Multiobjective Optimization of Non-Smooth PDE-Constrained 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 so-called 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 non-smooth 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 non-smooth 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 non-smooth PDEs. In the first step, optimality conditions for the non-smooth PDE-constrained 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 max-terms, 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 non-smooth 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

Sebastian Peitz, Katharina Bieker: On the Universal Transformation of Data-Driven Models to Control Systems, Automatica: Issue 149, Article No. 110840, 2023 (SPP1962-156).

Bennet Gebken, Katharina Bieker, Sebastian Peitz: On the Structure of Regularization Paths for Piecewise Differentiable Regularization Terms, J Glob Optim, 2022 (SPP1962-182).

Katharina Bieker, Bennet Gebken, Sebastian Peitz: On the Treatment of Optimization Problems with L1 Penalty Terms via Multiobjective Continuation, IEEE Transactions on Pattern Analysis and Machine Intelligence, vol. 44, no. 11, pp. 7797-7808, 2022 (SPP1962-153).

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 (SPP1962-140).

Bennet Gebken, Sebastian Peitz: Inverse Multiobjective Optimization: Inferring Decision Criteria from Data, Journal of Global Optimization volume 80, pages 3–29 , 2021 (SPP1962-141).

Stefan Klus, Feliks Nüske, Sebastian Peitz, Jan-Hendrik Niemannd, Cecilia Clement, Christof Schüttea: Data-Driven 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. 7593-7618, 2020.

Sebastian Peitz, Samuel E. Otto, Clarence W. Rowley: Data-Driven Model Predictive Control using Interpolated Koopman Generators, SIAM Journal on Applied Dynamical Systems 19(3), pp. 2162-2193, 2020.

Preprints

Bennet Gebken: Using Second-Order Information in Gradient Sampling Methods for Nonsmooth Optimization (SPP1962-196, 10/2022, [bib])

Bennet Gebken, Katharina Bieker, Sebastian Peitz: On the Structure of Regularization Paths for Piecewise Differentiable Regularization Terms (SPP1962-182, 11/2021, [bib])

Marco Bernreuther, Georg Müller, Stefan Volkwein: Stationarity Conditions and Scalarization in Multiobjective Optimal Control of a Nonsmooth PDE (SPP1962-167, 04/2021, [bib])

Marco Bernreuther, Georg Müller, Stefan Volkwein: Reduced Basis Model Order Reduction in Optimal Control of a Nonsmooth Semilinear Elliptic PDE (SPP1962-138r, 04/2021, [bib])

Stefan Banholzer, Bennet Gebken, Lena Reichle, Stefan Volkwein: ROM-Based Inexact Subdivision Methods for PDE-Constrained Multiobjective Optimization (SPP1962-166, 04/2021, [bib])

Sebastian Peitz, Katharina Bieker: On the Universal Transformation of Data-Driven Models to Control Systems (SPP1962-156, 02/2021, [bib])

Katharina Bieker, Bennet Gebken, Sebastian Peitz: On the Treatment of Optimization Problems with L1 Penalty Terms via Multiobjective Continuation (SPP1962-153, 12/2020, [bib])

Bennet Gebken, Sebastian Peitz: Inverse Multiobjective Optimization: Inferring Decision Criteria from Data (SPP1962-141, 07/2020, [bib])

Bennet Gebken, Sebastian Peitz: An Efficient Descent Method for Locally Lipschitz Multiobjective Optimization Problems (SPP1962-140, 04/2020, [bib])

Marco Bernreuther, Georg Müller, Stefan Volkwein: Reduced Basis Model Order Reduction in Optimal Control of a Nonsmooth Semilinear Elliptic PDE (SPP1962-138, 04/2020, [bib])

Stefan Banholzer, Giulia Fabrini, Lars Grüne, Stefan Volkwein: Multiobjective Model Predictive Control of a Parabolic Advection-Diffusion-Reaction Equation (SPP1962-139, 04/2020, [bib])

Constantin Christof, Georg Müller: Multiobjective Optimal Control of a Non-Smooth Semilinear Elliptic Partial Differential Equation (SPP1962-130, 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 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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