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

Title data

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) . - pp. 1-42.
ISSN 0323-3898
DOI: https://doi.org/10.1080/02331934.2020.1812605

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Project information

Project title:
Project's official titleProject's id
Algorithmische Lösungsansätze in der mengenwertigen Optimierung392195690

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: Article in a journal
Refereed: Yes
Additional notes: 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
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: 25 Nov 2020 06:43
Last Modified: 25 Nov 2020 06:43
URI: https://eref.uni-bayreuth.de/id/eprint/60398

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