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 infinite-dimensional 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 infinite-dimensional 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 finite-dimensional 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 finite-dimensional 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 (SPP1962-170).

Alberto De Marchi, Xiaoxi Jia, Christian Kanzow, Patrick Mehlitz: Constrained Structured Optimization and Augmented Lagrangian Proximal Methods, Mathematical Programming, 2023 (SPP1962-191).

Patrick Mehlitz, Gerd Wachsmuth: Subdifferentiation of Nonconvex Sparsity-Promoting Functionals on Lebesgue Spaces, eSIAM Journal on Control and Optimization Vol. 60, Iss. 3, 2022 (SPP1962-173).

Gerd Wachsmuth: Maximal Monotone Operators with Non-Maximal Graphical Limit, Examples and Counterexamples Volume 2, 2022 (SPP1962-174).

Felix Harder, Gerd Wachsmuth: M-stationarity for a class of MPCCs in Lebesgue spaces, Journal of Mathematical Analysis and Applications Volume 512, Issue 2, 2022 (SPP1962-180).

Daniel Wachsmuth, Gerd Wachsmuth: Second-Order Conditions for Non-Uniformly Convex Integrands: Quadratic Growth in $L^1$, Journal of Nonsmooth Analysis and Optimization, May 23, 2022, Volume 3, 2022 (SPP1962-184).

Constantin Christof, Gerd Wachsmuth: Lipschitz Stability and Hadamard Directional Differentiability for Elliptic and Parabolic Obstacle-type Quasi-variational Inequalities, SIAM Journal on Control and Optimization, Vol. 60, Iss. 6, 2022 (SPP1962-169).

Yu Deng, Patrick Mehlitz, Uwe Prüfert: Coupled versus Decoupled Penalization of Control Complementarity Constraints, ESAIM: COCV Volume 27, 2021 (SPP1962-125).

Felix Harder, Patrick Mehlitz, Gerd Wachsmuth: Reformulation of the M-stationarity Conditions as a System of Discontinuous Equations and its Solution by a Semismooth Newton Method, SIAM J. Optim., 31(2), 1459–1488., 2021 (SPP1962-135).

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 (SPP1962-128).

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 (SPP1962-202, 04/2023, [bib])

Alberto De Marchi, Xiaoxi Jia, Christian Kanzow, Patrick Mehlitz: Constrained Structured Optimization and Augmented Lagrangian Proximal Methods (SPP1962-191, 04/2022, [bib])

Daniel Wachsmuth, Gerd Wachsmuth: A Simple Proof of the Baillon--Haddad Theorem on Open Subsets of Hilbert Spaces (SPP1962-189, 04/2022, [bib])

Markus Friedemann, Felix Harder, Gerd Wachsmuth: Finding Global Solutions of Some Inverse Optimal Control Problems using Penalization and Semismooth Newton Methods (SPP1962-188, 03/2022, [bib])

Constantin Christof, Gerd Wachsmuth: Semismoothness for Solution Operators of Obstacle-Type Variational Inequalities with Applications in Optimal Control (SPP1962-186, 01/2022, [bib])

Daniel Wachsmuth, Gerd Wachsmuth: Second-Order Conditions for Non-Uniformly Convex Integrands: Quadratic Growth in $L^1$ (SPP1962-184, 11/2021, [bib])

Felix Harder, Gerd Wachsmuth: M-stationarity for a class of MPCCs in Lebesgue spaces (SPP1962-180, 10/2021, [bib])

Alexander Y. Kruger, Patrick Mehlitz: Optimality conditions, approximate stationarity, and applications --- a story beyond Lipschitzness (SPP1962-179, 10/2021, [bib])

Gerd Wachsmuth: From Resolvents to Generalized Equations and Quasi-variational Inequalities: Existence and Differentiability (SPP1962-177, 09/2021, [bib])

Felix Harder: New Stationarity Conditions between Strong and M-Stationarity for Mathematical Programs with Complementarity Constraints (SPP1962-175, 09/2021, [bib])

Gerd Wachsmuth: Maximal Monotone Operators with Non-Maximal Graphical Limit (SPP1962-174, 07/2021, [bib])

Patrick Mehlitz, Gerd Wachsmuth: Subdifferentiation of Nonconvex Sparsity-Promoting Functionals on Lebesgue Spaces (SPP1962-173, 07/2021, [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])

Constantin Christof, Gerd Wachsmuth: Lipschitz Stability and Hadamard Directional Differentiability for Elliptic and Parabolic Obstacle-type Quasi-variational Inequalities (SPP1962-169, 05/2021, [bib])

Ira Neitzel, Gerd Wachsmuth: First-order Conditions for the Optimal Control of the Obstacle Problem with State Constraints (SPP1962-154, 12/2020, [bib])

Felix Harder: A New Elementary Proof for M-stationarity under MPCC-GCQ for Mathematical Programs with Complementarity Constraints (SPP1962-151, 11/2020, [bib])

Felix Harder, Patrick Mehlitz, Gerd Wachsmuth: Reformulation of the M-stationarity Conditions as a System of Discontinuous Equations and its Solution by a Semismooth Newton Method (SPP1962-135, 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])

Yu Deng, Patrick Mehlitz, Uwe Prüfert: Coupled versus Decoupled Penalization of Control Complementarity Constraints (SPP1962-125, 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 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

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