Analysis and Solution Methods for Bilevel Optimal Control Problems

Description

We are going to consider bilevel optimal control problems governed by partial differential equations. Using different approaches (lower level optimality conditions, lower level optimal value function, differentiability properties of the lower level solution w.r.t. the upper level solution variables) we are going to construct necessary and sufficient optimality conditions for such problems. Therefore, we start considering a general bilevel optimization problem in Banach spaces before the obtained results are applied to bilevel optimal control problems. Furthermore, we want to analyze the numerical behaviour of such problems. Here it has to be considered whether and how theoretical approaches are realizable in practice. Especially, the sensitivity and stability analysis of finite-dimensional bilevel programming problems plays a crucial role when choosing appropriate reformulations and discretization strategies. Error bounds have to be obtained which can be used for a convergence analysis in the function space setting. Thus, both the theoretical as well as the numerical approach lead to problems of parametric and nonsmooth optimal control. We want to set up a collection of benchmark problems which can be used in order to test and compare the derived theoretical results and numerical methods.

Publications

Marc Herrmann, Roland Herzog, Stephan Schmidt, Josè Vidal, Gerd Wachsmuth: Discrete Total Variation with Finite Elements and Applications to Imaging, J Math Imaging Vis, 2018 (SPP1962-054).

Patrick Mehlitz: On the Sequential Normal Compactness Condition and its Restrictiveness in Selected Function Spaces, Set-Valued and Variational Analysis , 2018 (SPP1962-025).

Felix Harder, Gerd Wachsmuth: Comparison of Optimality Systems for the Optimal Control of the Obstacle Problem , GAMM‐Mitteilungen, 40: 312-338., 2018 (SPP1962-029).

Constantin Christof, Gerd Wachsmuth: On the Non-Polyhedricity of Sets with Upper and Lower Bounds in Dual Spaces, GAMM‐Mitteilungen, 40: 339-350., 2018 (SPP1962-031).

Constantin Christof, Gerd Wachsmuth: No-Gap Second-Order Conditions via a Directional Curvature Functional, SIAM J. Optim., 28(3), 2097–2130. , 2018 (SPP1962-026).

Patrick Mehlitz, Gerd Wachsmuth: The Weak Sequential Closure of Decomposable Sets in Lebesgue Spaces and its Application to Variational Geometry , Set-Valued and Variational Analysis , 2017 (SPP1962-016).

Ulisse Stefanelli, Daniel Wachsmuth, Gerd Wachsmuth: Optimal control of a rate-independent evolution equation via viscous regularization, Discrete and Continuous Dynamical Systems - Series S 10(6), 1467-1485 , 2017 (SPP1962-001).

Patrick Mehlitz, Gerd Wachsmuth: The Limiting Normal Cone to Pointwise Defined Sets in Lebesgue Spaces, Set-Valued and Variational Analysis, 2016 (SPP1962-004).

Preprints

Tobias Geiger, Daniel Wachsmuth, Gerd Wachsmuth: Optimal Control of ODEs with State Suprema (SPP1962-094, 10/2018, [bib])

Sören Bartel, Gerd Wachsmuth: Numerical Approximation of Optimal Convex Shapes (SPP1962-089, 10/2018, [bib])

Tommy Etling, Roland Herzog, Estefanía Loayza, Gerd Wachsmuth: First and Second Order Shape Optimization Based on Restricted Mesh Deformations (SPP1962-088, 10/2018, [bib])

Christian Clason, Yu Deng, Patrick Mehlitz, Uwe Prüfert: Optimal Control Problems with Control Complementarity Constraints (SPP1962-081, 10/2018, [bib])

Stephan Dempe, Patrick Mehlitz: Semivectorial Bilevel Programming versus Scalar Bilevel Programming (SPP1962-082, 10/2018, [bib])

Stephan Dempe, Felix Harder, Patrick Mehlitz, Gerd Wachsmuth: Solving Inverse Optimal Control Problems via Value Functions to Global Optimality (SPP1962-066, 07/2018, [bib])

Anne-Therese Rauls, Gerd Wachsmuth: Generalized Derivatives for the Solution Operator of the Obstacle Problem (SPP1962-057, 06/2018, [bib])

Marc Herrmann, Roland Herzog, Stephan Schmidt, Josè Vidal, Gerd Wachsmuth: Discrete Total Variation with Finite Elements and Applications to Imaging (SPP1962-054, 04/2018, [bib])

Yu Deng, Patrick Mehlitz, Uwe Prüfert: Optimal Control in First-order Sobolev Spaces with Inequality Constraints (SPP1962-053, 04/2018, [bib])

