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Numerical Optimization Methods for the Optimal Control of Elliptic Variational Inequalities

  • Thomas M. Surowiec
Chapter
Part of the The IMA Volumes in Mathematics and its Applications book series (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.

Notes

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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Authors and Affiliations

  1. 1.FB12 Mathematik und InformatikPhilipps-Universität MarburgMarburgGermany

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