BTU overview
Faculty 1
Chair of Optimal Control Prof. Dr. rer. nat. habil. Gerd Wachsmuth

Theoretical and numerical analysis of optimal control problems with two decision levels

General information

SPP1962

Keywords

  • Hierarchical optimisation and optimal control
  • Non-differentiable optimisation and optimal control
  • Optimisation problems and optimal control problems with complementarity constraints
  • infinite-dimensional optimisation
  • Optimisation via partial differential equations
  • necessary and sufficient optimality conditions
  • Numerical methods of optimisation
  • numerical analysis
  • Numerics of partial differential equations

Project description

The project aims to analyse hierarchical optimal control problems with partial differential equations and two decision levels. The derivation of necessary and sufficient optimality conditions for such problems is to be carried out by utilising different approaches (use of optimality conditions of the lower level, use of the optimal value function of the lower level, use of local differentiability properties of the solution of the lower level with respect to the decision variables of the upper level). First, a general hierarchical optimisation problem in Banach spaces is assumed before the results obtained are applied to hierarchical optimal control problems. Furthermore, the numerical behaviour of such problems will be investigated. Here, the practical feasibility of theoretical approaches must be scrutinised. In particular, the sensitivity and stability analysis of finite-dimensional hierarchical optimisation problems play a decisive role in the selection of suitable reformulation approaches and discretisation strategies. Error bounds are to be derived which allow a convergence analysis in the selected function spaces. Both the theoretical and the numerical treatment of hierarchical Optimal Control problems lead to problems in the context of parametric and non-smooth Optimal Control. A class of reference problems will be created to test and compare the derived theoretical results and numerical methods.

Project-related publications

Publications in Journals

  1. Daniel Wachsmuth and Gerd Wachsmuth
    A simple proof of the Baillon-Haddad theorem on open subsets of Hilbert spaces
    Journal of Convex Analysis, 30(4), p.1319-1328, 2023
    arXiv: 2204.00282
     
  2. Constantin Christof and Gerd Wachsmuth
    Semismoothness for Solution Operators of Obstacle-Type Variational Inequalities with Applications in Optimal Control
    SIAM Journal on Control and Optimization (SICON), 61(3), p.1162-1186, 2023
    DOI: 10.1137/21m1467365
    arXiv: 2112.12018
     
  3. Markus Friedemann, Felix Harder and Gerd Wachsmuth
    Finding global solutions of some inverse optimal control problems using penalization and semismooth Newton methods
    Journal of Global Optimization, 86(4), p.1025-1061, 2023
    DOI: 10.1007/s10898-023-01288-7
    arXiv: 2203.00325
     
  4. Ira Neitzel and Gerd Wachsmuth
    First-order conditions for the optimal control of the obstacle problem with state constraints
    Pure and Applied Functional Analysis, 7(5), p.1881-1911, 2022
    arXiv: 2012.15324
    Preprint SPP1962-154
     
  5. Alexander Y. Kruger and Patrick Mehlitz
    Optimality conditions, approximate stationarity, and applications - a story beyond Lipschitzness
    ESAIM: Control, Optimisation and Calculus of Variations (COCV), 28, p.42, 2022
    DOI: 10.1051/cocv/2022024
     
  6. Xiaoxi Jia, Christian Kanzow, Patrick Mehlitz and Gerd Wachsmuth
    An Augmented Lagrangian Method for Optimization Problems with Structured Geometric Constraints
    Mathematical Programming, 199, p.1365-1415, 2023
    DOI: 10.1007/s10107-022-01870-z
     
  7. Constantin Christof and Gerd Wachsmuth
    Lipschitz Stability and Hadamard Directional Differentiability for Elliptic and Parabolic Obstacle-Type Quasi-Variational Inequalities
    SIAM Journal on Control and Optimization (SICON), 60(6), p.3430-3456, 2022
    DOI: 10.1137/21M1419635
    arXiv: 2105.05895
    Preprint SPP1962-169
     
  8. Daniel Wachsmuth and Gerd Wachsmuth
    Second-order conditions for non-uniformly convex integrands: quadratic growth in L¹
    Journal of Nonsmooth Analysis and Optimizaton, 3, 2022
    DOI: 10.46298/jnsao-2022-8733
    arXiv: 2111.10238
    Preprint SPP1962-184
     
  9. Patrick Mehlitz and Gerd Wachsmuth
    Subdifferentiation of nonconvex sparsity-promoting functionals on Lebesgue spaces
    SIAM Journal on Control and Optimization (SICON), 60(3), p.1819-1839, 2022
    DOI: 10.1137/21m1435173
    arXiv: 2107.09340
    Preprint SPP1962-173
     
