Publications at the Institute of Mathematics

Results: 1922
Created on: Mon, 16 May 2022 23:09:03 +0200 in 0.0602 sec


De Santis, Marianna; Eichfelder, Gabriele; Patria, Daniele;
On the exactness of the ε-constraint method for biobjective nonlinear integer programming. - In: Operations research letters, ISSN 0167-6377, Bd. 50 (2022), 3, S. 356-361

The ε-constraint method is a well-known scalarization technique used for multiobjective optimization. We explore how to properly define the step size parameter of the method in order to guarantee its exactness when dealing with biobjective nonlinear integer problems. Under specific assumptions, we prove that the number of subproblems that the method needs to address to detect the complete Pareto front is finite. We report numerical results on portfolio optimization instances built on real-world data and show a comparison with an existing criterion space algorithm.



https://doi.org/10.1016/j.orl.2022.04.007
Grundel, Sara; Heyder, Stefan; Hotz, Thomas; Ritschel, Tobias K. S.; Sauerteig, Philipp; Worthmann, Karl;
How much testing and social distancing is required to control COVID-19? : some insight based on an age-differentiated compartmental model. - In: SIAM journal on control and optimization, ISSN 1095-7138, Bd. 60 (2022), 2, S. S145-S169

In this paper, we provide insights on how much testing and social distancing is required to control COVID-19. To this end, we develop a compartmental model that accounts for key aspects of the disease: incubation time, age-dependent symptom severity, and testing and hospitalization delays; the model's parameters are chosen based on medical evidence, and, for concreteness, adapted to the German situation. Then, optimal mass-testing and age-dependent social distancing policies are determined by solving optimal control problems both in open loop and within a model predictive control framework. We aim to minimize testing and/or social distancing until herd immunity sets in under a constraint on the number of available intensive care units. We find that an early and short lockdown is inevitable but can be slowly relaxed over the following months.



https://doi.org/10.1137/20M1377783
Grüne, Lars; Schaller, Manuel; Schiela, Anton;
Efficient model predictive control for parabolic PDEs with goal oriented error estimation. - In: SIAM journal on scientific computing, ISSN 1095-7197, Bd. 44 (2022), 1, S. A471-A500

We show how a posteriori goal oriented error estimation can be used to efficiently solve the subproblems occurring in a model predictive control (MPC) algorithm. In MPC, only an initial part of a computed solution is implemented as a feedback, which motivates grid refinement particularly tailored to this context. To this end, we present a truncated cost functional as an objective for goal oriented adaptivity and prove under stabilizability assumptions that error indicators decay exponentially outside the support of this quantity. This leads to very efficient time and space discretizations for MPC, which we will illustrate by means of various numerical examples.



https://doi.org/10.1137/20M1356324
Schmitz, Philipp; Faulwasser, Timm; Worthmann, Karl;
Willems' fundamental lemma for linear descriptor systems and its use for data-driven output-feedback MPC. - In: IEEE control systems letters, ISSN 2475-1456, Bd. 6 (2022), S. 2443-2448

In this letter we investigate data-driven predictive control of discrete-time linear descriptor systems. Specifically, we give a tailored variant of Willems' fundamental lemma, which shows that for descriptor systems the non-parametric modeling via a Hankel matrix requires less data compared to linear time-invariant systems without algebraic constraints. Moreover, we use this description to propose a data-driven framework for optimal control and predictive control of discrete-time linear descriptor systems. For the latter, we provide a sufficient stability condition for receding-horizon control before we illustrate our findings with an example.



https://doi.org/10.1109/LCSYS.2022.3161054
Babovsky, Hans; Bold, Lea;
Balanced states and closure relations: the fluid dynamic limit of kinetic models. - Ilmenau : Technische Universität Ilmenau, Institut für Mathematik, 2022. - 1 Online-Ressource (20 Seiten). - (Preprint ; M22,03)

