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An ℓ 1-Penalty Scheme for the Optimal Control of Elliptic Variational Inequalities

  • Conference paper
Numerical Analysis and Optimization

Part of the book series: Springer Proceedings in Mathematics & Statistics ((PROMS,volume 134))

Abstract

Anℓ 1-penalty scheme in function space for the optimal control of elliptic variational inequalities is proposed. In anL 2-tracking context, an iterative algorithm is proven to generate a sequence which converges to some weakly C-stationary point and, under certain conditions, even to a strongly stationary point of the original problem. In the case of point tracking control, where the objective contains pointwise function evaluations of the state variable, a modified model problem with constraints on the dual variable associated with the variational inequality constraint is introduced and an auxiliary problem that penalizes not only the complementarity, but also the state constraint, is analyzed. Passing to the limit with the penalty parameter in the stationarity system of the auxiliary problem yields some weak form of a C-stationarity system for the original problem if the additional dual constraints are not active. Finally, numerical results obtained by the new algorithms are documented.

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References

  1. Achdou, Y.: 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  MathSciNet  MATH  Google Scholar 

  2. Adams, R., Fournier, J.: Sobolev Spaces. Pure and Applied Mathematics, vol. 140. Elsevier, Amsterdam (2003)

    Google Scholar 

  3. Adams, R., Hedberg, L.: Function Spaces and Potential Theory. A Series of Comprehensive Studies in Mathematics, vol. 314. Springer, Berlin (1996)

    Google Scholar 

  4. Anitescu, M., Tseng, P., Wright, S.: Elastic-mode algorithms for mathematical programs with equilibrium constraints: global convergence and stationarity properties. Math. Program. 110, 337–371 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  5. Barbu, V.: Optimal Control of Variational Inequalities. Addison-Wesley, Reading (1984)

    MATH  Google Scholar 

  6. Bensoussan, A., Lions, J., Papanicolaou, G.: Asymptotic Analysis for Periodic Structures. North-Holland, Amsterdam (1978)

    MATH  Google Scholar 

  7. Bonnans, J., Shapiro, A.: Perturbation Analysis of Optimization Problems. Springer, New York (2000)

    Book  MATH  Google Scholar 

  8. Brett, C., Elliott, C., Hintermüller, M., Löbhard, C.: Mesh adaptivity in optimal control of elliptic variational inequalities with point-tracking of the state. Start project, IFB-Report No. 67 (09/2013), Institute of Mathematics and Scientific Computing, University of Graz (2013)

    Google Scholar 

  9. Facchinei, F., Pang, J.: Finite-Dimensional Variational Inequalities and Complementarity Problems. Springer Series in Operations Research, vol. 1. Springer, New York (2003)

    Google Scholar 

  10. Glowinski, R., Lions, J., Trémolières, R.: Numerical Analysis of Variational Inequalities. Studies in Mathematics and its Applications, vol. 8. North-Holland, Amsterdam (1981)

    Google Scholar 

  11. Gröger, K.: A W 1, p-estimate for solutions to mixed boundary value problems for second order elliptic differential equations. Math. Ann. 283, 679–687 (1989)

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  14. Hintermüller, M., Kopacka, I.: 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  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  16. Hintermüller, M., Surowiec, T.: A bundle-free implicit programming approach for a class of MPECs in function space. Start Project, IFB-Report No. 60 (09/2012), Institute of Mathematics and Scientific Computing, University of Graz (2013)

    Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  18. Hintermüller, M., Hoppe, R., Löbhard, C.: A dual-weighted residual approach to goal-oriented adaptivity for optimal control of elliptic variational inequalities. ESAIM Control Optim. Calc. Var. 20, 524–546 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  19. Hintermüller, M., Mordukhovich, B., Surowiec, T.: Several approaches for the derivation of stationarity conditions for elliptic MPECs with upper-level control constraints. Math. Program. 146, 555–582 (2014)

    Article  MathSciNet  MATH  Google Scholar 

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

    MATH  Google Scholar 

  21. Lions, J.: Optimal Control of Systems Governed by Partial Differential Equations. Grundlehren der mathematischen Wissenschaften. Springer, Heidelberg (1971)

    Book  MATH  Google Scholar 

  22. Löbhard, C.: Optimal control of variational inequalities: Numerical methods and point tracking. Ph.D. thesis, Humboldt-Universität zu Berlin, Institute of Mathematics (2014)

    Google Scholar 

  23. Luo, Z., Pang, J., Ralph, D.: Mathematical Programs with Equilibrium Constraints. Cambridge University Press, Cambridge (1996)

    Book  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  25. Mordukhovich, B.: Variational Analysis and Generalized Differentiation I: Basic Theory. Grundlehren Der Mathematischen Wissenschaften, vol. 330. Springer, Berlin (2006)

    Google Scholar 

  26. Mordukhovich, B.: Variational Analysis and Generalized Differentiation II: Applications. Grundlehren Der Mathematischen Wissenschaften, vol. 331. Springer, Berlin (2006)

    Google Scholar 

  27. Outrata, J., Kočvara, M., Zowe, J.: Nonsmooth Approach to Optimization Problems with Equilibrium Constraints: Theory, Applications, and Numerical Results. Nonconvex Optimization and Its Applications, vol. 152. Kluwer Academic, Dordrecht (1998)

    Google Scholar 

  28. Outrata, J., Jarušek, J., Stará, J.: On optimality conditions in control of elliptic variational inequalities. Set Valued Anal. 19, 23–42 (2011)

    Article  MATH  Google Scholar 

  29. Rodrigues, J.-F.: Obstacle Problems in Mathematical Physics. North-Holland, Amsterdam (1987)

    MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  32. Ulbrich, M.: Semismooth Newton methods for operator equations in function spaces. SIAM J. Optim. 13, 805–842 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  33. Wachsmuth, G.: Strong stationarity for optimal control of the obstacle problem with control constraints. SIAM J. Optim. 24, 1914–1932 (2014)

    Article  MathSciNet  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

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Author information

Authors and Affiliations

  1. Department of Mathematics, Humboldt-Universität zu Berlin, Unter den Linden 6, 10099, Berlin, Germany

    M. Hintermüller & C. Löbhard

  2. Faculté des Sciences et Technique, Université Sultan Moulay Slimane, Béni-Mellal, Morocco

    M. H. Tber

Authors
  1. M. Hintermüller
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  2. C. Löbhard
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  3. M. H. Tber
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Corresponding author

Correspondence to M. Hintermüller .

Editor information

Editors and Affiliations

  1. Department of Mathematics College of Science, Sultan Qaboos University, Muscat, Oman

    Mehiddin Al-Baali

  2. Faculty of Engineering, University of Calabria, Arcavacada di Rende, Italy

    Lucio Grandinetti

  3. Department Mathematics and Statistics College of Science, Sultan Qaboos University, Muscat, Oman

    Anton Purnama

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Hintermüller, M., Löbhard, C., Tber, M.H. (2015). An ℓ 1-Penalty Scheme for the Optimal Control of Elliptic Variational Inequalities. In: Al-Baali, M., Grandinetti, L., Purnama, A. (eds) Numerical Analysis and Optimization. Springer Proceedings in Mathematics & Statistics, vol 134. Springer, Cham. https://doi.org/10.1007/978-3-319-17689-5_7

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