Optimization Online


Optimality Conditions for Set Optimization using a Directional Derivative based on Generalized Steiner Sets

Robert Baier(robert.baier***at***uni-bayreuth.de)
Gabriele Eichfelder(gabriele.eichfelder***at***tu-ilmenau.de)
Tobias Gerlach(tobias.gerlach***at***tu-ilmenau.de)

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.

Keywords: Set optimization, Set relation, Set di erence, Support function, Directional derivative, Optimality condition, Steiner point, Generalized Steiner set

Category 1: Nonlinear Optimization (Unconstrained Optimization )

Category 2: Other Topics (Multi-Criteria Optimization )

Category 3: Convex and Nonsmooth Optimization (Nonsmooth Optimization )


Download: [PDF]

Entry Submitted: 11/15/2019
Entry Accepted: 11/15/2019
Entry Last Modified: 11/15/2019

Modify/Update this entry

  Visitors Authors More about us Links
  Subscribe, Unsubscribe
Digest Archive
Search, Browse the Repository


Coordinator's Board
Classification Scheme
Give us feedback
Optimization Journals, Sites, Societies
Mathematical Optimization Society