Publikationen

Baier, Robert; Eichfelder, Gabriele; Gerlach, Tobias
Optimality conditions for set optimization using a directional derivative based on generalized Steiner sets. - Ilmenau : Technische Universität Ilmenau, Institut für Mathematik, 2019. - 1 Online-Ressource (40 Seiten). - (Preprint ; M19,09)

Set-optimization has attracted increasing interest in the last years, as for instance uncertain multiobjective optimization problems lead to such problems with a set- valued objective function. Thereby, from a practical point of view, most of all the so-called set approach is of interest. However, optimality conditions for these problems, for instance using directional derivatives, are still very limited. The key aspect for a useful directional derivative is the definition of a useful set difference for the evaluation of the numerator in the difference quotient. We present here a new set difference which avoids the use of a convex hull and which applies to arbitrary convex sets, and not to strictly convex sets only. The new set difference is based on the new concept of generalized Steiner sets. We introduce the Banach space of generalized Steiner sets as well as an embedding of convex sets in this space using Steiner points. In this Banach space we can easily define a difference and a directional derivative. We use the latter for new optimality conditions for set optimization. Numerical examples illustrate the new concepts.

https://www.db-thueringen.de/receive/dbt_mods_00040057

Eichfelder, Gabriele; Gerlach, Tobias
On classes of set optimization problems which are reducible to vector optimization problems and its impact on numerical test instances. - In: Variational analysis and set optimization, (2019), S. 265-290

Set optimization with the set approach has recently gained increasing interest due to its practical relevance. In this problem class one studies optimization problems with a set-valued objective map and defines optimality based on a direct comparison of the images of the objective function, which are sets here. Meanwhile, in the literature a wide range of theoretical tools as scalarization approaches and derivative concepts as well as first numerical algorithms are available. These numerical algorithms require on the one hand test instances where the optimal solution sets are known. On the other hand, in most examples and test instances in the literature only set-valued maps with a very simple structure are used. We study in this paper such special set-valued maps and we show that some of them are such simple that they can equivalently be expressed as a vector optimization problem. Thus we try to start drawing a line between simple set-valued problems and such problems which have no representation as multiobjective problems. Those having a representation can be used for defining test instances for numerical algorithms with easy verifiable optimal solution set.

Huang, Junjie; Sun, Junfeng; Chen, Alatancang; Trunk, Carsten
Invertibility of 2 × 2 operator matrices. - In: Mathematische Nachrichten, ISSN 1522-2616, Bd. 292 (2019), 11, S. 2411-2426

https://doi.org/10.1002/mana.201800351

Grebinyk, Anna; Prylutska, Svitlana; Buchelnikov, Anatoliy; Tverdokhleb, Nina; Grebinyk, Sergii; Evstigneev, Maxim; Matyshevska, Olga; Cherepanov, Vsevolod; Prylutskyy, Yuriy; Yashchuk, Valeriy; Naumovets, Anton; Ritter, Uwe; Dandekar, Thomas; Frohme, Marcus
C60 fullerene as an effective nanoplatform of alkaloid Berberine delivery into leukemic cells. - In: Pharmaceutics, ISSN 1999-4923, Bd. 11 (2019), 11, 586, S. 1-23

https://doi.org/10.3390/pharmaceutics11110586

Zhong, Jinhui; Yi, Juemin; Wang, Dong; Korte, Anke; Chimeh, Abbas; Schaaf, Peter; Runge, Erich; Lienau, Christoph
Observing nonlinear plasmon-exciton coupling for enhanced harmonic generation. - In: 2019 Conference on Lasers and Electro-Optics Europe & European Quantum Electronics Conference (CLEO/Europe-EQEC), (2019), insges. 1 S.

https://doi.org/10.1109/CLEOE-EQEC.2019.8872963