Bilevel Optimal Control: Theory, Algorithms, and Applications
Description
The main goal of this project is to deepen the knowledge of bilevel optimal control problems (BOCPs) and infinitedimensional mathematical programs with complementarity constraints (MPCCs). In particular, we aim for theoretical results such as the development of new constraint qualifications in infinite dimensions and the sharpening of available optimality conditions. Moreover, we investigate algorithmic approaches which exploit the specific problem structures and study prototypical applications.
Our project consists of three work areas.
Theory of BOCPs (A): In this work area, we want to advance available optimization theory in infinite dimensions. In particular, we address optimality conditions of first and second order for MPCCs and aim for the construction of new constraint qualifications. Moreover, a discretization of infinitedimensional problems is unavoidable for numerical computations so we will also focus on results in numerical analysis.
Solution algorithms (B): On the one hand, we consider BOCPs where the lower level parametric optimization problem is an optimal control problem with a PDE constraint. Under some assumptions it is possible to exploit the lower level optimal value function in order to construct solution algorithms. In case where the overall problem data is fully convex while the upper level decision variable comes from a finitedimensional space, a global solution method is already available which exploits a piecewise affine upper approximate of the value function. Now, we want to tackle slightly different settings using related ideas. On the other hand, we aim for the development of an active set method for the numerical solution of MPCCs in the finitedimensional setting. Afterwards, a generalization to the function space setting will be discussed.
Prototypical applications (C): In this work area, some popular examples for BOCPs will be investigated. In particular, applications like the optimal measuring via a bilevel approach or parameter identification in optimal control problems are under consideration. We will apply the findings of work packages (A) and (B) in order to derive new optimality conditions and efficient solution algorithms.
Publications
Xiaoxi Jia, Christian Kanzow, Patrick Mehlitz, Gerd Wachsmuth: An Augmented Lagrangian Method for Optimization Problems with Structured Geometric Constraints, Mathematical Programming volume 199, pages 1365–1415, 2023 (SPP1962170).
Alberto De Marchi, Xiaoxi Jia, Christian Kanzow, Patrick Mehlitz: Constrained Structured Optimization and Augmented Lagrangian Proximal Methods, Mathematical Programming, 2023 (SPP1962191).
Constantin Christof, Gerd Wachsmuth: Lipschitz Stability and Hadamard Directional Differentiability for Elliptic and Parabolic Obstacletype Quasivariational Inequalities, SIAM Journal on Control and Optimization, Vol. 60, Iss. 6, 2022 (SPP1962169).
Patrick Mehlitz, Gerd Wachsmuth: Subdifferentiation of Nonconvex SparsityPromoting Functionals on Lebesgue Spaces, eSIAM Journal on Control and Optimization Vol. 60, Iss. 3, 2022 (SPP1962173).
Gerd Wachsmuth: Maximal Monotone Operators with NonMaximal Graphical Limit, Examples and Counterexamples Volume 2, 2022 (SPP1962174).
Felix Harder, Gerd Wachsmuth: Mstationarity for a class of MPCCs in Lebesgue spaces, Journal of Mathematical Analysis and Applications Volume 512, Issue 2, 2022 (SPP1962180).
Daniel Wachsmuth, Gerd Wachsmuth: SecondOrder Conditions for NonUniformly Convex Integrands: Quadratic Growth in $L^1$, Journal of Nonsmooth Analysis and Optimization, May 23, 2022, Volume 3, 2022 (SPP1962184).
Felix Harder, Patrick Mehlitz, Gerd Wachsmuth: Reformulation of the Mstationarity Conditions as a System of Discontinuous Equations and its Solution by a Semismooth Newton Method, SIAM J. Optim., 31(2), 1459–1488., 2021 (SPP1962135).
Yu Deng, Patrick Mehlitz, Uwe Prüfert: Coupled versus Decoupled Penalization of Control Complementarity Constraints, ESAIM: COCV Volume 27, 2021 (SPP1962125).
Patrick Mehlitz, Gerd Wachsmuth: Bilevel Optimal Control: Existence Results and Stationarity Conditions , In: Dempe S., Zemkoho A. (eds) Bilevel Optimization. Springer Optimization and Its Applications, vol 161. Springer, Cham. , 2020.
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 (SPP1962128).
Preprints
Christian Kanzow, Fabius Krämer, Patrick Mehlitz, Gerd Wachsmuth, Frank Werner: A Nonsmooth Augmented Lagrangian Method and its Application to Poisson Denoising and Sparse Control (SPP1962202, 04/2023, [bib])
Alberto De Marchi, Xiaoxi Jia, Christian Kanzow, Patrick Mehlitz: Constrained Structured Optimization and Augmented Lagrangian Proximal Methods (SPP1962191, 04/2022, [bib])
Daniel Wachsmuth, Gerd Wachsmuth: A Simple Proof of the BaillonHaddad Theorem on Open Subsets of Hilbert Spaces (SPP1962189, 04/2022, [bib])
Markus Friedemann, Felix Harder, Gerd Wachsmuth: Finding Global Solutions of Some Inverse Optimal Control Problems using Penalization and Semismooth Newton Methods (SPP1962188, 03/2022, [bib])
Constantin Christof, Gerd Wachsmuth: Semismoothness for Solution Operators of ObstacleType Variational Inequalities with Applications in Optimal Control (SPP1962186, 01/2022, [bib])
Daniel Wachsmuth, Gerd Wachsmuth: SecondOrder Conditions for NonUniformly Convex Integrands: Quadratic Growth in $L^1$ (SPP1962184, 11/2021, [bib])
Felix Harder, Gerd Wachsmuth: Mstationarity for a class of MPCCs in Lebesgue spaces (SPP1962180, 10/2021, [bib])
Alexander Y. Kruger, Patrick Mehlitz: Optimality conditions, approximate stationarity, and applications  a story beyond Lipschitzness (SPP1962179, 10/2021, [bib])
Gerd Wachsmuth: From Resolvents to Generalized Equations and Quasivariational Inequalities: Existence and Differentiability (SPP1962177, 09/2021, [bib])
Felix Harder: New Stationarity Conditions between Strong and MStationarity for Mathematical Programs with Complementarity Constraints (SPP1962175, 09/2021, [bib])
Gerd Wachsmuth: Maximal Monotone Operators with NonMaximal Graphical Limit (SPP1962174, 07/2021, [bib])
Patrick Mehlitz, Gerd Wachsmuth: Subdifferentiation of Nonconvex SparsityPromoting Functionals on Lebesgue Spaces (SPP1962173, 07/2021, [bib])
Xiaoxi Jia, Christian Kanzow, Patrick Mehlitz, Gerd Wachsmuth: An Augmented Lagrangian Method for Optimization Problems with Structured Geometric Constraints (SPP1962170, 05/2021, [bib])
Constantin Christof, Gerd Wachsmuth: Lipschitz Stability and Hadamard Directional Differentiability for Elliptic and Parabolic Obstacletype Quasivariational Inequalities (SPP1962169, 05/2021, [bib])
Ira Neitzel, Gerd Wachsmuth: Firstorder Conditions for the Optimal Control of the Obstacle Problem with State Constraints (SPP1962154, 12/2020, [bib])
Felix Harder: A New Elementary Proof for Mstationarity under MPCCGCQ for Mathematical Programs with Complementarity Constraints (SPP1962151, 11/2020, [bib])
Felix Harder, Patrick Mehlitz, Gerd Wachsmuth: Reformulation of the Mstationarity Conditions as a System of Discontinuous Equations and its Solution by a Semismooth Newton Method (SPP1962135, 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 (SPP1962128, 12/2019, [bib])
Yu Deng, Patrick Mehlitz, Uwe Prüfert: Coupled versus Decoupled Penalization of Control Complementarity Constraints (SPP1962125, 10/2019, [bib])
Research Area
Modeling, problem analysis, algorithm design and convergence analysis
The focus of this area is on the development and analysis of genuinely nonsmooth models in the sciences in order to properly capture realworld effects and to avoid comprising smoothing approaches. In simulation and optimization this requires to advance setvalued analysis and the design of robust algorithms for nonsmooth problems.Realization of algorithms, adaptive discretization and model reduction
As the target applications of this SPP involve nonsmooth structures and partial differential operators, the discretization of the associated problems and robust error estimation are important issues to be address, and proper modelreduction techniques need to be developed.Members
Project Related News

