On Classes of Set Optimization Problems which are Reducible to Vector Optimization Problems and its Impact on Numerical Test Instances

Eichfelder, Gabriele; Gerlach, Tobias

doi:10.1201/b22166-10

Cited by 5 publications

(5 citation statements)

References 13 publications

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“…Based on that, the proofs of (i), (ii), (iv) and (v) are easy and hence omitted. For the proof of (iii) we refer to [18,Lemma 10.3.17]. Statement (vi) follows by Lemma 3.1 and the Weierstrass theorem.…”

Section: Resultsmentioning

confidence: 99%

Convexity and continuity of specific set-valued maps and their extremal value functions

2023

J. Appl. Numer. Optim.

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In this paper, we study several classes of set-valued maps, which can be used in set-valued optimization and its applications, and their respective maximum and minimum value functions. The definitions of these maps are based on scalar-valued, vector-valued, and cone-valued maps. Moreover, we consider those extremal value functions which are obtained when optimizing linear functionals over the image sets of the set-valued maps. Such extremal value functions play an important role for instance for derivative concepts for set-valued maps or for algorithmic approaches in set-valued optimization. We formulate conditions under which the set-valued maps and their extremal value functions inherit properties like (Lipschitz-)continuity and convexity.

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Section: Resultsmentioning

confidence: 99%

Convexity and continuity of specific set-valued maps and their extremal value functions

2023

J. Appl. Numer. Optim.

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“…This is true for A − St B but not for the modification proposed by Jahn using metric differences. For the study of such specific set-valued maps we also refer to [31].…”

Section: Relation To Jahn's Set Difference and Directional Derivativementioning

confidence: 99%

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

2020

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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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“…In case the functions f i are linear, then we have f Z (x) = {f (x)}+{f (z) ∈ R m | z ∈ Z}. This simplifies the problem significantly, see [10,Theorem 23] or [9]. In case the functions f i and the set Z are convex, the sets f Z (x) for x ∈ S do not have be convex, as the following example shows:…”

Section: Decision Uncertaintymentioning

confidence: 99%

An algorithmic approach to multiobjective optimization with decision uncertainty

2019

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In real life applications optimization problems with more than one objective function are often of interest. Next to handling multiple objective functions, another challenge is to deal with uncertainties concerning the realization of the decision variables. One approach to handle these uncertainties is to consider the objectives as set-valued functions. Hence, the image of one variable is a whole set, which includes all possible outcomes of this variable. We choose a robust approach and thus these sets have to be compared using the so called upper-type less order relation.We propose a numerical method to calculate a covering of the set of optimal solutions of such an uncertain multiobjective optimization problem. We use a branchand-bound approach and lower and upper bound sets for being able to compare the arising sets. The calculation of these lower and upper bound sets uses techniques known from global optimization as convex underestimators as well as techniques used in convex multiobjective optimization as outer approximation techniques. We also give first numerical results for this algorithm.

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On Classes of Set Optimization Problems which are Reducible to Vector Optimization Problems and its Impact on Numerical Test Instances

Cited by 5 publications

References 13 publications

Convexity and continuity of specific set-valued maps and their extremal value functions

Convexity and continuity of specific set-valued maps and their extremal value functions

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

An algorithmic approach to multiobjective optimization with decision uncertainty

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