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Combinatorial integral approximation

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  • Sebastian Sager
  • Michael Jung
  • Christian Kirches

Abstract

We are interested in structures and efficient methods for mixed-integer nonlinear programs (MINLP) that arise from a first discretize, then optimize approach to time-dependent mixed-integer optimal control problems (MIOCPs). In this study we focus on combinatorial constraints, in particular on restrictions on the number of switches on a fixed time grid. We propose a novel approach that is based on a decomposition of the MINLP into a NLP and a MILP. We discuss the relation of the MILP solution to the MINLP solution and formulate bounds for the gap between the two, depending on Lipschitz constants and the control discretization grid size. The MILP solution can also be used for an efficient initialization of the MINLP solution process. The speedup of the solution of the MILP compared to the MINLP solution is considerable already for general purpose MILP solvers. We analyze the structure of the MILP that takes switching constraints into account and propose a tailored Branch and Bound strategy that outperforms cplex on a numerical case study and hence further improves efficiency of our novel method. Copyright Springer-Verlag 2011

Suggested Citation

  • 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.
    • Handle: RePEc:spr:mathme:v:73:y:2011:i:3:p:363-380
    DOI: 10.1007/s00186-011-0355-4
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    References listed on IDEAS

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    1. T. Christof & G. Reinelt, 1996. "Combinatorial optimization and small polytopes," TOP: An Official Journal of the Spanish Society of Statistics and Operations Research, Springer;Sociedad de Estadística e Investigación Operativa, vol. 4(1), pages 1-53, June.
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    Citations

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    Cited by:

    1. 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.
    2. 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.
    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. 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.
    5. 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.
    6. Bürger, Adrian & Bohlayer, Markus & Hoffmann, Sarah & Altmann-Dieses, Angelika & Braun, Marco & Diehl, Moritz, 2020. "A whole-year simulation study on nonlinear mixed-integer model predictive control for a thermal energy supply system with multi-use components," Applied Energy, Elsevier, vol. 258(C).
    7. Pipicelli, Michele & Muccillo, Massimiliano & Gimelli, Alfredo, 2023. "Influence of the control strategy on the performance of hybrid polygeneration energy system using a prescient model predictive control," Applied Energy, Elsevier, vol. 329(C).
    8. S. Göttlich & A. Potschka & C. Teuber, 2019. "A partial outer convexification approach to control transmission lines," Computational Optimization and Applications, Springer, vol. 72(2), pages 431-456, March.
    9. Sebastian Sager & Mathieu Claeys & Frédéric Messine, 2015. "Efficient upper and lower bounds for global mixed-integer optimal control," Journal of Global Optimization, Springer, vol. 61(4), pages 721-743, April.

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    More about this item

    Keywords

    MINLP; MIOCP; MILP; Optimal control; Integer programming; 90C11; 90C30; 49J15; 90C57;
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