Apr 07, 2022 : New preprint submitted
Christian Kanzow submitted the preprint SPP1962191 Constrained Structured Optimization and Augmented Lagrangian Proximal Methods.

Apr 01, 2022 : New preprint submitted
Gerd Wachsmuth submitted the preprint SPP1962189 A Simple Proof of the BaillonHaddad Theorem on Open Subsets of Hilbert Spaces.

Mar 02, 2022 : New preprint submitted
Gerd Wachsmuth submitted the preprint SPP1962188 Finding Global Solutions of Some Inverse Optimal Control Problems using Penalization and Semismooth Newton Methods.

Jan 10, 2022 : New preprint submitted
Gerd Wachsmuth submitted the preprint SPP1962186 Semismoothness for Solution Operators of ObstacleType Variational Inequalities with Applications in Optimal Control.

Nov 22, 2021 : New preprint submitted
Gerd Wachsmuth submitted the preprint SPP1962184 SecondOrder Conditions for NonUniformly Convex Integrands: Quadratic Growth in $L^1$.

Nov 09, 2021 : New preprint submitted
Hedwig Keller submitted the preprint SPP1962181 Numerical Approximation of Optimal Convex and Rotationally Symmetric Shapes for an Eigenvalue Problem arising in Optimal Insulation.

Oct 29, 2021 : New preprint submitted
Felix Harder submitted the preprint SPP1962180 Mstationarity for a class of MPCCs in Lebesgue spaces.

Oct 14, 2021 : New preprint submitted
Patrick Mehlitz submitted the preprint SPP1962179 Optimality conditions, approximate stationarity, and applications  a story beyond Lipschitzness.

Sep 30, 2021 : New preprint submitted
Gerd Wachsmuth submitted the preprint SPP1962177 From Resolvents to Generalized Equations and Quasivariational Inequalities: Existence and Differentiability.

Sep 06, 2021 : New preprint submitted
Felix Harder submitted the preprint SPP1962175 New Stationarity Conditions between Strong and MStationarity for Mathematical Programs with Complementarity Constraints.