Exploiting structure in multiobjective optimization and optimal control - Peitz, Sebastian
Multiobjective optimization plays an increasingly important role in modern applications, where several criteria are often of equal importance. The task in multiobjective optimization and multiobjective optimal control is therefore to compute the set of optimal compromises (the Pareto set) between the conflicting objectives. Since – in contrast to the solution of a single objective optimization problem – the Pareto set generally consists of an infinite number of solutions, the computational effort can quickly become challenging. This is even more the case when many problems have to be solved, when the number of objectives is high, or when the objectives are costly to evaluate. Consequently, this thesis is devoted to the identification and exploitation of structure both in the Pareto set and the dynamics of the underlying model as well as to the development of efficient algorithms for solving problems with additional parameters, with a high number of objectives or with PDE-constraints. These three challenges are addressed in three respective parts. In the first part, predictor-corrector methods are extended to entire Pareto sets. When certain smoothness assumptions are satisfied, then the set of parameter dependent Pareto sets possesses additional structure, i.e. it is a manifold. The tangent space can be approximated numerically which yields a direction for the predictor step. In the corrector step, the predicted set converges to the Pareto set at a new parameter value. The resulting algorithm is applied to an example from autonomous driving. In the second part, the hierarchical structure of Pareto sets is investigated. When considering a subset of the objectives, the resulting solution is a subset of the Pareto set of the original problem. Under additional smoothness assumptions, the respective subsets are located on the boundary of the Pareto set of the full problem. This way, the “skeleton” of a Pareto set can be computed and due to the exponential increase in computing time with the number of objectives, the computations of these subsets are significantly faster which is demonstrated using an example from industrial laundries. In the third part, PDE-constrained multiobjective optimal control problems are addressed by reduced order modeling methods. Reduced order models exploit the structure in the system dynamics, for example by describing the dynamics of only the most energetic modes. The model reduction introduces an error in both the function values and their gradients, which has to be taken into account in the development of algorithms. Both scalarization and set-oriented approaches are coupled with reduced order modeling. Convergence results are presented and the numerical benefit is investigated. The algorithms are applied to semi-linear heat flow problems as well as to the Navier-Stokes equations.

A Birkhoff–Lewis Type Theorem for the Nonlinear Wave Equation - Biasco, Luca; Di Gregorio, Laura (Archive for Rational Mechanics and Analysis)

A comparison of two predictive approaches to control the longitudinal dynamics of electric vehicles - Eckstein, Julian; Peitz, Sebastian; Schäfer, Kai; Friedel, Patrick; Köhler, Ulrich; Hessel von Molo, Mirko; Ober-Blöbaum, Sina; Dellnitz, Michael (Procedia Technology, 3rd International Conference on System-Integrated Intelligence: New Challenges for Product and Production Engineering)
In this contribution we compare two different approaches to the implementation of a Model Predictive Controller in an electric vehicle with respect to the quality of the solution and real-time applicability. The goal is to develop an intelligent cruise control in order to extend the vehicle range, i.e. to minimize energy consumption, by computing the optimal torque profile for a given track. On the one hand, a path-based linear model with strong simplifications regarding the vehicle dynamics is used. On the other hand, a nonlinear model is employed in which the dynamics of the mechanical and electrical subsystem are modeled.

A Conjecture on the Existence of Isolated Eigenvalues of the Perron-Frobenius Operator - Dellnitz, Michael; Froyland, Gary; Sertl, Stefan (Equadiff 99)

A Descent Method for Equality and Inequality Constrained Multiobjective Optimization Problems - Gebken, Bennet; Peitz, Sebastian; Dellnitz, Michael (Numerical and Evolutionary Optimization – NEO 2017)
In this article we propose a descent method for equality and inequality constrained multiobjective optimization problems (MOPs) which generalizes the steepest descent method for unconstrained MOPs by Fliege and Svaiter to constrained problems by using two active set strategies. Under some regularity assumptions on the problem, we show that accumulation points of our descent method satisfy a necessary condition for local Pareto optimality. Finally, we show the typical behavior of our method in a numerical example.

A Descent Method for Nonsmooth Multiobjective Optimization in Hilbert Spaces - Sonntag, Konstantin; Gebken, Bennet; Müller, Georg; Peitz, Sebastian; Volkwein, Stefan (arXiv:2402.06376)
The efficient optimization method for locally Lipschitz continuous multiobjective optimization problems from [1] is extended from finite-dimensional problems to general Hilbert spaces. The method iteratively computes Pareto critical points, where in each iteration, an approximation of the subdifferential is computed in an efficient manner and then used to compute a common descent direction for all objective functions. To prove convergence, we present some new optimality results for nonsmooth multiobjective optimization problems in Hilbert spaces. Using these, we can show that every accumulation point of the sequence generated by our algorithm is Pareto critical under common assumptions. Computational efficiency for finding Pareto critical points is numerically demonstrated for multiobjective optimal control of an obstacle problem.

