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Optimality conditions for set optimization using a directional derivative based on generalized Steiner sets

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Baier, Robert ; Eichfelder, Gabriele ; Gerlach, Tobias:
Optimality conditions for set optimization using a directional derivative based on generalized Steiner sets.
In: Optimization. (14 September 2020) . - S. 1-42.
ISSN 0323-3898
DOI: https://doi.org/10.1080/02331934.2020.1812605

Dies ist die aktuelle Version des Eintrags.

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Projekttitel:
Offizieller Projekttitel
Projekt-ID
Algorithmische Lösungsansätze in der mengenwertigen Optimierung
392195690

Projektfinanzierung: Deutsche Forschungsgemeinschaft

Abstract

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.

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Publikationsform: Artikel in einer Zeitschrift
Begutachteter Beitrag: Ja
Zusätzliche Informationen: online available since September 2020
Contents:
1. Introduction
2. Notation and basic results
3. Generalized Steiner sets as a tool in set optimization
3.1 The Banach spaces of generalized Steiner sets
3.2 Embeddings of convex sets via Steiner points
4. Optimality conditions for set optimization
4.1 Optimality conditions for set optimization problems based on visualization results
4.2 A new directional derivative for set optimization
4.3 Relation to Jahn's set difference and directional derivative
Keywords: set optimization; set difference; directional derivative; optimality condition; generalized Steiner set
Fachklassifikationen: Mathematics Subject Classification Code: 26E25 (49K99 49J53)
Institutionen der Universität: Fakultäten > Fakultät für Mathematik, Physik und Informatik > Mathematisches Institut > Lehrstuhl Mathematik V (Angewandte Mathematik)
Profilfelder > Advanced Fields > Nichtlineare Dynamik
Fakultäten
Fakultäten > Fakultät für Mathematik, Physik und Informatik
Fakultäten > Fakultät für Mathematik, Physik und Informatik > Mathematisches Institut
Profilfelder
Profilfelder > Advanced Fields
Titel an der UBT entstanden: Ja
Themengebiete aus DDC: 500 Naturwissenschaften und Mathematik > 510 Mathematik
Eingestellt am: 25 Nov 2020 06:43
Letzte Änderung: 25 Nov 2020 06:43
URI: https://eref.uni-bayreuth.de/id/eprint/60398

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