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An Augmented Lagrangian Method for State Constrained Linear Parabolic Optimal Control Problems

Author

Listed:
  • Hailing Wang

    (Shanghai University)

  • Changjun Yu

    (Shanghai University)

  • Yongcun Song

    (Friedrich-Alexander-Universität Erlangen-Nürnberg)

Abstract

In this paper, we consider a class of state constrained linear parabolic optimal control problems. Instead of treating the inequality state constraints directly, we reformulate the problem as an equality-constrained optimization problem, and then apply the augmented Lagrangian method (ALM) to solve it. We prove the convergence of the ALM without any existence or regularity assumptions on the corresponding Lagrange multipliers, which is an essential complement to the classical theoretical results for the ALM because restrictive regularity assumptions are usually required to guarantee the existence of the Lagrange multipliers associated with the state constraints. In addition, under an appropriate choice of penalty parameter sequence, we can obtain a super-linear non-ergodic convergence rate for the ALM. Computationally, we apply a semi-smooth Newton (SSN) method to solve the ALM subproblems and design an efficient preconditioned conjugate gradient method for solving the Newton systems. Some numerical results are given to illustrate the effectiveness and efficiency of our algorithm.

Suggested Citation

  • Hailing Wang & Changjun Yu & Yongcun Song, 2024. "An Augmented Lagrangian Method for State Constrained Linear Parabolic Optimal Control Problems," Journal of Optimization Theory and Applications, Springer, vol. 203(1), pages 196-226, October.
    • Handle: RePEc:spr:joptap:v:203:y:2024:i:1:d:10.1007_s10957-024-02494-3
    DOI: 10.1007/s10957-024-02494-3
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    References listed on IDEAS

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    1. Yurii Nesterov, 2018. "Lectures on Convex Optimization," Springer Optimization and Its Applications, Springer, edition 2, number 978-3-319-91578-4, December.
    2. Veronika Karl & Daniel Wachsmuth, 2018. "An augmented Lagrange method for elliptic state constrained optimal control problems," Computational Optimization and Applications, Springer, vol. 69(3), pages 857-880, April.
    3. Hamdullah Yücel & Peter Benner, 2015. "Adaptive discontinuous Galerkin methods for state constrained optimal control problems governed by convection diffusion equations," Computational Optimization and Applications, Springer, vol. 62(1), pages 291-321, September.
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