Bilevel Linear Programming, Multiobjective Programming, and Monotonic Reverse Convex Programming

  • Hoang Thy
Part of the Nonconvex Optimization and Its Applications book series (NOIA, volume 20)


The Bilevel Linear Programming problem and the problem of Linear Optimization over the Efficient Set are shown to be special forms of linear program with an additional reverse convex constraint having a monotonicity property. Exploiting this structure, one can convert the latter problem into a problem of much reduced dimension which can then be efficiently handled by d.c. programming decomposition methods.


Bilevel linear program optimization over the efficient set monotonic reverse convex program decomposition 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. [1]
    H.P. Benson: 1984,‘Optimization over the Efficient Set’ J. Math. anal. Appl. 98, 562–580.MathSciNetzbMATHCrossRefGoogle Scholar
  2. [2]
    S. Bolintineau: 1993, ‘Minimization of a quasiconcave function over an efficient set’, Mathematical Programming, 61, 89–110.MathSciNetCrossRefGoogle Scholar
  3. [3]
    J.P. Dauer and T.A. Fosnaugh: 1995, “Optimization over the efficient set”, Journal of Global Optimization, 7, 261–277.MathSciNetzbMATHCrossRefGoogle Scholar
  4. [4]
    J.G. Ecker and I. A. Kouda: 1975, ‘Finding efficient points for multiple objective programs’, Mathematical Programming, 8, 375–377.MathSciNetCrossRefGoogle Scholar
  5. [5]
    J. Fülöp: 1993, ‘On the equivalence between a bilinear programming problem and linear optimization over the efficient set’, Working paper WP 93–1, LORDS, Computer and Automation Institute, Budapest.Google Scholar
  6. [6]
    J. Fülöp: 1994, ‘On the Lagrange Duality of Convex Minimization Subject to Linear Constraints and an Additional Facial Reverse Convex Constraint’, Working Paper WP 94-, LORDS, Computer and Automation Institute, Budapest.Google Scholar
  7. [7]
    P. Hansen, B. Jaumard and G. Savard: 1992, ‘New branch and bound rules for linear bilevel programming’, SIAM J. Stat. Comput., 13, 1194–1217.MathSciNetzbMATHCrossRefGoogle Scholar
  8. [8]
    R. Horst and H. Thy: 1993, Global Optimization (Deterministic Approaches), second edition, Springer-Verlag, Berlin New York.Google Scholar
  9. [9]
    L.D. Muu: 1993, ‘Methods for Optimizing a Linear Function over the Efficient Set’, Preprint, Institute of Mathematics, Hanoi.Google Scholar
  10. [10]
    C.H. Papadimitriou and K. Steiglitz: 1982, Combinatorial Optimization: Algorithms and Complexity, Prentice-Hall, Inc., New Jersey.zbMATHGoogle Scholar
  11. [11]
    R.T. Rockafellar: 1970, Convex Analysis, Princeton University Press, Princeton.zbMATHGoogle Scholar
  12. [12]
    P.T. Thach: 1991, ‘Quasiconjugate of Functions, Duality Relationship between Quasi-Convex Minimization under a Reverse Convex Constraint and Quasi-Convex Maximization under a Convex Constraint, and Applications J. Math. Anal. Appl.. 159, 299–322.MathSciNetzbMATHCrossRefGoogle Scholar
  13. [13]
    P.T. Thach, H. Konno and D. Yokota: 1993 ‘Dual Approach to Minimization on the Set of Pareto-Optimal Solution’, to appear in J. Optim. Theory Appl. Google Scholar
  14. [14]
    H. Thy: 1995, ‘D.C. Optimization: Theory, Methods and Algorithms’, in Handbook of Global Optimization, eds. R. Horst and P.M. Pardalos, Kluwer Academic Publishers, Dordrecht/Boston/London, 149–216.Google Scholar
  15. [15]
    H. Thy: 1992, ‘On nonconvex optimization problems with separated nonconvex variables’, Journal of Global Optimization, 2, 133–144.CrossRefGoogle Scholar
  16. [16]
    H. Thy, A. Migdalas and P. Värbrand: 1993, ‘A Global Optimization Approach for the Linear Two-Level Program’, Journal of Global Optimization, 3, 1–23.CrossRefGoogle Scholar
  17. [17]
    H. Thy and S. Ghannadan: 1996, ‘A new branch and bound method for bilevel linear programs’, this volume.Google Scholar
  18. [18]
    H. Thy, A. Migdalas and P. Värbrand: 1994, ‘A Quasiconcave Minimization Method for Solving Linear Two Level Programs’, Journal of Global Optimization, 4, 243–264.CrossRefGoogle Scholar
  19. [19]
    H. Thy and B.T. Tam: 1994, ‘Polyhedral Annexation vs Outer Approximation for Decomposition of Monotonic Quasiconcave Minimization Problems’, Acta Mathematica Vietnamica. Google Scholar
  20. [20]
    D.J. White and G. Anandalingam: 1993, ‘A Penalty Function Approach for Solving Bilevel Linear Programs’, Journal of Global Optimization, 3, 397–420.MathSciNetzbMATHCrossRefGoogle Scholar

Copyright information

© Kluwer Academic Publishers 1998

Authors and Affiliations

  • Hoang Thy
    • 1
  1. 1.Institute of MathematicsHanoiVietnam

Personalised recommendations