IDEAS home Printed from https://ideas.repec.org/a/eee/matcom/v190y2021icp641-658.html
   My bibliography  Save this article

Substructure exploitation of a nonsmooth Newton method for large-scale optimal control problems with full discretization

Author

Listed:
  • Pablos, Blanca
  • Gerdts, Matthias

Abstract

We investigate the application of full discretization and a nonsmooth Newton method to large-scale optimal control problems. Based on a first discretize, then optimize approach, we discretize the state and control variables in time following a collocation method. Then, a nonsmooth Newton method combined with a line search globalization strategy is used to find a solution to the resulting finite-dimensional nonlinear optimization problem. In order to reduce the computational effort of solving the linear systems that arise from the application of the nonsmooth Newton method, we propose a structure exploitation strategy that results in a sparse banded matrix. We propose as well a substructure exploitation strategy based on a block LU decomposition. The different exploitation strategies combined with the use of appropriate linear solvers are demonstrated and compared for a quadratic 2D heat equation control problem discretized with the method of lines, and the approach that proved to be the most efficient is applied to a nonlinear version of the problem.

Suggested Citation

  • Pablos, Blanca & Gerdts, Matthias, 2021. "Substructure exploitation of a nonsmooth Newton method for large-scale optimal control problems with full discretization," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 190(C), pages 641-658.
    • Handle: RePEc:eee:matcom:v:190:y:2021:i:c:p:641-658
    DOI: 10.1016/j.matcom.2021.05.039
    as

    Download full text from publisher

    File URL: http://www.sciencedirect.com/science/article/pii/S0378475421002238
    Download Restriction: Full text for ScienceDirect subscribers only
    File URL: https://libkey.io/10.1016/j.matcom.2021.05.039?utm_source=ideas
    LibKey link: if access is restricted and if your library uses this service, LibKey will redirect you to where you can use your library subscription to access this item
    ---><---

    As the access to this document is restricted, you may want to search for a different version of it.

    References listed on IDEAS

    as
    1. Kurt Chudej & Hans Josef Pesch & Markus Wächter & Gottfried Sachs & Florent Bras, 2009. "Instationary Heat-Constrained Trajectory Optimization of a Hypersonic Space Vehicle by ODE–PDE-Constrained Optimal Control," Springer Optimization and Its Applications, in: Variational Analysis and Aerospace Engineering, chapter 0, pages 127-144, Springer.
    2. Houyuan Jiang, 1999. "Global Convergence Analysis of the Generalized Newton and Gauss-Newton Methods of the Fischer-Burmeister Equation for the Complementarity Problem," Mathematics of Operations Research, INFORMS, vol. 24(3), pages 529-543, August.
    3. Gerdts, Matthias & Greif, Günter & Pesch, Hans Josef, 2008. "Numerical optimal control of the wave equation: optimal boundary control of a string to rest in finite time," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 79(4), pages 1020-1032.
    4. Jong-Shi Pang, 1990. "Newton's Method for B-Differentiable Equations," Mathematics of Operations Research, INFORMS, vol. 15(2), pages 311-341, May.
    Full references (including those not matched with items on IDEAS)

