvement and forcing rule for quadratic binary programming. Comput. Optim. Appl. 33, 187–208 (2006)CrossRefGoogle Scholar
  14. 14.
    Iasemidis L.D., Pardalos P.M., Sackellares J.C., Shiau D.S.: Quadratic binary programming and dynamical system approach to determine the predictability of epileptic seizures. J. Comb. Optim. 5, 9–26 (2001)CrossRefGoogle Scholar
  15. 15.
    Jeyakumar V., Huy N.Q.: Global minimization of difference of quadratic and convex functions over box or binary constraints. Optim. Lett. 2, 223–238 (2008)CrossRefGoogle Scholar
  16. 16.
    Jeyakumar V., Wu Z.Y.: Conditions for global optimality of quadratic minimization problems with LMI and bound constraints, Special Issue of the International Conference, SJOM2005, Singapore. Asia-Pac. J. Oper. Res. 24(2), 149–160 (2007)CrossRefGoogle Scholar
  17. 17.
    Jeyakumar V., Rubinov A.M., Wu Z.Y.: Sufficient global optimality conditions for non-convex quadratic minimization problems with box constraints. J. Glob. Optim. 36, 471–481 (2006)CrossRefGoogle Scholar
  18. 18.
    Jeyakumar V., Rubinov A.M., Wu Z.Y.: Non-convex quadratic minimization problems with quadratic constraints: global optimality conditions. Math. Program. 110(3), 521–541 (2007)CrossRefGoogle Scholar
  19. 19.
    More J.J.: Generalizations of the trust region problem. Optim. Meth. Softw. 2, 189–209 (1993)CrossRefGoogle Scholar
  20. 20.
    Palubeckis G.: Multistart tabu search strategies for the unconstrained binary quadratic optimization problem. Ann. Oper. Res. 131, 259–282 (2004)CrossRefGoogle Scholar
  21. 21.
    Pardalos P.M., Prokopyev O.A., Shylo O., Shylo V.: Global equilibrium search applied to the unconstrained binary quadratic optimization problem. Optim. Meth. Softw. 23(1), 129–140 (2008)CrossRefGoogle Scholar
  22. 22.
    Pardalos P.M., Rodgers G.P.: Computational aspects of a branch and bound algorithm for quadratic zeroone programming. Computing 45, 131–144 (1990)CrossRefGoogle Scholar
  23. 23.
    Pardalos P.M., Chaovalitwongse W., Iasemidis L.D., Sackellares J.C., Shiau D.-S., Carney P.R., Prokopyev O.A., Yatsenko V.A.: Seizure warning algorithm based on optimization and nonlinear dynamics. Math. Program. 101(2), 365–385 (2004)CrossRefGoogle Scholar
  24. 24.
    Peng J.M., Yuan Y.X.: Optimality condtions for the minimization of a qudratic with two quadratic constraints. Siam. J. Optim. 7(3), 579–594 (1997)CrossRefGoogle Scholar
  25. 25.
    Strekalovsky A.S.: Global optimality conditions for nonconvex optimization. J. Glob. Optim. 12, 415–434 (1998)CrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC. 2009

Authors and Affiliations

  1. 1.Department of MathematicsShanghai UniversityShanghaiChina

Personalised recommendations