Optimization Problems in Banach Spaces with Non-Smooth Structure

Description

The aim of this project is to develop and analyse algorithms for the numerical solution of some highly difficult optimization problems in Banach spaces. This includes mathematical programs with complementarity constraints, switching constraints, or problems involving a sparsity term either in the objective function or the constraints. By exploiting the special structure of these problems, the goal is to derive solution methods with strong global and local convergence properties under realistic, problem-tailored assumptions. All methods will be implemented and tested extensively on several relevant examples.

Publications

Eike Börgens, Christian Kanzow, Patrick Mehlitz, Gerd Wachsmuth: New Constraint Qualifications for Optimization Problems in Banach Spaces Based on Cone Continuity Properties, SIAM J. Optim., 30(4), 2956–2982 , 2020 (SPP1962-128).

Preprints

Carolin Natemeyer, Daniel Wachsmuth: A Penalty Scheme to Solve Constrained Non-convex Optimization Problems in $BV(Omega)$ (SPP1962-178, 10/2021, [bib])

Xiaoxi Jia, Christian Kanzow, Patrick Mehlitz, Gerd Wachsmuth: An Augmented Lagrangian Method for Optimization Problems with Structured Geometric Constraints (SPP1962-170, 05/2021, [bib])

Carolin Natemeyer, Daniel Wachsmuth: A Proximal Gradient Method for Control Problems with Nonsmooth and Nonconvex Control Cost (SPP1962-142, 07/2020, [bib])

Nguyen Thanh Qui, Daniel Wachsmuth: Full Stability for Variational Nash Equilibriums of Parametric Optimal Control Problems of PDEs (SPP1962-134, 02/2020, [bib])

Eduardo Casas, Daniel Wachsmuth: Analysis of Optimal Control Problems with an $L^0$ Term in the Cost Functional (SPP1962-133, 02/2020, [bib])

Eike Börgens, Christian Kanzow, Patrick Mehlitz, Gerd Wachsmuth: New Constraint Qualifications for Optimization Problems in Banach Spaces Based on Cone Continuity Properties (SPP1962-128, 12/2019, [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. Christian Kanzow

    Julius-Maximilians-Universität Würzburg
    Principal Investigator
  • member's portrait

    Prof. Daniel Wachsmuth

    Julius-Maximilians-Universität Würzburg
    Principal Investigator
  • member's portrait

    Xiaoxi Jia

    Julius-Maximilians-Universität Würzburg
    Research Assistant
  • member's portrait

    Carolin Natemeyer

    Julius-Maximilians-Universität Würzburg
    Research Assistant

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