Bilevel Optimal Control: Theory, Algorithms, and Applications
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.
No publications from this project yet.
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])
Modeling, problem analysis, algorithm design and convergence analysisThe 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 reductionAs 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.
Prof. Stephan DempeTechnische Universität Bergakademie Freiberg
Prof. Gerd WachsmuthBrandenburgische Technische Universität Cottbus-Senftenberg
Dr. Patrick MehlitzBrandenburgische Technische Universität Cottbus-Senftenberg
Dr. Uwe PrüfertTechnische Universität Bergakademie Freiberg
Markus FriedemannTechnische Universität Bergakademie Freiberg
Felix HarderBrandenburgische Technische Universität Cottbus-Senftenberg
Project Related News
Feb 24, 2020 : New preprint submitted
Gerd Wachsmuth submitted the preprint SPP1962-135 Reformulation of the M-stationarity Conditions as a System of Discontinuous Equations and its Solution by a Semismooth Newton Method.
Dec 13, 2019 : New preprint submitted
Gerd Wachsmuth submitted the preprint SPP1962-128 New Constraint Qualifications for Optimization Problems in Banach Spaces Based on Cone Continuity Properties.
Oct 25, 2019 : New preprint submitted
Uwe Prüfert submitted the preprint SPP1962-125 Coupled versus Decoupled Penalization of Control Complementarity Constraints.