Abstract
The prime concern in this paper is the interface between efficient solutions and rational reaction sets in multilevel programming. It is shown that there need be no rational reaction solution which is also efficient, although in the bilevel case such a solution always exists, and in the generalK-level case such a solution exists if certain conditions hold. Some theoretical properties of an extension of known methods for generating efficient solutions are given, which are important algorithmically.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Von Stackelberg, H.,The Theory of the Market Economy, Oxford University Press, Oxford, England, 1952.
Bi, Z., Calami, P. H., andConn, A. R.,Exact Penalty Function Approach for the Linear Bilevel Programming Problem, Technical Report 167-0-310789, Department of Systems Engineering, Combinatorics, and Optimization, University of Waterloo, Waterloo, Ontario, Canada, 1980.
Wen, U. P.,Mathematical Methods for Multilevel Linear Programming, PhD Thesis, State University of New York at Buffalo, Buffalo, New York, 1981.
Wen, U. P., andHsu, S. T.,Linear Bilevel Programming Problems: A Review, Journal of the Operational Research Society, Vol. 42, pp. 125–134, 1991.
White, D. J., andAnandalingam, G.,A Penalty Function Approach for Solving Bilevel Linear Programs, Journal of Global Optimization, Vol. 3, pp. 397–419, 1993.
Bard, J. F.,An Investigation of the Linear Three-Level Programming Problem, IEEE Transactions on Systems, Man, and Cybernetics, Vol. 5, pp. 711–717, 1984.
Bialas, W. F., andKarwan, M. H.,Two-Level Linear Programming, Management Science, Vol. 30, pp. 1004–1020, 1984.
Bard, J. F.,An Efficient-Point Algorithm for a Linear Two-Stage Optimization Problem, Operations Research, Vol. 31, pp. 670–684, 1983.
Haurie, A., Savard, G., andWhite, D. J.,A Note on an Efficient Point Algorithm for a Linear Two-Stage Optimization Problem, Operations Research, Vol. 38, pp. 553–555, 1990.
White, D. J.,Optimality and Efficiency, John Wiley and Sons, Chichester, England, 1982.
Zeleny, M.,Linear Multiobjective Programming, Springer Verlag, Berlin, Germany, 1974.
Zangwill, W. I.,Nonlinear Programming, Prentice-Hall, Englewood Cliffs, New Jersey, 1969.
Author information
Authors and Affiliations
Additional information
Communicated by P. L. Yu
This work was carried out under the auspices of the University of Manchester, Manchester, England and the University of Virginia, Charlottesville, Virginia, USA. The author thanks the referees for their helpful comments on the original version.
Rights and permissions
About this article
Cite this article
White, D.J. Multilevel programming, rational reaction sets, and efficient solutions. J Optim Theory Appl 87, 727–746 (1995). https://doi.org/10.1007/BF02192141
Issue Date:
DOI: https://doi.org/10.1007/BF02192141