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Combinatorial optimal control of semilinear elliptic PDEs

Author

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  • Christoph Buchheim

    (TU Dortmund)

  • Renke Kuhlmann

    (TU Dortmund)

  • Christian Meyer

    (TU Dortmund)

Abstract

Optimal control problems (OCPs) containing both integrality and partial differential equation (PDE) constraints are very challenging in practice. The most wide-spread solution approach is to first discretize the problem, which results in huge and typically nonconvex mixed-integer optimization problems that can be solved to proven optimality only in very small dimensions. In this paper, we propose a novel outer approximation approach to efficiently solve such OCPs in the case of certain semilinear elliptic PDEs with static integer controls over arbitrary combinatorial structures, where we assume the nonlinear part of the PDE to be non-decreasing and convex. The basic idea is to decompose the OCP into an integer linear programming (ILP) master problem and a subproblem for calculating linear cutting planes. These cutting planes rely on the pointwise concavity or submodularity of the PDE solution with respect to the control variables. The decomposition allows us to use standard solution techniques for ILPs as well as for PDEs. We further benefit from reoptimization strategies for the PDE solution due to the iterative structure of the algorithm. Experimental results show that the new approach is capable of solving the combinatorial OCP of a semilinear Poisson equation with up to 180 binary controls to global optimality within a 5 h time limit. In the case of the screened Poisson equation, which yields semi-infinite integer linear programs, problems with as many as 1400 binary controls are solved.

Suggested Citation

  • Christoph Buchheim & Renke Kuhlmann & Christian Meyer, 2018. "Combinatorial optimal control of semilinear elliptic PDEs," Computational Optimization and Applications, Springer, vol. 70(3), pages 641-675, July.
  • Handle: RePEc:spr:coopap:v:70:y:2018:i:3:d:10.1007_s10589-018-9993-2
    DOI: 10.1007/s10589-018-9993-2
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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. M. Santis & G. Pillo & S. Lucidi, 2012. "An active set feasible method for large-scale minimization problems with bound constraints," Computational Optimization and Applications, Springer, vol. 53(2), pages 395-423, October.
    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. Björn Geißler & Oliver Kolb & Jens Lang & Günter Leugering & Alexander Martin & Antonio Morsi, 2011. "Mixed integer linear models for the optimization of dynamical transport networks," Mathematical Methods of Operations Research, Springer;Gesellschaft für Operations Research (GOR);Nederlands Genootschap voor Besliskunde (NGB), vol. 73(3), pages 339-362, June.
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    Cited by:

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    2. Carolin Natemeyer & Daniel Wachsmuth, 2021. "A proximal gradient method for control problems with non-smooth and non-convex control cost," Computational Optimization and Applications, Springer, vol. 80(2), pages 639-677, November.
    3. Jeffrey Larson & Sven Leyffer & Prashant Palkar & Stefan M. Wild, 2021. "A method for convex black-box integer global optimization," Journal of Global Optimization, Springer, vol. 80(2), pages 439-477, June.

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