Yu Deng, Patrick Mehlitz, Uwe Prüfert: On an Optimal Control Problem with Gradient Constraints (SPP1962-050, 02/2018, [bib])

Felix Harder, Gerd Wachsmuth: Optimality Conditions for a Class of Inverse Optimal Control Problems With Partial Differential Equations (SPP1962-048, 01/2018, [bib])

Constantin Christof, Gerd Wachsmuth: Differential Sensitivity Analysis of Variational Inequalities with Locally Lipschitz Continuous Solution Operators (SPP1962-039, 11/2017, [bib])

Constantin Christof, Gerd Wachsmuth: On the Non-Polyhedricity of Sets with Upper and Lower Bounds in Dual Spaces (SPP1962-031, 08/2017, [bib])

Felix Harder: Legendre Forms in Reflexive Banach Spaces (SPP1962-030, 08/2017, [bib])

Felix Harder, Gerd Wachsmuth: Comparison of Optimality Systems for the Optimal Control of the Obstacle Problem (SPP1962-029, 08/2017, [bib])

Constantin Christof, Gerd Wachsmuth: No-Gap Second-Order Conditions via a Directional Curvature Functional (SPP1962-026, 07/2017, [bib])

Patrick Mehlitz: On the Sequential Normal Compactness Condition and its Restrictiveness in Selected Function Spaces (SPP1962-025, 06/2017, [bib])

Felix Harder, Gerd Wachsmuth: The Limiting Normal Cone of a Complementarity Set in Sobolev Spaces (SPP1962-023, 06/2017, [bib])

Patrick Mehlitz, Gerd Wachsmuth: The Weak Sequential Closure of Decomposable Sets in Lebesgue Spaces and its Application to Variational Geometry (SPP1962-016, 04/2017, [bib])

Patrick Mehlitz, Gerd Wachsmuth: The Limiting Normal Cone to Pointwise Defined Sets in Lebesgue Spaces (SPP1962-004, 12/2016, [bib])

Ulisse Stefanelli, Daniel Wachsmuth, Gerd Wachsmuth: Optimal control of a rate-independent evolution equation via viscous regularization (SPP1962-001, 11/2016, [bib])

Research Area

Modeling, problem analysis, algorithm design and convergence analysis

The focus of this area is on the development and analysis of genuinely non-smooth models in the sciences in order to properly capture real-world effects and to avoid comprising smoothing approaches. In simulation and optimization this requires to advance set-valued analysis and the design of robust algorithms for non-smooth problems.

Realization of algorithms, adaptive discretization and model reduction

As the target applications of this SPP involve non-smooth structures and partial differential operators, the discretization of the associated problems and robust error estimation are important issues to be address, and proper model-reduction techniques need to be developed.

Members

  • member's portrait

    Prof. Stephan Dempe

    Technische Universität Bergakademie Freiberg
    Principal Investigator
  • member's portrait

    Prof. Gerd Wachsmuth

    Brandenburgische Technische Universität Cottbus-Senftenberg
    Principal Investigator
  • member's portrait

    Felix Harder

    Brandenburgische Technische Universität Cottbus-Senftenberg
    Research Assistant
  • member's portrait

    Prof. Anton Schiela

    Universität Bayreuth
    Cooperation Partner
  • member's portrait

    Prof. Michael Ulbrich

    Technische Universität München
    Cooperation Partner

Project Related News

  • Oct 26, 2018 : New preprint submitted

    Gerd Wachsmuth submitted the preprint SPP1962-094, Optimal Control of ODEs with State Suprema

  • Oct 09, 2018 : New preprint submitted

    Patrick Mehlitz submitted the preprint SPP1962-082, Semivectorial Bilevel Programming versus Scalar Bilevel Programming

  • Jun 13, 2018 : New preprint submitted

    Gerd Wachsmuth submitted the Preprint SPP1962-057, Generalized Derivatives for the Solution Operator of the Obstacle Problem

  • Feb 14, 2018 : New preprint submitted

    Patrick Mehlitz submitted the Preprint SPP1962-050, On an Optimal Control Problem with Gradient Constraints

  • Jan 27, 2018 : New preprint submitted

    Felix Harder submitted the Preprint SPP1962-048, Optimality Conditions for a Class of Inverse Optimal Control Problems with Partial Differential Equations

  • Jul 06, 2017 : New preprint submitted

    Patrick Mehlitz submitted the Preprint SPP1962-025, On the Sequential Normal Compactness Condition and its Restrictiveness in Selected Function Spaces

  • Oct 10, 2016 : Welcome to our new project member

    Felix Harder joins project 6.