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

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

Preprints

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

    Dr. Gerd Wachsmuth

    Technische Universität Chemnitz
    Principal Investigator
  • member's portrait

    Felix Harder

    Technische Universität Chemnitz
    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

  • 10. 10. 2016 : Welcome to our new project member

    Felix Harder joins project 6.

  • 06. 07. 2017 : New preprint submitted

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