Multiobjective Optimization of NonSmooth PDEConstrained Problems — Switches, State Constraints and Model Order Reduction
Description
In almost all technical applications, multiple criteria are of interest – both during development as well as operation. Examples are fast but energy efficient vehicles and constructions that have to be light as well as stable. The goal in the resulting multiobjective optimization problems is the computation of the set of optimal compromises – the socalled Pareto set. A decision maker can then select an appropriate solution from this set. In control applications, it is possible to quickly switch between different compromises as a reaction to changes in the external conditions.
The Pareto set generally consists of infinitely many compromise solutions, its numerical approximation is therefore considerably more expensive than the solution of scalar optimization problems. This can quickly result in prohibitively large computational cost, particularly in situations where solutions to the underlying system are computationally expensive. For instance, this is the case when the system is described by a partial differential equation (PDE). In this context, surrogate models that can be solved significantly faster than classical numerical approximations by the finite element method are frequently used. In the case of nonsmooth PDEs, reducing the computational cost is particularly important since these problems are often significantly more expensive to solve than smooth problems. However, the surrogate models introduce an approximation error intro the system, which has to be quantified and considered both in the analysis and the development of numerical algorithms. For nonsmooth problems, literature on this topic is currently scarce.
The goal of this project is the development of efficient numerical methods to solve multiobjective optimization problems that are constrained by nonsmooth PDEs. In the first step, optimality conditions for the nonsmooth PDEconstrained problems will be derived, and the (hierarchical) structure of the Pareto sets will be analyzed. Building on this, algorithms for the computation of Pareto sets will be developed for these problems. The methods will be used for the optimization of problems with maxterms, contact problems, and time dependent hybrid and switched systems. In order to handle the numerical effort, reduced order modeling techniques – such as Reduced Basis, Proper Orthogonal Decomposition, and more recent approaches based on the Koopman operator – will be extended to the nonsmooth setting. This requires the consideration of inexactness in the convergence analysis. Finally, the algorithms will be applied to several different problem settings in cooperation with other members of the Priority Programme.
Publications
Bennet Gebken, Sebastian Peitz: An Efficient Descent Method for Locally Lipschitz Multiobjective Optimization Problems, Journal of Optimization Theory and Applications volume 188, pages 696–723, 2021 (SPP1962140).
Bennet Gebken, Sebastian Peitz: Inverse Multiobjective Optimization: Inferring Decision Criteria from Data, Journal of Global Optimization volume 80, pages 3–29 , 2021 (SPP1962141).
Stefan Klus, Feliks Nüske, Sebastian Peitz, JanHendrik Niemannd, Cecilia Clement, Christof Schüttea: DataDriven Approximation of the Koopman Generator: Model Reduction, System Identification, and Control, Physica D: Nonlinear Phenomena, 406, 132416, 2020.
Carlos I Hernández Castellanos, Sina Ober‐Blöbaum, Sebastian Peitz: Explicit multiobjective model predictive control for nonlinear systems under uncertainty, International Journal on Robust and Nonlinear Control 30(17), pp. 75937618, 2020.
Sebastian Peitz, Samuel E. Otto, Clarence W. Rowley: DataDriven Model Predictive Control using Interpolated Koopman Generators, SIAM Journal on Applied Dynamical Systems 19(3), pp. 21622193, 2020.
Preprints
Bennet Gebken, Katharina Bieker, Sebastian Peitz: On the Structure of Regularization Paths for Piecewise Differentiable Regularization Terms (SPP1962182, 11/2021, [bib])
Marco Bernreuther, Georg Müller, Stefan Volkwein: Reduced Basis Model Order Reduction in Optimal Control of a Nonsmooth Semilinear Elliptic PDE (SPP1962138r, 04/2021, [bib])
Marco Bernreuther, Georg Müller, Stefan Volkwein: Stationarity Conditions and Scalarization in Multiobjective Optimal Control of a Nonsmooth PDE (SPP1962167, 04/2021, [bib])
Stefan Banholzer, Bennet Gebken, Lena Reichle, Stefan Volkwein: ROMBased Inexact Subdivision Methods for PDEConstrained Multiobjective Optimization (SPP1962166, 04/2021, [bib])
Sebastian Peitz, Katharina Bieker: On the Universal Transformation of DataDriven Models to Control Systems (SPP1962156, 02/2021, [bib])
Katharina Bieker, Bennet Gebken, Sebastian Peitz: On the Treatment of Optimization Problems with L1 Penalty Terms via Multiobjective Continuation (SPP1962153, 12/2020, [bib])
Bennet Gebken, Sebastian Peitz: Inverse Multiobjective Optimization: Inferring Decision Criteria from Data (SPP1962141, 07/2020, [bib])
Bennet Gebken, Sebastian Peitz: An Efficient Descent Method for Locally Lipschitz Multiobjective Optimization Problems (SPP1962140, 04/2020, [bib])
Marco Bernreuther, Georg Müller, Stefan Volkwein: Reduced Basis Model Order Reduction in Optimal Control of a Nonsmooth Semilinear Elliptic PDE (SPP1962138, 04/2020, [bib])
Stefan Banholzer, Giulia Fabrini, Lars Grüne, Stefan Volkwein: Multiobjective Model Predictive Control of a Parabolic AdvectionDiffusionReaction Equation (SPP1962139, 04/2020, [bib])
Constantin Christof, Georg Müller: Multiobjective Optimal Control of a NonSmooth Semilinear Elliptic Partial Differential Equation (SPP1962130, 01/2020, [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

