Journal of Global Optimization

, Volume 26, Issue 2, pp 199–219 | Cite as

A Global Optimization Method for Solving Convex Quadratic Bilevel Programming Problems

  • Le Dung Muu
  • Nguyen Van Quy


We use the merit function technique to formulate a linearly constrained bilevel convex quadratic problem as a convex program with an additional convex-d.c. constraint. To solve the latter problem we approximate it by convex programs with an additional convex-concave constraint using an adaptive simplicial subdivision. This approximation leads to a branch-and-bound algorithm for finding a global optimal solution to the bilevel convex quadratic problem. We illustrate our approach with an optimization problem over the equilibrium points of an n-person parametric noncooperative game.

Convex quadratic bilevel programming Merit function Saddle function Branch-and-bound algorithm Optimization over an equilibrium set 


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  1. 1.
    An, L.T.H., Pham, D.T. and Muu, L.D. (1999), Exact penalty in D.C. programming, Vietnam Journal of Mathematics, 27, 161-178.Google Scholar
  2. 2.
    Bank, B., Guddat J., Kummer B. and Tammer K. (1982), Non-Linear Parametric Optimization, Akademie-Verlag, Berlin.Google Scholar
  3. 3.
    Bard, J.F. and Falk, J.F. (1982), An explicit solution in the multilevel programming problem, Computers and Operations Research 9, 77-100.Google Scholar
  4. 4.
    Bard, J.F. (1988), Convex two-level optimization, Mathematical Programming 40, 15-27.Google Scholar
  5. 5.
    Bard, J.F. (1983), An algorithm for solving the general bilevel programming problem, Mathematics of Operations Research 8, 260-272.Google Scholar
  6. 6.
    Bard, J.F., (1983), An efficient algorithm for a linear two-stage optimization problem, Operations Research July-August, 670-684.Google Scholar
  7. 7.
    Bard, J.F. and Moore, J.T. (1990), A branch-and-bound algorithm for the bilevel programming problem, SIAM Journal of Scientific and Statistical Computing 11, 281-292.Google Scholar
  8. 8.
    Ben-Ayed, O. (1993), Bilevel linear programming, Computer and Operations Research 20, 485–509.Google Scholar
  9. 9.
    Calvete, H.L. and Gale, C. (1998), On the quasiconcave bilevel programming problem, Journal of Optimization Theory and Applications 98, 613-621.Google Scholar
  10. 10.
    Dempe, S. (1992), A necessary and sufficient optimality condition for bilevel programming problem, Optimization 25, 341-354.Google Scholar
  11. 11.
    Falk, J. and Liu, J. (1995), On bilevel programming: Part 1: General nonlinear cases, Mathematical Programming 70, 47-72.Google Scholar
  12. 12.
    Fukushima, M. (1996), A new merit function and a successive quadratic programming algorithm for variational inequalities problems, SIAM Journal Optimization 6, 703-713.Google Scholar
  13. 13.
    Fülöp, J. (1993), On the Equivalence between a Linear Bilevel Programming Problem Linear Optimization over the Efficient Set, Working Paper WP 93-1 LORDS Computer and Automation Institute, Budapest.Google Scholar
  14. 14.
    Hansen, B., Jaumard, B. and Savarg, G. (1992), New branch-and-bound rules for linear bilevel programming, SIAM Journal on Scientific and Statistical Computing 13, 1194-1217.Google Scholar
  15. 15.
    Horst, R. and Tuy, H. (1996), Global Optimization (Detreministic Approach), 3rd Edition, Springer, Berlin.Google Scholar
  16. 16.
    Horst, R., Muu, L.D. and Nast, M., (1994), Branch-and-bound decomposition approach for solving quasiconvex-concave programs, Journal of Optimization Theory and Applications 82, 267-293.Google Scholar
  17. 17.
    Judice, J. and Faustino, A.M. (1992), A sequencial LCP method for bilevel linear programming, Annals of Operations Research 34, 89-106.Google Scholar
  18. 18.
    Konnov, I. (2001), Combined Relaxation Methods for Variational Inequalities Lecture Notes in Economics and Mathematical Systems 495, Springer, Berlin.Google Scholar
  19. 19.
    Migdalas, A. (1995), Bilevel programming in traffic planning models: Methods and challenge, Journal of Global Optimization 7, 381-405.Google Scholar
  20. 20.
    Migdalas, A., Pardalos, P. and Värbrand, P., (eds.) (1998), Multilevel Optimization: Algorithms and Applications, Kluwer Academic Publishers, Dordrecht.Google Scholar
  21. 21.
    Muu, L.D. (1993), An algorithm for solving convex programs with an additional convexconcave constraint, Mathematical Programming 61, 75-87.Google Scholar
  22. 22.
    Muu L.D. (2000), On the construction of initial polyhedral convex set for optimization problem over the efficient set and bilevel linear program, Vietnam Journal of Mathematics 28, 177-182.Google Scholar
  23. 23.
    Muu, L.D. and Oettli W. (2001), Optimization over an equilibrium sets, Optimization 49, 179–189.Google Scholar
  24. 24.
    Quy, N.V. and Muu, L.D. (2001), On penalty function method for a class of nonconvex constrained optimization problems, Vietnam Journal of Mathematics 29, 235-256.Google Scholar
  25. 25.
    Savard, G. and Gauvin, J. (1994), The steepest descent direction for the nonlinear bilevel programming problems, Operations Research Letters 15, 265-272.Google Scholar
  26. 26.
    Tuy, H., Migdalas, A. and Värbrand, P. (1993), A global optimization approach for the linear two-level programs, Journal of Global Optimization 3, 1-23.Google Scholar
  27. 27.
    Tuy, H., Migdalas, A. and Värbrand, P. (1994), A quasiconcave minimization method for solving linear two-level programs, Journal of Global Optimization 4, 243-263.Google Scholar
  28. 28.
    Vicente, L., Savarg, G. and Judice J. (1994), Descent approaches for quadratic bilevel programming, Journal of Optimization Theory and Applications 81, 379-399.Google Scholar
  29. 29.
    White, D.J. and Anandalingam, G. (1993), A penalty function approach for solving bilevel linear program, Journal of Global Optimization 3, 397–419.Google Scholar

Copyright information

© Kluwer Academic Publishers 2003

Authors and Affiliations

  • Le Dung Muu
    • 1
  • Nguyen Van Quy
    • 2
  1. 1.Hanoi Institute of MathematicsBo HoVietnam
  2. 2.The Accounting and Finance University of HanoiDong Ngac, Tu Liem, HanoiVietnam

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