The paper is concerned with closure relations for moment hierarchies of gaskinetic systems in the uid dynamic limit. We develop the concept of balanced solutions which provides a more detailed description of kinetic solutions that the classical approaches. This allows to compare di_erent models in use like the nonlinear Boltzmann equation, its linearization, and the BGK model and their relation to the classical Navier-Stokes equations.



https://nbn-resolving.org/urn:nbn:de:gbv:ilm1-2022200188
Behrndt, Jussi; Schmitz, Philipp; Teschl, Gerald; Trunk, Carsten;
Relative oscillation theory and essential spectra of Sturm-Liouville operators. - Ilmenau : Technische Universität Ilmenau, Institut für Mathematik, 2022. - 1 Online-Ressource (15 Seiten). - (Preprint ; M22,02)

The relationship between linear relations and matrix pencils is investigated. Given a linear relation, we introduce its Weyr characteristic. If the linear relation is the range (or the kernel) representation of a given matrix pencil, we show that there is a correspondence between this characteristic and the Kronecker canonical form of the pencil. This relationship is exploited to obtain estimations on the invariant characteristics of matrix pencils under rank one perturbations.



https://nbn-resolving.org/urn:nbn:de:gbv:ilm1-2022200169
Gernandt, Hannes; Martínez Pería, Francisco; Philipp, Friedrich; Trunk, Carsten;
On characteristic invariants of matrix pencils and linear relations. - Ilmenau : Technische Universität Ilmenau, Institut für Mathematik, 2022. - 1 Online-Ressource (35 Seiten). - (Preprint ; M22,01)

The relationship between linear relations and matrix pencils is investigated. Given a linear relation, we introduce its Weyr characteristic. If the linear relation is the range (or the kernel) representation of a given matrix pencil, we show that there is a correspondence between this characteristic and the Kronecker canonical form of the pencil. This relationship is exploited to obtain estimations on the invariant characteristics of matrix pencils under rank one perturbations.



https://nbn-resolving.org/urn:nbn:de:gbv:ilm1-2022200140
Bang-Jensen, Jørgen; Havet, Frederic; Kriesell, Matthias; Yeo, Anders;
Low chromatic spanning sub(di)graphs with prescribed degree or connectivity properties. - In: Journal of graph theory, ISSN 1097-0118, Bd. 99 (2022), 4, S. 615-636

https://doi.org/10.1002/jgt.22755
Eichfelder, Gabriele; Groetzner, Patrick;
A note on completely positive relaxations of quadratic problems in a multiobjective framework. - In: Journal of global optimization, ISSN 1573-2916, Bd. 82 (2022), 3, S. 615-626

In a single-objective setting, nonconvex quadratic problems can equivalently be reformulated as convex problems over the cone of completely positive matrices. In small dimensions this cone equals the cone of matrices which are entrywise nonnegative and positive semidefinite, so the convex reformulation can be solved via SDP solvers. Considering multiobjective nonconvex quadratic problems, naturally the question arises, whether the advantage of convex reformulations extends to the multicriteria framework. In this note, we show that this approach only finds the supported nondominated points, which can already be found by using the weighted sum scalarization of the multiobjective quadratic problem, i.e. it is not suitable for multiobjective nonconvex problems.



https://doi.org/10.1007/s10898-021-01091-2
Bouza, Gemayqzel; Quintana, Ernest; Tammer, Christiane;
On Clarke's subdifferential of marginal functions. - In: Applied set-valued analysis and optimization, ISSN 2562-7783, Bd. 3 (2021), 3, S. 281-292

In this paper, we derive an upper estimate of the Clarke subdifferential of marginal functions in Banach spaces. The structure of the upper estimate is very similar to other results already obtained in the literature. The novelty lies on the fact that we derive our assertions in general Banach spaces, and avoid the use of the Asplund assumption.



https://doi.org/10.23952/asvao.3.2021.3.03