Algorithms for Quasi-Variational Inequalities in Infinite-Dimensional Spaces

Description

The aim of this project is to develop and analyse algorithms for the numerical solution of some classes of quasi-variational inequalities. Such inequalities occur, e.g., in connection with generalized Nash equilibria in multi-player control problems. Moreover, they are widely used to describe the value function in stochastic control problems. Our goal is twofold: a) Transfer existing solution methods from finite-dimensional to infinite-dimensional problems. b) Develop problem-tailored solution methods by taking into account the particular structure of certain quasi-variational inequalities. All methods should have a strong theoretical background and will be tested extensively on suitable examples.

Publications

Christian Kanzow, Alexandra Schwarz: Spieltheorie. Theorie und Verfahren zur Lösung von Nash- und verallgemeinerten Nash-Gleichgewichtsproblemen, Birkhäuser-Verlag, 2018.

Christian Kanzow, Daniel Steck: Augmented Lagrangian and Exact Penalty Methods for Quasi-Variational Inequalities, Computational Optimization and Applications 69(3):801-824 , 2018.

Daniel Steck: Brezis Pseudomonotonicity is Strictly Weaker than Ky-Fan Hemicontinuity, Journal of Optimization Theory and Applications, 2018.

Christian Kanzow, Daniel Steck, Daniel Wachsmuth: An Augmented Lagrangian Method for Optimization Problems in Banach Spaces, SIAM J. Control Optim. 56(1):272-291, 2018 (SPP1962-003).

Veronika Karl, Daniel Wachsmuth: An Augmented Lagrange Method for Elliptic State Constrained Optimal Control Problems, Comp. Opt. Appl. 69(3), 857-880, 2018 (SPP1962-008).

Christian Kanzow, Daniel Steck: Improved Local Convergence Results for Augmented Lagrangian Methods in $C^2$-Cone Reducible Constrained Optimization, Math. Programming, 2018 (SPP1962-040).

Christian Kanzow, Daniel Steck: On Error Bounds and Multiplier Methods for Variational Problems in Banach Spaces, SIAM Journal on Control and Optimization, 2018 (SPP1962-034).

Christian Kanzow, Daniel Steck: A Generalized Proximal-Point Method for Convex Optimization Problems in Hilbert Spaces , A Journal of Mathematical Programming and Operations Research , 2017 (SPP1962-002).

Christian Kanzow, Daniel Steck: An Example Comparing the Standard and Safeguarded Augmented Lagrangian Methods, Operations Research Letters , 2017 (SPP1962-013).

Christian Kanzow, Daniel Steck: Augmented Lagrangian Methods for the Solution of Generalized Nash Equilibrium Problems, SIAM Journal on Optimization 26 (4), 2016.

Preprints

Christian Kanzow, Daniel Steck: Quasi-Variational Inequalities in Banach Spaces: Theory and Augmented Lagrangian Methods (SPP1962-100, 12/2018, [bib])

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

Veronika Karl, Ira Neitzel, Daniel Wachsmuth: A Lagrange Multiplier Method for Semilinear Elliptic State Constrained Optimal Control Problems (SPP1962-087, 10/2018, [bib])

Christian Kanzow, Daniel Steck: Improved Local Convergence Results for Augmented Lagrangian Methods in $C^2$-Cone Reducible Constrained Optimization (SPP1962-040, 11/2017, [bib])

Christian Kanzow, Daniel Steck: On Error Bounds and Multiplier Methods for Variational Problems in Banach Spaces (SPP1962-034, 09/2017, [bib])

Christian Kanzow, Daniel Steck, Veronika Karl, Daniel Wachsmuth: The Multiplier-Penalty Method for Generalized Nash Equilibrium Problems in Banach Spaces (SPP1962-028, 07/2017, [bib])

Veronika Karl, Frank Pörner: An Augmented Lagrange Method for Ill-Posed Elliptic State Constrained Optimal Control Problems with Sparse Controls (SPP1962-027, 07/2017, [bib])

Christian Kanzow, Daniel Steck: An Example Comparing the Standard and Modified Augmented Lagrangian Methods (SPP1962-013, 03/2017, [bib])

Veronika Karl, Daniel Wachsmuth: An Augmented Lagrange Method for Elliptic State Constrained Optimal Control Problems (SPP1962-008, 02/2017, [bib])

Christian Kanzow, Daniel Steck: A Generalized Proximal-Point Method for Convex Optimization Problems in Hilbert Spaces (SPP1962-002, 12/2016, [bib])

Christian Kanzow, Daniel Steck, Daniel Wachsmuth: An Augmented Lagrangian Method for Optimization Problems in Banach Spaces (SPP1962-003, 12/2016, [bib])

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

    Veronika Karl

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

    Daniel Steck

    Julius-Maximilians-Universität Würzburg
    Research Assistant

Project Related News

  • Dec 04, 2018 : Book publication

    Christian Kanzow and Alexandra Schwarz published the book "Spieltheorie. Theorie und Verfahren zur Lösung von Nash- und verallgemeinerten Nash-Gleichgewichtsproblemen", Birkhäuser Verlag, 2018 (in German).

  • Dec 04, 2018 : New publication

    Preprint SPP1962-034 has been published.

  • Dec 03, 2018 : New preprint submitted

    Christian Kanzow submitted the preprint SPP1962-100, Quasi-Variational Inequalities in Banach Spaces: Theory and Augmented Lagrangian Methods

  • Oct 24, 2018 : New preprint submitted

    Veronika Karl submitted the preprint SPP1962-087, A Lagrange Multiplier Method for Semilinear Elliptic State Constrained Optimal Control Problems

  • Feb 17, 2017 : Welcome to our new project member

    Veronika Karl joins project 14.