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

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Trends in PDE Constrained Optimization

Part of the book series: International Series of Numerical Mathematics ((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.

This work was completed with the support of 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”.

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References

  1. Y. Achdou, An inverse problem for a parabolic variational inequality arising in volatility calibration with american options. SIAM J. Control Optim. 43(5), 1583–1615 (2005)

    Article  MATH  MathSciNet  Google Scholar 

  2. V. Barbu, Optimal Control of Variational Inequalities. Volume 100 of Research Notes in Mathematics (Pitman Advanced Publishing, Boston, 1984)

    Google Scholar 

  3. V. Barbu, A. Friedman, Optimal design of domains with free-boundary problems. SIAM J. Control Optim. 29(3), 623–637 (1991)

    Article  MATH  MathSciNet  Google Scholar 

  4. R. Becker, H. Kapp, R. Rannacher, Adaptive finite element methods for optimal control of partial differential equations: basic concept. SIAM J. Control Optim. 39(1), 113–132 (2000)

    Article  MATH  MathSciNet  Google Scholar 

  5. A. Bensoussan, J.-L. Lions, Controle impulsionnel et inéquations quasi-variationnelles d’évolutions. C. R. Acad. Sci. Paris 276, 1333–1338 (1974)

    MathSciNet  Google Scholar 

  6. C. Brett, C. Elliott, M. Hintermüller, C. Löbhard, A dual-weighted residual approach to adaptivity for optimal control of elliptic variational inequalities with pointwise objective functionals. IFB-Report, 67, 1–28 (2013)

    Google Scholar 

  7. H. Brézis, Monotonicity methods in Hilbert spaces and some applications to nonlinear partial differential equations, in Contributions to Nonlinear Functional Analysis (Academic, New York, 1971), pp. 101–156

    Google Scholar 

  8. H. Brézis, G. Stampacchia, Sur la regularité de la solution d’inéquations elliptiques. Bulletin de la Société Mathématique de France 96, 153–180 (1968)

    MATH  Google Scholar 

  9. C.M. Elliott, On a variational inequality formulation of an electrochemical machining moving boundary problem and its approximation by the finite element method. J. Inst. Math. Appl. 25(2), 121–131 (1980)

    Article  MATH  MathSciNet  Google Scholar 

  10. G. Fichera, Problemi elettrostatici con vincoli unilaterali: il problema di Signorini con ambigue condizioni al contorno. Memorie dell’Accademia Nazionale dei Lincei 8, 91–140 (1964)

    MathSciNet  Google Scholar 

  11. A. Gaevskaya, M. Hintermüller, R. Hoppe, Adaptive finite elements for optimally controlled elliptic variational inequalities of obstacle type. IFB-Report No. 68, 09 2013

    Google Scholar 

  12. A. Günther, M. Hinze, A posteriori error control of a state constrained elliptic control problem. J. Numer. Math. 16(4), 307–322 (2008)

    Article  MATH  MathSciNet  Google Scholar 

  13. A. Henrot, M. Pierre, Variation et optimisation de formes. Volume 48 of Mathématiques et Applications (Springer, Berlin, 1992). Une analyse géométrique

    Google Scholar 

  14. R. Herzog, C. Meyer, G. Wachsmuth, C-stationarity for optimal control of static plasticity with linear kinematic hardening. SIAM J. Control Optim. 50(5), 3052–3082 (2012)

    Article  MATH  MathSciNet  Google Scholar 

  15. R. Herzog, C. Meyer, G. Wachsmuth, B- and strong stationarity for optimal control of static plasticity with hardening. SIAM J. Optim. 23(1), 321–352 (2013)

    Article  MATH  MathSciNet  Google Scholar 

  16. M. Hintermüller, R. Hoppe, Goal-oriented adaptivity in control constrained optimal control of partial differential equations. SIAM J. Control Optim. 47, 1721–1743 (2008)

    Article  MATH  MathSciNet  Google Scholar 

  17. M. Hintermüller, I. Kopacka, Mathematical programs with complementarity constraints in function space: C- and strong stationarity and a path-following algorithm. SIAM J. Optim. 20, 868–902 (2009)

    Article  MATH  MathSciNet  Google Scholar 

  18. M. Hintermüller, I. Kopacka, A smooth penalty approach and a nonlinear multigrid algorithm for elliptic MPECs. Comput. Optim. Appl. 50(1), 111–145 (2011)

