Location via proxy:   [ UP ]  
[Report a bug]   [Manage cookies]                
Skip to main content

A quasiconcave minimization method for solving linear two-level programs

  • Published:
Journal of Global Optimization Aims and scope Submit manuscript

Abstract

In this paper the linear two-level problem is considered. The problem is reformulated to an equivalent quasiconcave minimization problem, via a reverse convex transformation. A branch and bound algorithm is developed which takes the specific structure into account and combines an outer approximation technique with a subdivision procedure.

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

Access this article

Subscribe and save

Springer+ Basic
$34.99 /Month
  • Get 10 units per month
  • Download Article/Chapter or eBook
  • 1 Unit = 1 Article or 1 Chapter
  • Cancel anytime
Subscribe now

Buy Now

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Similar content being viewed by others

References

  1. E. Aiyoshi and K. Shimisu (1981), Hierarchical Decentralized System and Its New Solution by a Barrier Method,IEEE Transactions on Systems, Man and Cybernetics SMC-11, pp. 444–449.

  2. G. Anandalingam and D. J. White (1990). A Solution Method for the Linear Stackelberg Problem,IEEE Trans. Auto. Contr. AC-35, pp. 1170–1175.

  3. G. Anandalingam and T. L. Friesz (eds.) (1992), Hierarchical Optimization,Annals of Operations Research 34. J.C.Baltzer AG, Basel.

    Google Scholar 

  4. J. F. Bard and J. E. Falk (1982), An Explicit Solution to the Multi-level Programming Problem,Computers and Oper. Res. 9(1), 77–100.

    Google Scholar 

  5. J. F. Bard (1983), An Algorithm for Solving the General Bi-level Programming Problem,Math. of Ops. Res. 8(2), 260–272.

    Google Scholar 

  6. J. F. Bard and J. T. Moore (1990), A Branch and Bound Algorithm for the Bilevel Programming Problem,SIAM J. Sci. Statist. Comput. 11, 281–292.

    Google Scholar 

  7. J. T. Moore and J. F. Bard (1990), The Mixed Integer Linear Bilevel Programming Problem,Ops. Res. 38(5), 911–921.

    Google Scholar 

  8. O. Ben-Ayed (1993), Bilevel Linear Programming,Computers and Oper. Res. 20(5), 485–501.

    Google Scholar 

  9. W. F. Bialas and M. H. Karwan (1984), Two-level Linear Programming,Management Science 30, 1004–1020.

    Google Scholar 

  10. W. F. Bialas, M. H. Karwan, and J. P. Shaw (1980), A Parametric Complementarity Pivot Approach for Two-level Linear Programming, Research Report No. 80-2, Operations Research Program, Dept. of Industrial Eng., State University of New York at Buffalo.

    Google Scholar 

  11. C. Blair (1992), The Computational Complexity of Multi-level Linear Programs,Annals of Operations Research 34, 13–19.

    Google Scholar 

  12. W. Candler and R. Townsley (1982), A Linear Two-level Programming Problem,Comput. & Ops. Res. 9(1), 59–76.

    Google Scholar 

  13. J. E. Falk (1973), A Linear Max-Min Problem,Mathematical Programming 5, 169–188.

    Google Scholar 

  14. J. Fortuny-Amat and B. McCarl (1981), A Representation and Economic Interpretation of a Two-level Programming Problem,J. Ops. Res. Soc. 32, 783–792.

    Google Scholar 

  15. P. Hansen, B. Jaumard, and G. Savard (1990), New Branching and Bounding Rules for Linear Bilevel Programming, To appear inSIAM Journal on Scientific and Statistical Computing.

  16. R. Horst and H. Tuy (1990),Global Optimization: deterministic Approaches, Springer Verlag, Berlin.

    Google Scholar 

  17. R. Horst, N. V. Thoai, and J. de Vries (1988), On Finding New Vertices and Redundant Constraints in Cutting Plane Algorithms for Global Optimization,Operations Research Letters 7, 85–90.

    Google Scholar 

  18. R. Horst and N. V. Thoai (1989), Modification, Implementation and Comparison of Three Algorithms for Globally Solving Linearly Constrained Concave Minimization Problems,Computing 42, 271–289.

    Google Scholar 

  19. J. J. Judice and A. M. Faustino (1988), The Solution of the Linear Bilevel Programming Problem by Using the Linear Complementarity Problem,Invest. Opnl. 8, 75–95.

    Google Scholar 

  20. J. J. Judice and A. M. Faustino (1992), A Sequential LCP Method for Bilevel Linear Programming,Annals of Operations Research 34, 89–106.

    Google Scholar 

  21. C. D. Kolstad and L. S. Lasdon (1986), Derivative Evaluation and Computational Experience with Large Bi-level Mathematical Programs, Faculty Working Paper No. 1266, College of Commerce and Business Administration, University of Illinois at Urbana-Champaign.

    Google Scholar 

  22. P. Loridan and J. Morgan (1989), New Results of Approximate Solutions in Two-level Optimization,Optimization 20(6), 819–836.

    Google Scholar 

  23. S. C. Narula and A. D. Nwosu (1983), Two-level Hierarchical programming problem,Multiple Criteria Decision Making — Theory and Application, edited by P. Hansen, New York: Springer Verlag, pp. 290–299.

    Google Scholar 

  24. S. C. Narula and A. D. Nwosu (1985), An Algorithm to Solve a Two-level Resource Control Preemptive hierarchical Programming Problem, InMathematics of Multi-Objective Optimization (Edited by P. Serafini), pp. 353–373, Springer, New York.

    Google Scholar 

  25. C. H. Papadimitriou and K. Steiglitz (1982),Combinatorial Optimization: Algorithms and Complexity, Prentice-Hall, Inc., New Jersey (1982).

    Google Scholar 

  26. P. T. Thach and H. Tuy, Dual Outer Approximation Methods for Concave Programs and Reverse Convex Programs, Submitted.

  27. H. Tuy (1985), Concave Minimization Under Linear Constraints with Special structure, Optimization 16, pp. 335–352.

    Google Scholar 

  28. H. Tuy (1991), Effect of Subdivision Strategy on Convergence and Efficiency of Some Global Optimization Algorithms,Journal of Global Optimization 1(1), 23–36.

    Google Scholar 

  29. H. Tuy, The Normal Conical Algorithm for Concave Minimization over Polytopes, To appear inMathematical Programming.

  30. H. Tuy, S. Migdalas, and P. Värbrand (1993), A Global Optimization Approach for the Linear Two-level Program,Journal of Global Optimization 3, 1–23.

    Google Scholar 

  31. G. Ünlü (1987), A Linear Bi-level Programming Algorithm Based on Bicriteria Programming,Comput. Ops. Res. 14(2), 173–179.

    Google Scholar 

  32. U.-P. Wen and S.-T. Hsu (1989). A Note on a Linear Bilevel Programming Algorithm Based on Bicriteria Programming,Comput. & Ops. Res. 16(1), 79–83.

    Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Rights and permissions

Reprints and permissions

About this article

Cite this article

Tuy, H., Migdalas, A. & Värbrand, P. A quasiconcave minimization method for solving linear two-level programs. J Glob Optim 4, 243–263 (1994). https://doi.org/10.1007/BF01098360

Download citation

  • Received:

  • Accepted:

  • Issue Date:

  • DOI: https://doi.org/10.1007/BF01098360

Key words