A Memetic PSO Algorithm for Scalar Optimization Problems - Schütze, Oliver; Talbi, El-ghazali; Pulido, Gregorio Toscano; Coello, Carlos Coello; Santana-Quintero, Luis Vicente (2007 IEEE Swarm Intelligence Symposium)

A multi-objective approach to the design of low thrust space trajectories using optimal control - Dellnitz, Michael; Ober-Blöbaum, Sina; Post, Marcus; Schütze, Oliver; Thiere, Bianca (Celestial Mechanics and Dynamical Astronomy)

A multiobjective MPC approach for autonomously driven electric vehicles - Peitz, Sebastian; Schäfer, Kai; Ober-Blöbaum, Sina; Eckstein, Julian; Köhler, Ulrich; Dellnitz, Michael (Proceedings of the 20th World Congress of the International Federation of Automatic Control (IFAC))
We present a new algorithm for model predictive control of non-linear systems with respect to multiple, conflicting objectives. The idea is to provide a possibility to change the objective in real-time, e.g. as a reaction to changes in the environment or the system state itself. The algorithm utilises elements from various well-established concepts, namely multiobjective optimal control, economic as well as explicit model predictive control and motion planning with motion primitives. In order to realise real-time applicability, we split the computation into an online and an offline phase and we utilise symmetries in the open-loop optimal control problem to reduce the number of multiobjective optimal control problems that need to be solved in the offline phase. The results are illustrated using the example of an electric vehicle where the longitudinal dynamics are controlled with respect to the concurrent objectives arrival time and energy consumption.

A Multiobjective MPC Approach for Autonomously Driven Electric Vehicles * *This research was funded by the German Federal Ministry of Education and Research (BMBF) within the Leading-Edge Cluster Intelligent Technical Systems OstWestfalenLippe (it’s OWL). - Peitz, Sebastian; Schäfer, Kai; Ober-Blöbaum, Sina; Eckstein, Julian; Köhler, Ulrich; Dellnitz, Michael (IFAC-PapersOnLine)

A multiobjective optimization approach for optimal control problems of mechanical systems with uncertainties - Ober-Blöbaum, Sina; Seifried, Albert (2013 European Control Conference (ECC))

A new approach for online multiobjective optimization of mechatronic systems - Witting, Katrin; Schulz, Bernd; Dellnitz, Michael; Böcker, Joachim; Fröhleke, Norbert (International Journal on Software Tools for Technology Transfer)

A New Data Structure for the Nondominance Problem in Multi-objective Optimization - Schütze, Oliver (Lecture Notes in Computer Science)

A note on the convergence of deterministic gradient sampling in nonsmooth optimization - Gebken, Bennet (Computational Optimization and Applications)
Abstract Approximation of subdifferentials is one of the main tasks when computing descent directions for nonsmooth optimization problems. In this article, we propose a bisection method for weakly lower semismooth functions which is able to compute new subgradients that improve a given approximation in case a direction with insufficient descent was computed. Combined with a recently proposed deterministic gradient sampling approach, this yields a deterministic and provably convergent way to approximate subdifferentials for computing descent directions.
Keywords: Applied Mathematics; Computational Mathematics; Control and Optimization

A note on the shadowing lemma and symmetric periodic points - Dellnitz, M; Melbourne, I (Nonlinearity)

A parametrisation method for high-order phase reduction in coupled oscillator networks - von der Gracht, Sören; Nijholt, Eddie; Rink, Bob (arXiv:2306.03320)
We present a novel method for high-order phase reduction in networks of weakly coupled oscillators and, more generally, perturbations of reducible normally hyperbolic (quasi-)periodic tori. Our method works by computing an asymptotic expansion for an embedding of the perturbed invariant torus, as well as for the reduced phase dynamics in local coordinates. Both can be determined to arbitrary degrees of accuracy, and we show that the phase dynamics may directly be obtained in normal form. We apply the method to predict remote synchronisation in a chain of coupled Stuart-Landau oscillators.

A Rigorous Numerical Method for the Global Analysis of Infinite-Dimensional Discrete Dynamical Systems - Day, S.; Junge, O.; Mischaikow, K. (SIAM Journal on Applied Dynamical Systems)

A set oriented approach to global optimal control - Junge, Oliver; Osinga, Hinke M. (ESAIM: Control, Optimisation and Calculus of Variations)

A set oriented approach to optimal feedback stabilization - Grüne, Lars; Junge, Oliver (Systems & Control Letters)

A Set-Oriented Numerical Approach for Dynamical Systems with Parameter Uncertainty - Dellnitz, Michael; Klus, Stefan; Ziessler, Adrian (SIAM Journal on Applied Dynamical Systems)