Optimization methods for mathematical programs with equilibrium constraints in function spaces based on adaptive error control and reduced order or low rank tensor approximations

Description

This project investigates optimization methods for mathematical programs with equilibrium constraints (MPECs) in function space that adaptively control the accuracy of the underlying discretization and of inexact subproblem solves in such a way that convergence is ensured. This enables the use of adaptive discretizations, reduced order models, and low rank tensor methods, thus making the solution of MPECs with high dimensional equilibrium constraints tractable and efficient. Two prototype classes of MPECs in function space are considered in the project: One with a family of parametric variational inequalities as constraints and the other constrained by a parabolic variational inequality. Based on a rigorous analytical foundation in function space, the project will develop and analyze inexact bundle methods combined with an implicit programming approach. In addition, inexact all-at-once methods will be considered. In both cases, the evaluation of cost function, constraints, and derivatives is carried out on discretizations which are adaptively refined during optimization and can further be approximated by reduced order models or low rank tensor methods. We will develop implementable control mechanisms for the inexactness, which are tailored to the needs of the optimization methods and can be based on a posteriori error estimators. The algorithms will be implemented and tested for the considered prototype classes of MPECs.

Publications

No publications from this project yet.

Preprints

Lukas Hertlein, Michael Ulbrich: An Inexact Bundle Algorithm for Nonconvex Nondifferentiable Functions in Hilbert Space (SPP1962-084, 10/2018, [bib])

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

Anne-Therese Rauls, Stefan Ulbrich: Subgradient Computation for the Solution Operator of the Obstacle Problem (SPP1962-056, 05/2018, [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.

Incorporation of parameter dependencies and robustness

In many applications the robustness of solutions with respect to a given parameter range (uncertainty set) is highly relevant. Correspondingly, in this research area of the SPP, bi- or multilevel optimization approaches will be studied in order to robustify problem solutions against uncertain parameters.

Members

  • member's portrait

    Prof. Michael Ulbrich

    Technische Universität München
    Principal Investigator
  • member's portrait

    Prof. Stefan Ulbrich

    Technische Universität Darmstadt
    Principal Investigator
  • member's portrait

    Lukas Hertlein

    Technische Universität München
    Research Assistant
  • member's portrait

    Anne-Therese Rauls

    Technische Universität Darmstadt
    Research Assistant

Project Related News

  • Oct 12, 2018 : New preprint submitted

    Lukas Hertlein submitted the preprint SPP1962-084, An Inexact Bundle Algorithm for Nonconvex Nondifferentiable Functions in Hilbert Space

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

    Lukas Hertlein joins project 23.

  • Aug 18, 2016 : Welcome to our new project member

    Anne-Therese Rauls joins Project 23