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Construction of Explicit Optimal Value Functions by a Symbolic-Numeric Cylindrical Algebraic Decomposition

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Computer Algebra in Scientific Computing (CASC 2011)

Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 6885))

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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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References

  1. Anai, H.: A symbolic-numeric approach to multi-parametric programming for control design. In: Proc. ICROS-SICE International Conference 2009, pp. 3525–3530 (2009)

    Google Scholar 

  2. 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)

    Article  MathSciNet  MATH  Google Scholar 

  3. Bellman, R.: Dynamic Programming. Princeton Univ. Press, Princeton (1957)

    Google Scholar 

  4. Bemporad, A., Morari, M., Dua, V., Pistikopoulos, E.N.: The explicit linear quadratic regulator for constrained systems. Automatica 38(1), 3–20 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  5. Bertsekas, D.: Dynamic Programming and Optimal Control, 3rd edn. Athena Scientific, Belmont (2005)

    MATH  Google Scholar 

  6. Bonnans, J.F., Shapiro, A.: Perturbation Analysis of Optimization Problems. Springer, Heidelberg (2000)

    Book  MATH  Google Scholar 

  7. Caviness, B., Johnson, J. (eds.): Quantifier Elimination and Cylindrical Algebraic Decomposition. Texts and monographs in symbolic computation. Springer, Heidelberg (1998)

    MATH  Google Scholar 

  8. 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)

    Google Scholar 

  9. Collins, G.E., Hong, H.: Partial cylindrical algebraic decomposition for quantifier elimination. Journal of Symbolic Computation 12(3), 299–328 (1991)

    Article  MathSciNet  MATH  Google Scholar 

  10. 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)

    Article  Google Scholar 

  11. Dorato, P., Yang, W., Abdallah, C.: Robust multi-objective feedback design by quantifier elimination. J. Symb. Comput. 24(2), 153–159 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  12. 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)

    Article  MATH  Google Scholar 

  13. 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)

    Article  MathSciNet  MATH  Google Scholar 

  14. Fiacco, A.V.: Introduction to Sensitivity and Stability Analysis in Nonlinear Programming. Academic Press, London (1983)

    MATH  Google Scholar 

  15. 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)

    Article  MathSciNet  MATH  Google Scholar 

  16. 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)

    Google Scholar 

  17. 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)

    Google Scholar 

  18. 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)

    Google Scholar 

  19. Pistikopoulos, E., Georgiadis, M., Dua, V. (eds.): Multi-parametric programming: theory, algorithms, and applications, vol. 1. Wiley-VCH, Chichester (2007)

    Google Scholar 

  20. Pistikopoulos, E., Georgiadis, M., Dua, V. (eds.): Multi-parametric model-based control: theory and applications, vol. 2. Wiley-VCH, Chichester (2007)

    Google Scholar 

  21. Yanami, H., Anai, H.: The Maple package SyNRAC and its application to robust control design. Future Generation Comp. Syst. 23(5), 721–726 (2007)

    Article  Google Scholar 

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Author information

Authors and Affiliations

  1. IT System Laboratories, Fujitsu Laboratories Ltd., Kamikodanaka 4-1-1, Nakahara-ku, Kawasaki, 211-8588, Japan

    Hidenao Iwane & Hirokazu Anai

  2. Graduate School of Mathematics, Kyushu University, 744 Motooka, Nishi-ku, Fukuoka, 819-0395, Japan

    Akifumi Kira & Hirokazu Anai

Authors
  1. Hidenao Iwane
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  2. Akifumi Kira
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  3. Hirokazu Anai
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Editor information

Editors and Affiliations

  1. Laboratory of Information Technologies (LIT), Joint Institute for Nuclear Research (JINR), 141980, Dubna, Russia

    Vladimir P. Gerdt

  2. Institut für Mathematik, Universität Kassel, Heinrich-Plett-Straße 40, 34132, Kassel, Germany

    Wolfram Koepf

  3. Institut für Informatik, Lehrstuhl für Effiziente Algorithmen, Technische Universität München, Boltzmannstraße 3, 85748, Garching, Germany

    Ernst W. Mayr

  4. Institute of Theoretical and Applied Mechanics, Russian Academy of Sciences, Novosibirsk, 630090, Russia

    Evgenii V. Vorozhtsov

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© 2011 Springer-Verlag Berlin Heidelberg

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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

  • Print ISBN: 978-3-642-23567-2

  • Online ISBN: 978-3-642-23568-9

  • eBook Packages: Computer ScienceComputer Science (R0)

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