Title data
Baier, Robert ; Eichfelder, Gabriele ; Gerlach, Tobias:
Optimality conditions for set optimization using a directional derivative based on generalized Steiner sets.
Ilmenau
:
Technische Universität Ilmenau, Fakultät für Mathematik und Naturwissenschaften
,
2019
. - 40 p.
- (Preprint M / Technische Universität Ilmenau, Institut für Mathematik
; 19/09
)
Project information
Project title: |
Project's official title Project's id Algorithmische Lösungsansätze in der mengenwertigen Optimierung 392195690 |
---|---|
Project financing: |
Deutsche Forschungsgemeinschaft |
Abstract in another language
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.
Further data
Item Type: | Preprint, postprint |
---|---|
Additional notes: | Accepted for publication in the journal "Optimization", online available in the Digital Library Thuringa since November 2019
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 relation; set difference; support functions; directional derivative; optimality condition; Steiner point; generalized Steiner set |
Subject classification: | Mathematics Subject Classification Code: 26E25 (49K99 49J53) |
Institutions of the University: | Faculties > Faculty of Mathematics, Physics und Computer Science > Department of Mathematics > Chair Mathematics V (Applied Mathematics) Profile Fields > Advanced Fields > Nonlinear Dynamics Faculties Faculties > Faculty of Mathematics, Physics und Computer Science Faculties > Faculty of Mathematics, Physics und Computer Science > Department of Mathematics Profile Fields Profile Fields > Advanced Fields |
Result of work at the UBT: | Yes |
DDC Subjects: | 500 Science > 510 Mathematics |
Date Deposited: | 20 Aug 2020 09:14 |
Last Modified: | 14 Sep 2020 12:08 |
URI: | https://eref.uni-bayreuth.de/id/eprint/56595 |
Available Versions of this Item
- Optimality conditions for set optimization using a directional derivative based on generalized Steiner sets. (deposited 20 Aug 2020 09:14) [Currently Displayed]