Prof. Michael Dellnitz
Universität Paderborn
Principal Investigator 
Prof. Sebastian Peitz
Universität Paderborn
Principal Investigator 
Prof. Stefan Volkwein
Universität Konstanz
Principal Investigator 
Marco Bernreuther
Universität Konstanz
Research Assistant 
Bennet Gebken
Universität Paderborn
Research Assistant 
Dr. Georg Müller
Universität Konstanz
Research Assistant
Project Related News

Nov 12, 2021 : New preprint submitted
Bennet Gebken submitted the preprint SPP1962182 On the Structure of Regularization Paths for Piecewise Differentiable Regularization Terms.

Apr 09, 2021 : New preprint submitted
Georg Müller submitted the preprint SPP1962167 Stationarity Conditions and Scalarization in Multiobjective Optimal Control of a Nonsmooth PDE.

Apr 09, 2021 : New revised preprint submitted
Georg Müller submitted the revised preprint SPP1962138r Reduced Basis Model Order Reduction in Optimal Control of a Nonsmooth Semilinear Elliptic PDE.

Apr 07, 2021 : New preprint submitted
Stefan Volkwein submitted the preprint SPP1962166 ROMBased Inexact Subdivision Methods for PDEConstrained Multiobjective Optimization.

Feb 08, 2021 : New preprint submitted
Sebastian Peitz submitted the preprint SPP1962156 On the Universal Transformation of DataDriven Models to Control Systems.

Dec 14, 2020 : New preprint submitted
Sebastian Peitz submitted the preprint SPP1962153 On the Treatment of Optimization Problems with L1 Penalty Terms via Multiobjective Continuation.

Jul 03, 2020 : New preprint submitted
Bennet Gebken submitted the preprint SPP1962141 Inverse Multiobjective Optimization: Inferring Decision Criteria from Data.

Apr 24, 2020 : New preprint submitted
Sebastian Peitz submitted the preprint SPP1962140 An Efficient Descent Method for Locally Lipschitz Multiobjective Optimization Problems.

Apr 18, 2020 : New preprint submitted
Stefan Volkwein submitted the preprint SPP1962138 Reduced Basis Model Order Reduction in Optimal Control of a Nonsmooth Semilinear Elliptic PDE.

Apr 18, 2020 : New preprint submitted
Stefan Volkwein submitted the preprint SPP1962139 Multiobjective Model Predictive Control of a Parabolic AdvectionDiffusionReaction Equation.