    Most related items

    These are the items that most often cite the same works as this one and are cited by the same works as this one.
    1. H. Xu & B. M. Glover, 1997. "New Version of the Newton Method for Nonsmooth Equations," Journal of Optimization Theory and Applications, Springer, vol. 93(2), pages 395-415, May.
    2. Jong-Shi Pang & Defeng Sun & Jie Sun, 2003. "Semismooth Homeomorphisms and Strong Stability of Semidefinite and Lorentz Complementarity Problems," Mathematics of Operations Research, INFORMS, vol. 28(1), pages 39-63, February.
    3. Jean-Pierre Dussault & Mathieu Frappier & Jean Charles Gilbert, 2019. "A lower bound on the iterative complexity of the Harker and Pang globalization technique of the Newton-min algorithm for solving the linear complementarity problem," EURO Journal on Computational Optimization, Springer;EURO - The Association of European Operational Research Societies, vol. 7(4), pages 359-380, December.
    4. Zhang, Xiang & Li, Lingfei & Zhang, Gongqiu, 2021. "Pricing American drawdown options under Markov models," European Journal of Operational Research, Elsevier, vol. 293(3), pages 1188-1205.
    5. H. Xu & X. W. Chang, 1997. "Approximate Newton Methods for Nonsmooth Equations," Journal of Optimization Theory and Applications, Springer, vol. 93(2), pages 373-394, May.
    6. Lorenzo Stella & Andreas Themelis & Panagiotis Patrinos, 2017. "Forward–backward quasi-Newton methods for nonsmooth optimization problems," Computational Optimization and Applications, Springer, vol. 67(3), pages 443-487, July.
    7. Enzo Busseti & Walaa M. Moursi & Stephen Boyd, 2019. "Solution refinement at regular points of conic problems," Computational Optimization and Applications, Springer, vol. 74(3), pages 627-643, December.
    8. Azam Asanjarani, 2023. "A Finsler Geometrical Programming Approach to the Nonlinear Complementarity Problem of Traffic Equilibrium," Journal of Optimization Theory and Applications, Springer, vol. 196(3), pages 797-809, March.
    9. J. Burke & S. Xu, 2002. "Complexity of a Noninterior Path-Following Method for the Linear Complementarity Problem," Journal of Optimization Theory and Applications, Springer, vol. 112(1), pages 53-76, January.
    10. L. W. Zhang & Z. Q. Xia, 2001. "Newton-Type Methods for Quasidifferentiable Equations," Journal of Optimization Theory and Applications, Springer, vol. 108(2), pages 439-456, February.
    11. Gerasimos G. Rigatos, 2016. "Boundary Control Of The Black–Scholes Pde For Option Dynamics Stabilization," Annals of Financial Economics (AFE), World Scientific Publishing Co. Pte. Ltd., vol. 11(02), pages 1-29, June.
    12. Michael Patriksson, 2004. "Sensitivity Analysis of Traffic Equilibria," Transportation Science, INFORMS, vol. 38(3), pages 258-281, August.
    13. J. H. Wu, 1998. "Long-Step Primal Path-Following Algorithm for Monotone Variational Inequality Problems," Journal of Optimization Theory and Applications, Springer, vol. 99(2), pages 509-531, November.
    14. D. H. Li & N. Yamashita & M. Fukushima, 2001. "Nonsmooth Equation Based BFGS Method for Solving KKT Systems in Mathematical Programming," Journal of Optimization Theory and Applications, Springer, vol. 109(1), pages 123-167, April.
    15. Axel Kröner & Karl Kunisch, 2014. "A minimum effort optimal control problem for the wave equation," Computational Optimization and Applications, Springer, vol. 57(1), pages 241-270, January.
    16. Long, Qiang & Wu, Changzhi & Wang, Xiangyu, 2015. "A system of nonsmooth equations solver based upon subgradient method," Applied Mathematics and Computation, Elsevier, vol. 251(C), pages 284-299.
    17. Shenglong Hu & Guoyin Li, 2021. "$${\text {B}}$$ B -subdifferentials of the projection onto the matrix simplex," Computational Optimization and Applications, Springer, vol. 80(3), pages 915-941, December.
    18. Marlon Azinovic & Jan v{Z}emliv{c}ka, 2023. "Economics-Inspired Neural Networks with Stabilizing Homotopies," Papers 2303.14802, arXiv.org.
    19. Roman Sznajder & M. Seetharama Gowda, 1998. "Nondegeneracy Concepts for Zeros of Piecewise Smooth Functions," Mathematics of Operations Research, INFORMS, vol. 23(1), pages 221-238, February.
    20. Matúš Benko & Patrick Mehlitz, 2024. "Isolated Calmness of Perturbation Mappings and Superlinear Convergence of Newton-Type Methods," Journal of Optimization Theory and Applications, Springer, vol. 203(2), pages 1587-1621, November.

    Corrections

    All material on this site has been provided by the respective publishers and authors. You can help correct errors and omissions. When requesting a correction, please mention this item's handle: RePEc:eee:matcom:v:190:y:2021:i:c:p:641-658. See general information about how to correct material in RePEc.

    If you have authored this item and are not yet registered with RePEc, we encourage you to do it here. This allows to link your profile to this item. It also allows you to accept potential citations to this item that we are uncertain about.

    If CitEc recognized a bibliographic reference but did not link an item in RePEc to it, you can help with this form .

    If you know of missing items citing this one, you can help us creating those links by adding the relevant references in the same way as above, for each refering item. If you are a registered author of this item, you may also want to check the "citations" tab in your RePEc Author Service profile, as there may be some citations waiting for confirmation.

    For technical questions regarding this item, or to correct its authors, title, abstract, bibliographic or download information, contact: Catherine Liu (email available below). General contact details of provider: http://www.journals.elsevier.com/mathematics-and-computers-in-simulation/ .

    Please note that corrections may take a couple of weeks to filter through the various RePEc services.

    IDEAS is a RePEc service. RePEc uses bibliographic data supplied by the respective publishers.