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Numerical Optimization Methods for the Optimal Control of Elliptic Variational Inequalities

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Frontiers in PDE-Constrained Optimization

Part of the book series: The IMA Volumes in Mathematics and its Applications ((IMA,volume 163))

Abstract

The optimal control of variational inequalities introduces a number of additional challenges to PDE-constrained optimization problems both in terms of theory and algorithms. The purpose of this article is to first introduce the theoretical underpinnings and then to illustrate various types of numerical methods for the optimal control of variational inequalities. For a generic problem class, sufficient conditions for the existence of a solution are discussed and subsequently, the various types of multiplier-based optimality conditions are introduced. Finally, a number of function-space-based algorithms for the numerical solution of these control problems are presented. This includes adaptive methods based on penalization or regularization as well as non-smooth approaches based on tools from non-smooth optimization and set-valued analysis. A new type of projected subgradient method based on an approximation of limiting coderivatives is proposed. Moreover, several existing methods are extended to include control constraints. The computational performance of the algorithms is compared and contrasted numerically.

TMS’s research was sponsored in part by the DFG grant no. SU 963/1-1 as well as Research Center matheon supported by the Einstein Foundation Berlin within project OT1.

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Acknowledgements

This paper is an extension of a short course given by the author at the “Frontiers in PDE-constrained Optimization” workshop on June 6–10, 2016 at the Institute for Mathematics and its Applications at the University of Minnesota, Minneapolis, which was sponsored by ExxonMobil. The author would therefore like to express his gratitude for the financial support and the opportunity to write this article. In addition, the author would like to thank Harbir Antil, Patrick Farrell, and the two anonymous reviewers for their helpful comments and thought-provoking questions on the text.

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  1. FB12 Mathematik und Informatik, Philipps-Universität Marburg, Marburg, Germany

    Thomas M. Surowiec

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  1. Department of Mathematical Sciences, George Mason University, Fairfax, VA, USA

    Harbir Antil

  2. Center for Computing Research, Sandia National Laboratories, Albuquerque, NM, USA

    Drew P. Kouri

  3. Corporate Strategic Research, ExxonMobil Research and Engineering Company, Annandale, NJ, USA

    Martin-D. Lacasse

  4. Center for Computing Research, Sandia National Laboratories, Albuquerque, NM, USA

    Denis Ridzal

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Surowiec, T.M. (2018). Numerical Optimization Methods for the Optimal Control of Elliptic Variational Inequalities. In: Antil, H., Kouri, D.P., Lacasse, MD., Ridzal, D. (eds) Frontiers in PDE-Constrained Optimization. The IMA Volumes in Mathematics and its Applications, vol 163. Springer, New York, NY. https://doi.org/10.1007/978-1-4939-8636-1_4

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