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Graph-based algorithms for the efficient solution of optimization problems involving monotone functions

  • Luca Consolini
  • Mattia Laurini
  • Marco LocatelliEmail author
Article
  • 164 Downloads

Abstract

In this paper, we address a class of specially structured problems that include speed planning, for mobile robots and robotic manipulators, and dynamic programming. We develop two new numerical procedures, that apply to the general case and to the linear subcase. With numerical experiments, in the linear case we show that the proposed algorithms outperform generic commercial solvers.

Keywords

Computational methods Acceleration of convergence Dynamic programming Linear programming Complete lattices 

Mathematics Subject Classification

90C35 90-08 90-04 65B99 90C39 06B23 

Notes

References

  1. 1.
    Al-Tamimi, A., Lewis, F.L., Abu-Khalaf, M.: Discrete-time nonlinear HJB solution using approximate dynamic programming: convergence proof. IEEE Trans. Syst. Man Cybern. Part B (Cybern.) 38(4), 943–949 (2008)CrossRefGoogle Scholar
  2. 2.
    Barabási, A.-L., Albert, R.: Emergence of scaling in random networks. Science 286, 509–512 (1999)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Bardi, M., Capuzzo-Dolcetta, I.: Optimal Control and Viscosity Solutions of Hamilton–Jacobi–Bellman Equations. Springer, Berlin (2008)zbMATHGoogle Scholar
  4. 4.
    Belotti, P., Lee, J., Liberti, L., Margot, F., Wächter, A.: Branching and bounds tightening techniques for non-convex MINLP. Optim. Methods Softw. 24(4–5), 597–634 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Cabassi, F., Consolini, L., Locatelli, M.: Time-optimal velocity planning by a bound-tightening technique. Comput. Optim. Appl. 70(1), 61–90 (2018)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Consolini, L., Locatelli, M., Minari, A., Nagy, A., Vajk, I.: Optimal time-complexity speed planning for robot manipulators. CoRR (2018). arXiv:1802.03294
  7. 7.
    Consolini, L., Locatelli, M., Minari, A., Piazzi, A.: A linear-time algorithm for minimum-time velocity planning of autonomous vehicles. In: Proceedings of the 24th Mediterranean Conference on Control and Automation (MED), IEEE (2016)Google Scholar
  8. 8.
    Consolini, L., Locatelli, M., Minari, A., Piazzi, A.: An optimal complexity algorithm for minimum-time velocity planning. Syst. Control Lett. 103, 50–57 (2017)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Davey, B., Priestley, H.: Introduction to Lattices and Order. Cambridge University Press, Cambridge (2002)CrossRefzbMATHGoogle Scholar
  10. 10.
    Granas, A., Dugundji, J.: Fixed Point Theory. Springer Monographs in Mathematics. Springer, Berlin (2003)CrossRefzbMATHGoogle Scholar
  11. 11.
    Hagberg, A. A., Schult, D. A., Swart, P. J.: Exploring network structure, dynamics, and function using networkx. In Varoquaux, G., Vaught, T., Millman, J. (eds), Proceedings of the 7th Python in Science Conference (SciPy2008), pp. 11–15 (2008)Google Scholar
  12. 12.
    Heinonen, J.: Lectures on Lipschitz Analysis. Bericht (Jyväskylän yliopisto. Matematiikan ja tilastotieteen laitos). University of Jyväskylä (2005)Google Scholar
  13. 13.
    Holme, P., Kim, B.J.: Growing scale-free networks with tunable clustering. Phys. Rev. E 65(026107), 1–4 (2002)Google Scholar
  14. 14.
    Knuth, D.E.: A generalization of Dijkstra’s algorithm. Inf. Process. Lett. 6(1), 1–5 (1976)MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Laurini, M., Micelli, P., Consolini, L., Locatelli, M.: A Jacobi-like acceleration for dynamic programming. In: 2016 IEEE 55th Conference on Decision and Control (CDC), pp. 7371–7376 (2016)Google Scholar
  16. 16.
    Liu, D., Wei, Q.: Finite-approximation-error-based optimal control approach for discrete-time nonlinear systems. IEEE Trans. Cybern. 43(2), 779–789 (2013)CrossRefGoogle Scholar
  17. 17.
    Newman, M.E.J., Watts, D.J.: Renormalization group analysis of the small-world network model. Phys. Lett. A 263(4–6), 341–346 (1999)MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    Wang, S., Gao, F., Teo, K.L.: An upwind finite-difference method for the approximation of viscosity solutions to Hamilton–Jacobi–Bellman equations. IMA J. Math. Control Inf. 17(2), 167–178 (2000)MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC, part of Springer Nature 2019

Authors and Affiliations

  1. 1.Dipartimento di Ingegneria e ArchitetturaUniversità degli Studi di ParmaParmaItaly

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