Abstract
Recently parametric treatment of constraint solving and optimization problems has received considerable attention in science and engineering. In this paper we show an efficient and systematic algorithm for parametric programming, i.e. computing exact optimal value functions, based on a specialized symbolic-numeric cylindrical algebraic decomposition. We also present some practical application examples from system and control theory.
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Anai, H.: A symbolic-numeric approach to multi-parametric programming for control design. In: Proc. ICROS-SICE International Conference 2009, pp. 3525–3530 (2009)
Anai, H., Hara, S., Kanno, M., Yokoyama, K.: Parametric polynomial spectral factorization using the sum of roots and its application to a control design problem. J. Symb. Comput. 44(7), 703–725 (2009)
Bellman, R.: Dynamic Programming. Princeton Univ. Press, Princeton (1957)
Bemporad, A., Morari, M., Dua, V., Pistikopoulos, E.N.: The explicit linear quadratic regulator for constrained systems. Automatica 38(1), 3–20 (2002)
Bertsekas, D.: Dynamic Programming and Optimal Control, 3rd edn. Athena Scientific, Belmont (2005)
Bonnans, J.F., Shapiro, A.: Perturbation Analysis of Optimization Problems. Springer, Heidelberg (2000)
Caviness, B., Johnson, J. (eds.): Quantifier Elimination and Cylindrical Algebraic Decomposition. Texts and monographs in symbolic computation. Springer, Heidelberg (1998)
Collins, G.E.: Quantifier elimination for real closed fields by cylindrical algebraic decomposition. In: Becvar, J. (ed.) MFCS 1975. LNCS, vol. 32. Springer, Heidelberg (1975)
Collins, G.E., Hong, H.: Partial cylindrical algebraic decomposition for quantifier elimination. Journal of Symbolic Computation 12(3), 299–328 (1991)
Domingueza, L.F., Narcisoa, D.A., Pistikopoulos, E.N.: Recent advances in multiparametric nonlinear programming. Computers & Chemical Engineering 34(5), 707–716 (2010); (Selected Paper of Symposium ESCAPE 19, June 14-17, 2009, Krakow, Poland)
Dorato, P., Yang, W., Abdallah, C.: Robust multi-objective feedback design by quantifier elimination. J. Symb. Comput. 24(2), 153–159 (1997)
Dua, P., Kouramas, K., Dua, V., Pistikopoulos, E.: MPC on a chip – recent advances on the application of multi-parametric model-based control. Computers & Chemical Engineering 32(4-5), 754–765 (2008)
Fotiou, I.A., Rostalski, P., Parrilo, P.A., Morari, M.: Parametric optimization and optimal control using algebraic geometry methods. International Journal of Control 79(11), 1340–1358 (2006)
Fiacco, A.V.: Introduction to Sensitivity and Stability Analysis in Nonlinear Programming. Academic Press, London (1983)
Hyodo, N., Hong, M., Yanami, H., Hara, S., Anai, H.: Solving and visualizing nonlinear parametric constraints in control based on quantifier elimination. Appl. Algebra Eng. Commun. Comput. 18(6), 497–512 (2007)
Iwane, H., Yanami, H., Anai, H., Yokoyama, K.: An effective implementation of a symbolic-numeric cylindrical algebraic decomposition for quantifier elimination. In: Proceedings of the 2009 International Workshop on Symbolic-Numeric Computation, vol. 1, pp. 55–64 (2009)
Iwane, H., Yanami, H., Anai, H.: An effective implementation of a symbolic-numeric cylindrical algebraic decomposition for optimization problems. In: Proceedings of the 2011 International Workshop on Symbolic-Numeric Computation (2011)
Kanno, M., Yokoyama, K., Anai, H., Hara, S.: Parametric optimization in control using the sum of roots for parametric polynomial spectral factorization. In: Wang, D. (ed.) ISSAC, pp. 211–218. ACM, New York (2007)
Pistikopoulos, E., Georgiadis, M., Dua, V. (eds.): Multi-parametric programming: theory, algorithms, and applications, vol. 1. Wiley-VCH, Chichester (2007)
Pistikopoulos, E., Georgiadis, M., Dua, V. (eds.): Multi-parametric model-based control: theory and applications, vol. 2. Wiley-VCH, Chichester (2007)
Yanami, H., Anai, H.: The Maple package SyNRAC and its application to robust control design. Future Generation Comp. Syst. 23(5), 721–726 (2007)
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Iwane, H., Kira, A., Anai, H. (2011). Construction of Explicit Optimal Value Functions by a Symbolic-Numeric Cylindrical Algebraic Decomposition. In: Gerdt, V.P., Koepf, W., Mayr, E.W., Vorozhtsov, E.V. (eds) Computer Algebra in Scientific Computing. CASC 2011. Lecture Notes in Computer Science, vol 6885. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-23568-9_19
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DOI: https://doi.org/10.1007/978-3-642-23568-9_19
Publisher Name: Springer, Berlin, Heidelberg
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