Publikationen am Institut für Mathematik

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Eichfelder, Gabriele; Kirst, Peter; Meng, Laura; Stein, Oliver;
A general branch-and-bound framework for continuous global multiobjective optimization. - In: Journal of global optimization. - Dordrecht [u.a.] : Springer Science + Business Media B.V, ISSN 1573-2916, (2021), insges. 33 S.
- Published: 19 January 2021

Current generalizations of the central ideas of single-objective branch-and-bound to the multiobjective setting do not seem to follow their train of thought all the way. The present paper complements the various suggestions for generalizations of partial lower bounds and of overall upper bounds by general constructions for overall lower bounds from partial lower bounds, and by the corresponding termination criteria and node selection steps. In particular, our branch-and-bound concept employs a new enclosure of the set of nondominated points by a union of boxes. On this occasion we also suggest a new discarding test based on a linearization technique. We provide a convergence proof for our general branch-and-bound framework and illustrate the results with numerical examples.



https://doi.org/10.1007/s10898-020-00984-y
Derkach, Volodymyr; Hassi, Seppo; Malamud, Mark;
Generalized boundary triples, II : some applications of generalized boundary triples and form domain invariant Nevanlinna functions. - Ilmenau : Technische Universität Ilmenau, Institut für Mathematik, 2021. - 1 Online-Ressource (54 Seiten). . - (Preprint. - M21,03)
https://nbn-resolving.org/urn:nbn:de:gbv:ilm1-2021200058
Derkach, Volodymyr; Schmitz, Philipp; Trunk, Carsten;
PT-symmetric Hamiltonians as couplings of dual pairs. - Ilmenau : Technische Universität Ilmenau, Institut für Mathematik, 2021. - 1 Online-Ressource (15 Seiten). . - (Preprint. - M21,02)
https://nbn-resolving.org/urn:nbn:de:gbv:ilm1-2021200042
Leben, Leslie; Martínez Pería, Francisco; Philipp, Friedrich; Trunk, Carsten; Winkler, Henrik;
Finite rank perturbations of linear relations and matrix pencils. - In: Complex analysis and operator theory. - Cham (ZG) : Springer International Publishing AG, ISSN 1661-8262, Bd. 15 (2021), 2, S. 1-37

We elaborate on the deviation of the Jordan structures of two linear relations that are finite-dimensional perturbations of each other. We compare their number of Jordan chains of length at least n. In the operator case, it was recently proved that the difference of these numbers is independent of n and is at most the defect between the operators. One of the main results of this paper shows that in the case of linear relations this number has to be multiplied by n+1 and that this bound is sharp. The reason for this behavior is the existence of singular chains. We apply our results to one-dimensional perturbations of singular and regular matrix pencils. This is done by representing matrix pencils via linear relations. This technique allows for both proving known results for regular pencils as well as new results for singular ones.



https://doi.org/10.1007/s11785-021-01082-x
Berger, Thomas; Ilchmann, Achim; Ryan, Eugene P.;
Funnel control of nonlinear systems. - In: Mathematics of control, signals, and systems : MCSS.. - London : Springer, ISSN 1435-568X, Bd. 33 (2021), 1, S. 151-194

Tracking of reference signals is addressed in the context of a class of nonlinear controlled systems modelled by r-th-order functional differential equations, encompassing inter alia systems with unknown "control direction" and dead-zone input effects. A control structure is developed which ensures that, for every member of the underlying system class and every admissible reference signal, the tracking error evolves in a prescribed funnel chosen to reflect transient and asymptotic accuracy objectives. Two fundamental properties underpin the system class: bounded-input bounded-output stable internal dynamics, and a high-gain property (an antecedent of which is the concept of sign-definite high-frequency gain in the context of linear systems).



https://doi.org/10.1007/s00498-021-00277-z
Janse van Rensburg, Dawie B.; van Straaten, Madelein; Theron, Frieda; Trunk, Carsten;
Square roots of H-nonnegative matrices. - In: Linear algebra and its applications : LAA.. - New York, NY : American Elsevier Publ., Bd. 621 (2021), S. 29-49

https://doi.org/10.1016/j.laa.2021.03.006
Bang-Jensen, Jørgen; Bessy, Stéphane; Huang, Jing; Kriesell, Matthias;
Good orientations of unions of edge-disjoint spanning trees. - In: Journal of graph theory. - New York, NY [u.a.] : Wiley, ISSN 1097-0118, Bd. 96 (2021), 4, S. 594-618

In this paper, we exhibit connections between the following subjects: Tree packing in graphs and digraphs (both behave completely different), the rigidity matroid of a graph, Henneberg moves on trees, the conjectures of Thomassen and Matthews and Sumner, and (s,t)-orderings of digraphs. We do this by studying graphs which admit acyclic orientations that contain an out-branching and in-branching which are arc-disjoint (such an orientation is called good). A 2T-graph is a graph whose edge set can be decomposed into two edge-disjoint spanning trees. It is a well-known result due to Tutte and Nash-Williams, respectively, that every 4-edge-connected graph contains a spanning 2T-graph. Vertex-minimal 2T-graphs with at least two vertices which are known as generic circuits play an important role in rigidity theory for graphs. We prove that every generic circuit has a good orientation. Using this result we prove that if G is 2T-graph whose vertex set has a partition V1,V2, ,Vk so that each Vi induces a generic circuit Gi of G and the set of edges between different Gi's form a matching in G, then G has a good orientation. We also obtain a characterization for the case when the set of edges between different Gi's form a double tree, that is, if we contract each Gi to one vertex, and delete parallel edges we obtain a tree. All our proofs are constructive and imply polynomial algorithms for finding the desired good orderings and the pairs of arc-disjoint branchings which certify that the orderings are good. We identify a structure which can be used to certify that a given 2T-graph does not have a good orientation.



https://doi.org/10.1002/jgt.22633
Baidiuk, Dmytro; Derkach, Volodymyr; Hassi, Seppo;
Unitary boundary pairs for isometric operators in Pontryagin spaces and generalized coresolvents. - In: Complex analysis and operator theory. - Cham (ZG) : Springer International Publishing AG, ISSN 1661-8262, Bd. 15 (2021), 2, S. 1-52

https://doi.org/10.1007/s11785-020-01073-4
Eichfelder, Gabriele; Jahn, Johannes;
Optimality conditions in discrete-continuous nonlinear optimization. - In: Minimax theory and its applications. - [Lemgo] : Heldermann Verlag, ISSN 2199-1413, Bd. 6 (2021), 1, S. 127-144

Schweser, Thomas;
Generalized hypergraph coloring. - In: Discussiones mathematicae. - Warsaw : De Gruyter Open, ISSN 2083-5892, Bd. 41 (2021), 1, S. 103-121

https://doi.org/10.7151/dmgt.2168