  10. Gerd Wachsmuth
    From resolvents to generalized equations and quasi-variational inequalities: existence and differentiability
    Journal of Nonsmooth Analysis and Optimizaton, 3, 2022
    DOI: 10.46298/jnsao-2022-8537
    arXiv: 2109.13569
    Preprint SPP1962-177
     
  11. Gerd Wachsmuth
    Maximal monotone operators with non-maximal graphical limit
    Examples and Counterexamples, 2, 2022
    DOI: 10.1016/j.exco.2022.100073
    arXiv: 2107.10055
    Preprint SPP1962-174
     
  12. Felix Harder and Gerd Wachsmuth
    M-stationarity for a class of MPCCs in Lebesgue spaces
    Journal of Mathematical Analysis and Applications (JMAA), 512(2), p.1-28, 2022
    DOI: 10.1016/j.jmaa.2022.126167
    arXiv: 2110.11693
    Preprint SPP1962-180
     
  13. Felix Harder
    A new elementary proof for M-stationarity under MPCC-GCQ for mathematical programs with complementarity constraints
    Journal of Nonsmooth Analysis and Optimizaton, 2, 2021
    DOI: 10.46298/jnsao-2021-6903
    arXiv: 2011.04474
    Preprint SPP1962-151
     
  14. Felix Harder, Patrick Mehlitz and Gerd Wachsmuth
    Reformulation of the M-stationarity conditions as a system of discontinuous equations and its solution by a semismooth Newton method
    SIAM Journal on Optimization (SIOPT), 31(2), p.1459-1488, 2021
    DOI: 10.1137/20m1321413
    arXiv: 2002.10124
    Preprint SPP1962-135
     
  15. Yu Deng, Patrick Mehlitz and Uwe Prüfert
    Coupled versus Decoupled Penalization of Control Complementarity Constraints
    ESAIM: Control, Optimisation and Calculus of Variations (COCV), 27, p.1-31, 2021
    DOI: 10.1051/cocv/2021022
     
  16. Tobias Geiger, Daniel Wachsmuth and Gerd Wachsmuth
    Optimal control of ODEs with state suprema
    Mathematical Control and Related Fields, 11(3), p.555-578, 2021
    DOI: 10.3934/mcrf.2021012
     
  17. Patrick Mehlitz and Gerd Wachsmuth
    Bilevel optimal control: existence results and stationarity conditions
    in: Bilevel Optimization: Advances and Next Challenges, Optimization and Its Applications, p.451-484, Springer, 2020
    DOI: 10.1007/978-3-030-52119-6_16
     
  18. Gerd Wachsmuth
    Elliptic quasi-variational inequalities under a smallness assumption: Uniqueness, differential stability and optimal control
    Calculus of Variations and Partial Differential Equations, 59(2), 2020
    DOI: 10.1007/s00526-020-01743-3
    arXiv: 1909.02934
    Preprint SPP1962-118
     
  19. Eike Börgens, Christian Kanzow, Patrick Mehlitz and Gerd Wachsmuth
    New Constraint Qualifications for Optimization Problems in Banach Spaces Based on Asymptotic KKT Conditions
    SIAM Journal on Optimization (SIOPT), 30(4), p.2956-2982, 2020
    DOI: 10.1137/19M1306804
    arXiv: 1912.06531
    Preprint SPP1962-128
     
  20. Constantin Christof and Gerd Wachsmuth
    On Second-Order Optimality Conditions for Optimal Control Problems Governed by the Obstacle Problem
    Optimization, 70(10), p.2247-2287, 2021
    DOI: 10.1080/02331934.2020.1778686
    arXiv: 1906.09065
    Preprint SPP1962-113
     
  21. Christian Clason, Yu Deng, Patrick Mehlitz and Uwe Prüfert
    Optimal control problems with control complementarity constraints
    Optimization Methods and Software, 35(2), p.142-170, 2020
    DOI: 10.1080/10556788.2019.1604705
     
  22. Stephan Dempe and Patrick Mehlitz
    Semivectorial Bilevel Programming versus Scalar Bilevel Programming
    Optimization, 69(4), p.657-679, 2020
    DOI: 10.1080/02331934.2019.1625900
     
  23. Stephan Dempe, Felix Harder, Patrick Mehlitz and Gerd Wachsmuth
    Solving Inverse Optimal Control Problems via Value Functions to Global Optimality
    Journal of Global Optimization, 74(2), p.297-325, 2019
    DOI: 10.1007/s10898-019-00758-1
    Preprint SPP1962-066
     
