Set Optimization—A Rather Short Introduction

Hamel, Andreas H.; Heyde, Frank; Löhne, Andreas; Rudloff, Birgit; Schrage, Carola

doi:10.1007/978-3-662-48670-2_3

Cited by 71 publications

(83 citation statements)

References 190 publications

(265 reference statements)

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“…For another solution notion, we refer the reader to [3], where solution denotes a special set consisting of solutions in the sense of Definition 2.4 w.r.t. the l-less order relation.…”

Section: Definition 24mentioning

confidence: 99%

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Optimality Conditions for Set-Valued Optimisation Problems Using a Modified Demyanov Difference

Dempe

Pilecka

2015

J Optim Theory Appl

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The aim of this paper was to provide optimality conditions for set-valued optimisation problems with respect to the set less order relation. For this purpose, we use a slightly modified Demyanov difference in order to introduce a sort of directional derivative for set-valued maps, which allows us to derive optimality conditions. Some results on existence and boundedness of the directional derivative are also given.

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“…For another solution notion, we refer the reader to [3], where solution denotes a special set consisting of solutions in the sense of Definition 2.4 w.r.t. the l-less order relation.…”

Section: Definition 24mentioning

confidence: 99%

“…Therefore, only one element may decide about optimality of the whole set [2]. In the lattice approach, conlinear spaces are considered which carry the structure of a complete lattice [3]. In this paper, we use the so-called set approach and exploit one of the well-known set relations [2,4]-the set less order relation (cf.…”

Section: Introductionmentioning

confidence: 99%

Optimality Conditions for Set-Valued Optimisation Problems Using a Modified Demyanov Difference

Dempe

Pilecka

2015

J Optim Theory Appl

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“…In this generality, the preorder l was probably discussed for the first time by Hamel in [9, (2.6)], at least in the context of variational analysis/optimization. However, there are many precursors: the interested reader is referred to [10] for a more thorough survey and detailed references.…”

Section: The Brezis-browder Principlementioning

confidence: 99%

Minimal element theorems revisited

Hamel

Zălinescu

2020

Journal of Mathematical Analysis and Applications

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Starting with the Brezis-Browder principle, we give stronger versions of many variational principles and minimal element theorems which appeared in the recent literature. Relationships among the elements of different sets of assumptions are discussed and clarified, i.e., assumptions to the metric structure of the underlying space and boundedness assumptions. New results involving set-valued maps and the increasingly popular set relations are obtained along the way. applications consist in a mere check if the order relation in question satisfies the required assumptions which links it to the metric structure. Therefore, the mentioned results can be proven without involving the Brezis-Browder principle as shown in the alternative proof of Theorem 3.7 below.The second line of research is concerned with order relations on arbitrary sets (without a metric structure, for example). Of course, alternative requirements have to be added. A prominent result of this type is the Brezis-Browder principle [1] (BB-principle) where the existence of a real-valued function is assumed which is bounded from below and increasing wrt the order relation. Further examples for such results can be found in [25,27] and also in [31]. Of course, BB-type theorems can also be used to obtain the results on metric spaces discussed in the previous paragraph with a suitable monotone function; the proof of the BB-principle also involves a countable induction argument (see the first proof of Theorem 3.2 below).A third goal is to lift the results from order relations on a set X to such relations on a product set X × Z. Results in this direction have been obtained first by Göpfert and Tammer (see [5, Section 3.10] and the references 140-144 therein) and are usually called minimal element theorems.In this note, all three of the above lines are followed providing very general results and discussing the relationships between different sets of assumptions in detail. Consequently, most relevant results in the literature are obtained as special cases. In particular, it is shown that variational principles and minimal element theorems involving (very general) set relations can be obtained (see, for example, Corollary 3.14 below). Such results can be found in [8] for complete metric spaces (manuscript version [7] from 2002) and more general ones in [12] (preprint version [11] from 2002), the latter reference already giving a proof based on the BB-principle and nonlinear scalarization.

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“…For this paper, the main results simplify when using the dual representation w.r.t. the positive convex conjugate introduced in [36]. This dual representation will be deduced below.…”

Section: Dual Representationmentioning

confidence: 99%

“…Proof of Theorem 2.9. Recall from [36,Remark 5.5] that the conjugate and negative conjugate are related via β t (Q, w) = G M t (w)− . (−β t (Q, w)) and −β t (Q, w) = G M t (w)− .…”

Section: B2 Proof Of Theorem 29mentioning

confidence: 99%

A supermartingale relation for multivariate risk measures

Feinstein

Rudloff

2018

Quantitative Finance

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The equivalence between multiportfolio time consistency of a dynamic multivariate risk measure and a supermartingale property is proven. Furthermore, the dual variables under which this set-valued supermartingale is a martingale are characterized as the worst-case dual variables in the dual representation of the risk measure. Examples of multivariate risk measures satisfying the supermartingale property are given. Crucial for obtaining the results are dual representations of scalarizations of set-valued dynamic risk measures, which are of independent interest in the fast growing literature on multivariate risks.

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Set Optimization—A Rather Short Introduction

Cited by 71 publications

References 190 publications

Optimality Conditions for Set-Valued Optimisation Problems Using a Modified Demyanov Difference

Optimality Conditions for Set-Valued Optimisation Problems Using a Modified Demyanov Difference

Minimal element theorems revisited

A supermartingale relation for multivariate risk measures

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