    Article  MATH  MathSciNet  Google Scholar 

  19. M. Hintermüller, A. Laurain, Optimal shape design subject to elliptic variational inequalities. SIAM J. Control Optim. 49(3), 1015–1047 (2011)

    Article  MATH  MathSciNet  Google Scholar 

  20. M. Hintermüller, C.N. Rautenberg, A sequential minimization technique for elliptic quasi-variational inequalities with gradient constraints. SIAM J. Optim. 22(4), 1224–1257 (2012)

    Article  MATH  MathSciNet  Google Scholar 

  21. M. Hintermüller, T. Surowiec, First order optimality conditions for elliptic mathematical programs with equilibrium constraints via variational analysis. SIAM J. Optim. 21(4), 1561–1593 (2011)

    Article  MATH  MathSciNet  Google Scholar 

  22. M. Hintermüller, T. Surowiec, A bundle-free implicit programming approach for a class of MPECs in function space. IFB-Report, 60, 2012

    Google Scholar 

  23. M. Hintermüller, K. Ito, K. Kunisch, The primal-dual active set strategy as a semismooth Newton method. SIAM J. Optim. 13, 865–888 (2002)

    Article  MathSciNet  Google Scholar 

  24. M. Hintermüller, R. Hoppe, C. Löbhard, ESAIM: Control, Optimization and Calculus of Variations 20(2), 524–546 (2014)

    Article  MATH  Google Scholar 

  25. K. Ito, K. Kunisch, An active set strategy based on the augmented lagrangian formulation for image restoration. ESAIM: Math. Model. Numer. Anal. 33(1), 1–21 (1999)

    Article  MATH  MathSciNet  Google Scholar 

  26. T. Keil, A smooth penalty approach for MPECs with gradient constrained lower-level problems in Banach spaces. Master’s thesis, Humboldt-Universität zu Berlin, 2013

    Google Scholar 

  27. D. Kinderlehrer, G. Stampacchia, An Introduction to Variational Inequalities and Their Applications (Academic, New York, 1980)

    MATH  Google Scholar 

  28. M. Kunze, J. Rodrigues, An elliptic quasi-variational inequality with gradient constraints and some of its applications. Math. Methods Appl. Sci. 23, 897–908 (2000)

    Article  MATH  MathSciNet  Google Scholar 

  29. J. Lions, G. Stampacchia, Variational inequalities. Commun. Pure Appl. Math. 20, 493–519 (1967)

    Article  MATH  MathSciNet  Google Scholar 

  30. J.-L. Lions, Sur le côntrole optimal des systemes distribuées. Enseigne 19, 125–166 (1973)

    MATH  Google Scholar 

  31. J.-L. Lions, Various topics in the theory of optimal control of distributed systems, in optimal control theory and its applications, part I. Lect. Notes Econ. Math. Syst. 105, 166–309 (1974)

    Article  Google Scholar 

  32. Z. Luo, J. Pang, D. Ralph, Mathematical Programs with Equilibrium Constraints (Cambridge University Press, Cambridge/New York, 1996)

    Book  Google Scholar 

  33. F. Mignot, Controle dans les inéquations variationnelles elliptiques. Funct. Anal. 22, 130–185 (1976)

    Article  MATH  MathSciNet  Google Scholar 

  34. F. Mignot, J. Puel, Optimal control in some variational inequalities. SIAM J. Control Optim. 22(3), 466–476 (1984)

    Article  MATH  MathSciNet  Google Scholar 

  35. P. Neittaanmäki, J. Sokołowski, J.-P. Zolésio, Optimization of the domain in elliptic variational inequalities. Appl. Math. Optim. 18(1), 85–98 (1988)

    Article  MATH  MathSciNet  Google Scholar 

  36. P. Neittaanmaki, J. Sprekels, D. Tiba, Optimization of Elliptic Systems: Theory and Applications. Springer Monographs in Mathematics (Springer, New York, 2006)

    Google Scholar 

  37. J. Outrata, M. Kočvara, J. Zowe, Nonsmooth Approach to Optimization Problems with Equilibrium Constraints: Theory, Applications, and Numerical Results. No. 152 in Nonconvex Optimization and Its Applications (Kluwer Academic, Dordrecht/Boston, 1998)

