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Efficient Solution of Discrete Subproblems Arising in Integer Optimal Control with Total Variation Regularization

Author

Listed:
  • Marvin Severitt

    (Faculty of Mathematics, TU Dortmund University, Dortmund 44227, Germany)

  • Paul Manns

    (Faculty of Mathematics, TU Dortmund University, Dortmund 44227, Germany)

Abstract

We consider a class of integer linear programs (IPs) that arise as discretizations of trust-region subproblems of a trust-region algorithm for the solution of control problems, where the control input is an integer-valued function on a one-dimensional domain and is regularized with a total variation term in the objective, which may be interpreted as a penalization of switching costs between different control modes. We prove that solving an instance of the considered problem class is equivalent to solving a resource-constrained shortest-path problem (RCSPP) on a layered directed acyclic graph. This structural finding yields an algorithmic solution approach based on topological sorting and corresponding run-time complexities that are quadratic in the number of discretization intervals of the underlying control problem, the main quantifier for the size of a problem instance. We also consider the solution of the RCSPP with an A * algorithm. Specifically, the analysis of a Lagrangian relaxation yields a consistent heuristic function for the A * algorithm and a preprocessing procedure, which can be employed to accelerate the A * algorithm for the RCSPP without losing optimality of the computed solution. We generate IP instances by executing the trust-region algorithm on several integer optimal control problems. The numerical results show that the accelerated A * algorithm and topological sorting outperform a general-purpose IP solver significantly. Moreover, the accelerated A * algorithm is able to outperform topological sorting for larger problem instances. We also give computational evidence that the performance of the superordinate trust-region algorithm may be improved if it is initialized with a solution obtained with the combinatorial integral approximation.

Suggested Citation

  • Marvin Severitt & Paul Manns, 2023. "Efficient Solution of Discrete Subproblems Arising in Integer Optimal Control with Total Variation Regularization," INFORMS Journal on Computing, INFORMS, vol. 35(4), pages 869-885, July.
    • Handle: RePEc:inm:orijoc:v:35:y:2023:i:4:p:869-885
    DOI: 10.1287/ijoc.2023.1294
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    References listed on IDEAS

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    1. Sebastian Sager & Michael Jung & Christian Kirches, 2011. "Combinatorial integral approximation," Mathematical Methods of Operations Research, Springer;Gesellschaft für Operations Research (GOR);Nederlands Genootschap voor Besliskunde (NGB), vol. 73(3), pages 363-380, June.
    2. Sebastian Sager & Clemens Zeile, 2021. "On mixed-integer optimal control with constrained total variation of the integer control," Computational Optimization and Applications, Springer, vol. 78(2), pages 575-623, March.
    3. Falk Hante & Sebastian Sager, 2013. "Relaxation methods for mixed-integer optimal control of partial differential equations," Computational Optimization and Applications, Springer, vol. 55(1), pages 197-225, May.
    4. Sven Leyffer & Paul Manns & Malte Winckler, 2021. "Convergence of sum-up rounding schemes for cloaking problems governed by the Helmholtz equation," Computational Optimization and Applications, Springer, vol. 79(1), pages 193-221, May.
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