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.
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References
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.
G. Anandalingam and D. J. White (1990). A Solution Method for the Linear Stackelberg Problem,IEEE Trans. Auto. Contr. AC-35, pp. 1170–1175.
G. Anandalingam and T. L. Friesz (eds.) (1992), Hierarchical Optimization,Annals of Operations Research 34. J.C.Baltzer AG, Basel.
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.
J. F. Bard (1983), An Algorithm for Solving the General Bi-level Programming Problem,Math. of Ops. Res. 8(2), 260–272.
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.
J. T. Moore and J. F. Bard (1990), The Mixed Integer Linear Bilevel Programming Problem,Ops. Res. 38(5), 911–921.
O. Ben-Ayed (1993), Bilevel Linear Programming,Computers and Oper. Res. 20(5), 485–501.
W. F. Bialas and M. H. Karwan (1984), Two-level Linear Programming,Management Science 30, 1004–1020.
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.
C. Blair (1992), The Computational Complexity of Multi-level Linear Programs,Annals of Operations Research 34, 13–19.
W. Candler and R. Townsley (1982), A Linear Two-level Programming Problem,Comput. & Ops. Res. 9(1), 59–76.
J. E. Falk (1973), A Linear Max-Min Problem,Mathematical Programming 5, 169–188.
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.
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.
R. Horst and H. Tuy (1990),Global Optimization: deterministic Approaches, Springer Verlag, Berlin.
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.
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.
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.
J. J. Judice and A. M. Faustino (1992), A Sequential LCP Method for Bilevel Linear Programming,Annals of Operations Research 34, 89–106.
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.
P. Loridan and J. Morgan (1989), New Results of Approximate Solutions in Two-level Optimization,Optimization 20(6), 819–836.
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.
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.
C. H. Papadimitriou and K. Steiglitz (1982),Combinatorial Optimization: Algorithms and Complexity, Prentice-Hall, Inc., New Jersey (1982).
P. T. Thach and H. Tuy, Dual Outer Approximation Methods for Concave Programs and Reverse Convex Programs, Submitted.
H. Tuy (1985), Concave Minimization Under Linear Constraints with Special structure, Optimization 16, pp. 335–352.
H. Tuy (1991), Effect of Subdivision Strategy on Convergence and Efficiency of Some Global Optimization Algorithms,Journal of Global Optimization 1(1), 23–36.
H. Tuy, The Normal Conical Algorithm for Concave Minimization over Polytopes, To appear inMathematical Programming.
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.
G. Ünlü (1987), A Linear Bi-level Programming Algorithm Based on Bicriteria Programming,Comput. Ops. Res. 14(2), 173–179.
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.
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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
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DOI: https://doi.org/10.1007/BF01098360