    Google Scholar 

  38. H. Scheel, S. Scholtes, Mathematical programs with complementarity constraints: stationarity, optimality, and sensitivity. Math. Oper. Res. 25(1), 1–22 (2000)

    Article  MATH  MathSciNet  Google Scholar 

  39. A. Schiela, D. Wachsmuth, Convergence analysis of smoothing methods for optimal control of stationary variational inequalities with control constraints. ESAIM: Math. Model. Numer. Anal. 47(5), 771–787 (2013)

    Article  MATH  MathSciNet  Google Scholar 

  40. J. Sokołowski, J.-P. Zolésio, Introduction to Shape Optimization: Shape Sensitivity Analysis. Volume 16 of Springer Series in Computational Mathematics (Springer, Berlin, 1992)

    Google Scholar 

  41. D. Tiba, Propriétés de controlabilité pour les systèmes elliptiques, la méthode des domaines fictifs et problèmes de design optimal, in Optimization, Optimal Control and Partial Differential Equations (Iaşi, 1992). Volume 107 of International Series Numerical Mathematics (Birkhäuser, Basel, 1992), pp. 251–261

    Google Scholar 

  42. B. Vexler, W. Wollner, Adaptive finite elements for elliptic optimization problems with control constraints. SIAM J. Control Optim. 47(1), 509–534 (2008)

    Article  MATH  MathSciNet  Google Scholar 

  43. J. Yvon, Contrôle optimal de systèmes gouvernés par des inéquations variationnelles. PhD thesis, Université de Compiegne, Paris, 1973

    Google Scholar 

  44. J. Yvon, Optimal control of systems governed by variational inequalities, in 5th Conference on Optimization Techniques. Lecture Notes in Computer Science (Springer, Berlin/Heidelberg, 1973), Rome, Italy, pp. 265–275

    Google Scholar 

  45. J. Zowe, S. Kurcyusz, Regularity and stability for the mathematical programming problem in Banach spaces. Appl. Math. Optim. 5, 49–62 (1979)

    Article  MATH  MathSciNet  Google Scholar 

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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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Authors and Affiliations

  1. Institut für Mathematik, Humboldt-Universität zu Berlin, Unter den Linden 6, D-10099, Berlin, Germany

    Michael Hintermüller, Caroline Löbhard & Thomas M. Surowiec

  2. Institut für Mathematik, Technische Universität Berlin, Straße des 17. Juni 136, D-10623, Berlin, Germany

    Antoine Laurain

  3. Department of Mathematics and Scientific Computing, Karl-Franzens-University of Graz, Heinrichstrasse 36, A-8010, Graz, Austria

    Carlos N. Rautenberg

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  1. Michael Hintermüller
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  2. Antoine Laurain
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  3. Caroline Löbhard
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  4. Carlos N. Rautenberg
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  5. Thomas M. Surowiec
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Correspondence to Michael Hintermüller .

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Editors and Affiliations

  1. Department Mathematik, Universität Erlangen-Nürnberg, Erlangen, Germany

    Günter Leugering

  2. Institut für Dynamik komplexer technisch, Max-Planck-Institut, Magdeburg, Germany

    Peter Benner

  3. Fakultät Bio- und Chemieingenieurwesen, Technische Universität Dortmund, Dortmund, Germany

    Sebastian Engell

  4. Institut für Mathematik, Humboldt-Universität zu Berlin, Berlin, Germany

    Andreas Griewank

  5. Mathematisches Institut, Universität Basel, Basel, Switzerland

    Helmut Harbrecht

  6. Fachbereich Mathematik Optimierung und Approximation, Universität Hamburg, Hamburg, Germany

    Michael Hinze

  7. Institut für Angewandte Mathematik, Ruprecht-Karls-Universität Heidelberg, Heidelberg, Germany

    Rolf Rannacher

  8. Fachbereich Mathematik, Technische Universität Darmstadt, Darmstadt, Germany

    Stefan Ulbrich

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Hintermüller, M., Laurain, A., Löbhard, C., Rautenberg, C.N., Surowiec, T.M. (2014). Elliptic Mathematical Programs with Equilibrium Constraints in Function Space: Optimality Conditions and Numerical Realization. In: Leugering, G., et al. Trends in PDE Constrained Optimization. International Series of Numerical Mathematics, vol 165. Birkhäuser, Cham. https://doi.org/10.1007/978-3-319-05083-6_9

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