Abstract
We propose a novel algorithm for solving multiparametric linear programming problems. Rather than visiting different bases of the associated LP tableau, we follow a geometric approach based on the direct exploration of the parameter space. The resulting algorithm has computational advantages, namely the simplicity of its implementation in a recursive form and an efficient handling of primal and dual degeneracy. Illustrative examples describe the approach throughout the paper. The algorithm is used to solve finite-time constrained optimal control problems for discrete-time linear dynamical systems.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Gass, S., and Saaty, T., The Computational Algorithm for the Parametric Objective Function, Naval Research Logistics Quarterly, Vol. 2, pp. 39-45, 1955.
Gal, T., Postoptimal Analyses, Parametric Programming, and Related Topics, 2nd Edition, de Gruyter, Berlin, Germany, 1995.
Gal, T., and Greenberg, H., Advances in Sensitivity Analysis and Parametric Programming, International Series in Operations Research and Management Science, Kluwer Academic Publishers, Dordrecht, Netherlands, Vol. 6, 1997.
Gal, T., and Nedoma, J., Multiparametric Linear Programming, Management Science, Vol. 18, pp. 406-442, 1972.
Filippi, C., On the Geometry of Optimal Partition Sets in Multiparametric Linear Programming, Technical Report 12, Department of Pure and Applied Mathematics, University of Padova, Padova, Italy, 1997.
Schechter, M., Polyhedral Functions and Multiparametric Linear Programming, Journal of Optimization Theory and Applications, Vol. 53, pp. 269-280, 1987.
Fiacco, A.V., Introduction to Sensitivity and Stability Analysis in Nonlinear Programming, Academic Press, London, UK, 1983.
Bemporad, A., Morari, M., Dua, V., and Pistikopoulos, E., The Explicit Linear-Quadratic Regulator for Constrained Systems, Automatica, Vol. 38, pp. 3-20, 2002.
Dua, V., and Pistikopoulos, E., An Algorithm for the Solution of Multiparametric Mixed Integer Linear Programming Problems, Annals of Operations Research, Vol. 99, pp. 123-139, 2000.
Zadeh, L., and Whalen, L., On Optimal Control and Linear Programming, IRE Transaction on Automatic Control, Vol. 7, pp. 45-46, 1962.
Morari, M., and Lee, J., Model Predictive Control: Past, Present, and Future, Computers and Chemical Engineering, Vol. 23, pp. 667-682, 1999.
Mayne, D., Rawlings, J., Rao, C., and Scokaert, P., Constrained Model Predictive Control: Stability and Optimality, Automatica, Vol. 36, pp. 789-814, 2000.
Bemporad, A., Borrelli, F., and Morari, M., Model Predictive Control Based on Linear Programming: The Explicit Solution, IEEE Transactions on Automatic Control, Vol. 47, pp. 1974-1985, 2002.
Borrelli, F., Discrete-Time Constrained Optimal Control, PhD Thesis, Automatic Control Labotatory, ETH, Zurich, Switzerland, 2002.
Adler, I., and Monteiro, R., A Geometric View of Parametric Linear Programming, Algorithmica, Vol. 8, pp. 161-176, 1992.
Mehrotra, S., and Monteiro, R., Parametric and Range Analysis for Interior-Point Methods., Technical Report, Department of Systems and Industrial Engineering, University of Arizona, Tucson, Arizona, 1992.
Berkelaar, A., Roos, K., and Terlaky, T., The Optimal Set and Optimal Partition Approach to Linear and Quadratic Programming, Advances in Sensitivity Analysis and Parametric Programming, Edited by T. Gal and H. Greenberg, International Series in Operations Research and Management Science, Kluwer Academic Publishers, Dordrecht, Netherlands, Vol. 6, 1997.
Bemporad, A., Fukuda, K., and Torrisi, F., Convexity Recognition of the Union of Polyhedra, Computational Geometry, Vol. 18, pp. 141-154, 2001.
Fukuda, K., cdd/cddC Reference Manual, Institute for Operations Research, ETH-Zentrum, Zurich, Switzerland, 0.61 (cdd) and 0.75 (cdd+) editions, 1997.
Murty, K.G., Computational Complexity of Parametric Linear Programming, Mathematical Programming, Vol. 19, pp. 213-219, 1980.
Goodman, J., and O'Rourke, J., Handbook of Discrete and Computational Geometry, Discrete Mathematics and Its Applications, CRC Press, New York, NY, 1997.
Borrelli, F., Bemporad, A., Fodor, M., and Hrovat, D., A Hybrid Approach to Traction Control, Hybrid Systems: Computation and Control, Edited by A. Sangiovanni-Vincentelli and M. D. Benedetto, Lecture Notes in Computer Science, Springer Verlag, Berlin, Germany, Vol. 2034, pp. 162-174, 2001.
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Borrelli, F., Bemporad, A. & Morari, M. Geometric Algorithm for Multiparametric Linear Programming. Journal of Optimization Theory and Applications 118, 515–540 (2003). https://doi.org/10.1023/B:JOTA.0000004869.66331.5c
Issue Date:
DOI: https://doi.org/10.1023/B:JOTA.0000004869.66331.5c