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Finding global solutions of some inverse optimal control problems using penalization and semismooth Newton methods

Author

Listed:
  • Markus Friedemann

    (Brandenburgische Technische Universität Cottbus-Senftenberg)

  • Felix Harder

    (Brandenburgische Technische Universität Cottbus-Senftenberg)

  • Gerd Wachsmuth

    (Brandenburgische Technische Universität Cottbus-Senftenberg)

Abstract

We present a method to solve a special class of parameter identification problems for an elliptic optimal control problem to global optimality. The bilevel problem is reformulated via the optimal-value function of the lower-level problem. The reformulated problem is nonconvex and standard regularity conditions like Robinson’s CQ are violated. Via a relaxation of the constraints, the problem can be decomposed into a family of convex problems and this is the basis for a solution algorithm. The convergence properties are analyzed. It is shown that a penalty method can be employed to solve this family of problems while maintaining convergence speed. For an example problem, the use of the identity as penalty function allows for the solution by a semismooth Newton method. Numerical results are presented. Difficulties and limitations of our approach to solve a nonconvex problem to global optimality are discussed.

Suggested Citation

  • Markus Friedemann & Felix Harder & Gerd Wachsmuth, 2023. "Finding global solutions of some inverse optimal control problems using penalization and semismooth Newton methods," Journal of Global Optimization, Springer, vol. 86(4), pages 1025-1061, August.
    • Handle: RePEc:spr:jglopt:v:86:y:2023:i:4:d:10.1007_s10898-023-01288-7
    DOI: 10.1007/s10898-023-01288-7
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    References listed on IDEAS

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    1. Vyacheslav V. Kalashnikov & Francisco Benita & Patrick Mehlitz, 2015. "The Natural Gas Cash-Out Problem: A Bilevel Optimal Control Approach," Mathematical Problems in Engineering, Hindawi, vol. 2015, pages 1-17, October.
    2. Stephan Dempe & Felix Harder & Patrick Mehlitz & Gerd Wachsmuth, 2019. "Solving inverse optimal control problems via value functions to global optimality," Journal of Global Optimization, Springer, vol. 74(2), pages 297-325, June.
    3. Patrick Mehlitz & Gerd Wachsmuth, 2020. "Bilevel Optimal Control: Existence Results and Stationarity Conditions," Springer Optimization and Its Applications, in: Stephan Dempe & Alain Zemkoho (ed.), Bilevel Optimization, chapter 0, pages 451-484, Springer.
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