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Implicit Peer Triplets in Gradient-Based Solution Algorithms for ODE Constrained Optimal Control

Author

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  • Jens Lang

    (Technical University of Darmstadt)

  • Bernhard A. Schmitt

    (Philipps-Universität Marburg)

Abstract

It is common practice to apply gradient-based optimization algorithms to numerically solve large-scale ODE constrained optimal control problems. Gradients of the objective function are most efficiently computed by approximate adjoint variables. High accuracy with moderate computing time can be achieved by such time integration methods that satisfy a sufficiently large number of adjoint order conditions and supply gradients with higher orders of consistency. In this paper, we upgrade our former implicit two-step Peer triplets constructed in [Algorithms, 15:310, 2022] to meet those new requirements. Since Peer methods use several stages of the same high stage order, a decisive advantage is their lack of order reduction as for semi-discretized PDE problems with boundary control. Additional order conditions for the control and certain positivity requirements now intensify the demands on the Peer triplet. We discuss the construction of 4-stage methods with order pairs (3, 3) and (4, 3) in detail and provide three Peer triplets of practical interest. We prove convergence of order $$s-1$$ s - 1 , at least, for s-stage methods if state, adjoint and control satisfy the corresponding order conditions. Numerical tests show the expected order of convergence for the new Peer triplets.

Suggested Citation

  • Jens Lang & Bernhard A. Schmitt, 2024. "Implicit Peer Triplets in Gradient-Based Solution Algorithms for ODE Constrained Optimal Control," Journal of Optimization Theory and Applications, Springer, vol. 203(1), pages 985-1026, October.
    • Handle: RePEc:spr:joptap:v:203:y:2024:i:1:d:10.1007_s10957-024-02541-z
    DOI: 10.1007/s10957-024-02541-z
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    References listed on IDEAS

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    1. Rico Buchholz & Harald Engel & Eileen Kammann & Fredi Tröltzsch, 2013. "On the optimal control of the Schlögl-model," Computational Optimization and Applications, Springer, vol. 56(1), pages 153-185, September.
    2. Jens Lang & Bernhard A. Schmitt, 2023. "Exact Discrete Solutions of Boundary Control Problems for the 1D Heat Equation," Journal of Optimization Theory and Applications, Springer, vol. 196(3), pages 1106-1118, March.
    3. Albi, G. & Herty, M. & Pareschi, L., 2019. "Linear multistep methods for optimal control problems and applications to hyperbolic relaxation systems," Applied Mathematics and Computation, Elsevier, vol. 354(C), pages 460-477.
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