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Elliptic Mathematical Programs with Equilibrium Constraints in Function Space: Optimality Conditions and Numerical Realization

  • Michael HintermüllerEmail author
  • Antoine Laurain
  • Caroline Löbhard
  • Carlos N. Rautenberg
  • Thomas M. Surowiec
Chapter
Part of the International Series of Numerical Mathematics book series (ISNM, volume 165)

Abstract

Recent advances in the analytical as well as numerical treatment of classes of elliptic mathematical programs with equilibrium constraints (MPECs) in function space are discussed. In particular, stationarity conditions for control problems with point tracking objectives and subject to the obstacle problem as well as for optimization problems with variational inequality constraints and pointwise constraints on the gradient of the state are derived. For the former problem class including the case of L 2-tracking-type objectives (rather than pointwise ones) a bundle-free solution method as well as adaptive finite element discretizations are introduced. Moreover, the analytical and numerical treatment of shape design problems subject to elliptic variational inequality constraints is highlighted. With respect to problems involving gradient constraints, the paper ends with a fixed-point-Moreau-Yosida-based semismooth Newton solver for a class of nonlinear elliptic quasi-variational inequality problems.

Keywords

MPECs and MPCCs in function space Gradient constraints Point-tracking Adaptive finite element methods Nonlinear elliptic quasivariational inequality problem Optimal shape design subject to elliptic variational inequalities 

Mathematics Subject Classification (2010)

49K20 49M25 65K10 90C33 

Notes

Acknowledgements

The authors acknowledge support by DFG-Project “Elliptic Mathematical Programs with Equilibrium Constraints (MPECs) in Function Space: Optimality Conditions and Numerical Realization” within the DFG Priority Program SPP 1253 on “Optimization with Partial Differential Equations”, project C28 of the DFG Research Center “Matheon” as well as the Austrian Science Fund FWF under START-Project Y305 “Interfaces and Free Boundaries”.

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Copyright information

© Springer International Publishing Switzerland 2014

Authors and Affiliations

  • Michael Hintermüller
    • 1
    Email author
  • Antoine Laurain
    • 2
  • Caroline Löbhard
    • 1
  • Carlos N. Rautenberg
    • 3
  • Thomas M. Surowiec
    • 1
  1. 1.Institut für MathematikHumboldt-Universität zu BerlinBerlinGermany
  2. 2.Institut für MathematikTechnische Universität BerlinBerlinGermany
  3. 3.Department of Mathematics and Scientific ComputingKarl-Franzens-University of GrazGrazAustria

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