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A decomposition method for MINLPs with Lipschitz continuous nonlinearities

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Abstract

Many mixed-integer optimization problems are constrained by nonlinear functions that do not possess desirable analytical properties like convexity or factorability or cannot even be evaluated exactly. This is, e.g., the case for many problems constrained by differential equations or for models that rely on black-box simulation runs. For these problem classes, we present, analyze, and test algorithms that solve mixed-integer problems with Lipschitz continuous nonlinearities. Our theoretical results depend on the assumptions made on the (in)exactness of function evaluations and on the knowledge of Lipschitz constants. If Lipschitz constants are known, we prove finite termination at approximate globally optimal points both for the case of exact and inexact function evaluations. If only approximate Lipschitz constants are known, we prove finite termination and derive additional conditions under which infeasibility can be detected. A computational study for gas transport problems and an academic case study show the applicability of our algorithms to real-world problems and how different assumptions on the constraint functions up- or downgrade the practical performance of the methods.

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Notes

  1. Short pipes are pipes with very short length such that the pressure loss is negligible.

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Acknowledgements

This research has been performed as part of the Energie Campus Nürnberg and is supported by funding of the Bavarian State Government. We thank the DFG for their support within Project A05, A08, and B08 in CRC TRR 154.

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  1. Discrete Optimization, Friedrich-Alexander-Universität Erlangen-Nürnberg (FAU), Cauerstr. 11, 91058, Erlangen, Germany

    Martin Schmidt & Mathias Sirvent

  2. Energie Campus Nürnberg, Fürther Str. 250, 90429, Nuremberg, Germany

    Martin Schmidt

  3. Fachbereich Mathematik, Technische Universität Darmstadt, Dolivostr. 15, 64293, Darmstadt, Germany

    Winnifried Wollner

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Schmidt, M., Sirvent, M. & Wollner, W. A decomposition method for MINLPs with Lipschitz continuous nonlinearities. Math. Program. 178, 449–483 (2019). https://doi.org/10.1007/s10107-018-1309-x

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