A bilevel program is a mathematical program involving functions defined implicitly as solutions to another mathematical program. We discuss a method for extracting derivative information on the implicit function, which is especially efficient when the lower-level problem has simple bounds on the variables and/or many inactive constraints. Computational experience on problems with up to 230 variables and 30 constraints is presented.
This is a preview of subscription content, log in to check access.
Buy single article
Instant access to the full article PDF.
Price includes VAT for USA
Subscribe to journal
Immediate online access to all issues from 2019. Subscription will auto renew annually.
This is the net price. Taxes to be calculated in checkout.
Chen, C. I. andCruz, J. B. Jr.,Stackelberg Solution for Two-Person Games with Biased Information Patterns, IEEE Transactions on Automatic Control, Vol. AC-17, pp. 791–798, 1972.
Papavassilopoulos, G. P.,Solution of Some Stochastic Nash and Leader-Follower Games, SIAM Journal on Control and Optimization, Vol. 19, pp. 651–666, 1981.
Grossman, S. J., andHart, O. D.,An Analysis of the Principal-Agent Problem, Econometrica, Vol. 51, pp. 7–45, 1983.
Danskin, J. W.,The Theory of Max-Min with Applications, SIAM Journal on Applied Mathematics, Vol. 14, pp. 641–664, 1966.
Bialas, W. F., andKarwan, M. H.,Two-Level Linear Programming, Management Science, Vol. 30, pp. 1004–1020, 1984.
Fiacco, A. V.,Sensitivity Analysis for Nonlinear Programming Using Penalty Methods, Mathematical Programming, Vol. 10, pp. 287–311, 1976.
Fiacco, A. V.,Introduction to Sensitivity and Stability Analysis in Nonlinear Programming, Academic Press, New York, New York, 1983.
Fiacco, A. V., andMcCormick, G. P.,Nonlinear Programming: Sequential Unconstrained Minimization Techniques, John Wiley, New York, New York, 1968.
Kolstad, C. D.,A Review of the Literature on Bilevel Mathematical Programming, Report No. LA-10284-MS, Los Alamos National Laboratory, Los Alamos, New Mexico, 1985.
Candler, W., andTownsley, R.,A Linear Two-Level Programming Problem, Computers and Operations Research, Vol. 9, pp. 59–76, 1982.
Takayama, T., andJudge, G.,Spatial and Temporal Price and Allocation Models, North-Holland, Amsterdam, Holland, 1971.
Candler, W., andNorton, R. D.,Multilevel Programming, Discussion Paper No. 20, World Bank Development Research Center, Washington, DC, 1977.
Candler, W., Fortuny-Amat, J., andMcCarl, B.,The Potential Role of Multilevel Programming in Agricultural Economics, American Journal of Agricultural Economics, Vol. 63, pp. 521–531, 1981.
Schenk, G. W.,A Multilevel Programming Model to Determine Optimal Pollution Control Policies, M.S. Thesis, Department of Industrial Engineering, State University of New York at Buffalo, 1980.
Kolstad, C. D.,Empirical Properties of Economic Incentives and Command-and-Control Regulations for Air Pollution Control, Land Economics, Vol. 62, pp. 250–268, 1986.
Falk, J. E., andMcCormick, G. P.,Mathematical Structure of the International Coal Trade Model, Report No. DOE/NBB-0025, US Department of Energy, Washington, DC, 1982.
Falk, J. E., andMcCormick, G. P.,Computational Aspects of the International Coal Trade Model, Spatial Price Equilibrium: Advances in Theory and Applications, Edited by P. T. Harker, Springer-Verlag, Berlin, Germany, 1985.
Bialas, W. F., andKarwan, M. H.,On Two-Level Optimization, IEEE Transactions on Automatic Control, Vol. AC-27, pp. 211–214, 1982.
Bard, J.,An Algorithm for Solving the General Bilevel Programming Problem, Mathematics of Operations Research, Vol. 8, pp. 260–272, 1983.
Bard, J.,An Efficient Point Algorithm for a Linear Two-Stage Optimization Problem, Operations Research, Vol. 31, pp. 670–684, 1983.
Bard, J. F., andFalk, J. E.,An Explicit Solution to the Multi-Level Programming Problem, Computers and Operations Research, Vol. 9, pp. 77–100, 1982.
Fortuny-Amat, J. andMcCarl, B.,A Representation of a Two-Level Programming Problem, Journal of the Operational Research Society, Vol. 32, pp. 783–792, 1981.
Shimizu, K., andAiyoshi, E.,A New Computational Method for Stackelberg and Min-Max Problems by Use of a Penalty Method, IEEE Transactions on Automatic Control, Vol. AC-26, pp. 460–466, 1981.
De Silva, A. H.,Sensitivity Formulas for Nonlinear Factorable Programming and Their Application to the Solution of an Implicitly Defined Optimization Model of US Crude Oil Production, D.Sc. Dissertation, George Washington University, 1978.
Hogan, W. W.,Point-to-Set Maps in Mathematical Programming, SIAM Review, Vol. 15, pp. 591–603, 1973.
Tobin, R. L.,General Spatial Price Equilibria: Sensitivity Analysis for Variational Inequality and Nonlinear Complementarity Formulations, Spatial Price Equilibrium: Advance in Theory and Applications, Edited by P. T. Harker, Springer-Verlag, Berlin, Germany, 1985.
Tobin, R. L., andFriesz, T. L.,A New Look at Spatially Competitive Facility Location Models, Spatial Price Equilibrium: Advances in Theory and Applications, Edited by P. T. Harker, Springer-Verlag, Berlin, Germany, 1985.
Kolstad, C. D., andAbbey, D. S.,The Effect of Market Conduct on International Steam Coal Trade, European Economic Review, Vol. 24, pp. 39–59, 1984.
Murtagh, B. A., andSaunders, M. A.,A Projected Lagrangian Algorithm and its Implementation for Sparse Nonlinear Constraints, Report SOL 80-1R, Systems Optimization Laboratory, Department of Operations Research, Stanford University, Stanford, California, 1981.
Computational support from Robert Bivins and Myron Stein is gratefully acknowledged. We have also appreciated comments from Jon Bard and an anonymous referee. This work was supported in part by the US Department of Energy through the Los Alamos National Laboratory.
Communicated by M. Avriel
About this article
Cite this article
Kolstad, C.D., Lasdon, L.S. Derivative evaluation and computational experience with large bilevel mathematical programs. J Optim Theory Appl 65, 485–499 (1990). https://doi.org/10.1007/BF00939562
- Bilevel programming
- economic planning
- hierarchical decision making
- multilevel programming
- sensitivity analysis
- economic models