  24. Anne-Therese Rauls and Gerd Wachsmuth
    Generalized derivatives for the solution operator of the obstacle problem
    Set-Valued and Variational Analysis (SVAA), 28(2), p.259-285, 2020
    DOI: 10.1007/s11228-019-0506-y
     
  25. Constantin Christof and Gerd Wachsmuth
    Differential Sensitivity Analysis of Variational Inequalities with Locally Lipschitz Continuous Solution Operators
    Applied Mathematics & Optimization (AMO), 81(1), p.23-62, 2020
    DOI: 10.1007/s00245-018-09553-y
     
  26. Marc Herrmann, Roland Herzog, Stephan Schmidt, José Vidal-Núñez and Gerd Wachsmuth
    Discrete Total Variation with Finite Elements and Applications to Imaging
    Journal of Mathematical Imaging and Vision (JMIV), 61(4), p.411-431, 2019
    DOI: 10.1007/s10851-018-0852-7
     
  27. Patrick Mehlitz
    On the sequential normal compactness condition and its restrictiveness in selected function spaces
    Set-Valued and Variational Analysis (SVAA), 27, p.763-782, 2019
    DOI: 10.1007/s11228-018-0475-6
     
  28. Felix Harder
    Legendre Forms in Reflexive Banach Spaces
    Zeitschrift für Analysis und ihre Anwendungen (ZAA), 37(4), p.377-388, 2018
    DOI: 10.4171/zaa/1619
    Preprint SPP1962-030
     
  29. Felix Harder and Gerd Wachsmuth
    Optimality conditions for a class of inverse optimal control problems with partial differential equations
    Optimization, 68(2-3), p.615-643, 2018
    DOI: 10.1080/02331934.2018.1495205
    Preprint SPP1962-048
     
  30. Felix Harder and Gerd Wachsmuth
    The limiting normal cone of a complementarity set in Sobolev spaces
    Optimization, 67(5), p.1579-1603, 2018
    DOI: 10.1080/02331934.2018.1484467
    Preprint SPP1962-023
     
  31. Constantin Christof and Gerd Wachsmuth
    No-Gap Second-Order Conditions via a Directional Curvature Functional
    SIAM Journal on Optimization (SIOPT), 28(3), p.2097-2130, 2018
    DOI: 10.1137/17M1140418
    arXiv: 1707.07579
    Preprint SPP1962-026
     
  32. Constantin Christof and Gerd Wachsmuth
    On the Non-Polyhedricity of Sets with Upper and Lower Bounds in Dual Spaces
    GAMM Reports, 40(4), p.339-350, 2018
    DOI: 10.1002/gamm.201740005
    arXiv: 1711.02588
    Preprint SPP1962-031
     
  33. Felix Harder and Gerd Wachsmuth
    Comparison of Optimality Systems for the Optimal Control of the Obstacle Problem
    GAMM Reports, 40(4), p.312-338, 2018
    DOI: 10.1002/gamm.201740004
    Preprint SPP1962-029
     
  34. Patrick Mehlitz and Gerd Wachsmuth
    The weak sequential closure of decomposable sets in Lebesgue spaces and its application to variational geometry
    Set-Valued and Variational Analysis (SVAA), 27(1), p.265-294, 2019
    DOI: 10.1007/s11228-017-0464-1
    Preprint SPP1962-016
     
  35. Ulisse Stefanelli, Daniel Wachsmuth and Gerd Wachsmuth
    Optimal control of a rate-independent evolution equation via viscous regularization
    Discrete and Continuous Dynamical Systems - Series S (DCDS-S), 10(6), p.1467-1485, 2017
    DOI: 10.3934/dcdss.2017076
    Preprint SPP1962-001
     
  36. Patrick Mehlitz and Gerd Wachsmuth
    The limiting normal cone to pointwise defined sets in Lebesgue spaces
    Set-Valued and Variational Analysis (SVAA), 26(3), p.449-467, 2018
    DOI: 10.1007/s11228-016-0393-4
    Preprint SPP1962-004

Book Chapters

  1. Stephan Dempe, Felix Harder, Patrick Mehlitz and Gerd Wachsmuth
    Analysis and Solution Methods for Bilevel Optimal Control Problems
    in: Non-Smooth and Complementarity-Based Distributed Parameter Systems, p.77-99, Birkhäuser Basel, 2021
    DOI: 10.1007/978-3-030-79393-7_4

Preprints