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Erschienen in:
A Comparison of Techniques for Dynamic Multivariate Risk Measures Kapitel lesen Erstes Kapitel lesen

2015 | OriginalPaper | Buchkapitel

Set Optimization—A Rather Short Introduction

verfasst von : Andreas H. Hamel, Frank Heyde, Andreas Löhne, Birgit Rudloff, Carola Schrage

Erschienen in: Set Optimization and Applications - The State of the Art

Verlag: Springer Berlin Heidelberg

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Abstract

Recent developments in set optimization are surveyed and extended including various set relations as well as fundamental constructions of a convex analysis for set- and vector-valued functions, and duality for set optimization problems. Extensive sections with bibliographical comments summarize the state of the art. Applications to vector optimization and financial risk measures are discussed along with algorithmic approaches to set optimization problems.

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Vorheriges Kapitel Nonlinear Scalarizations of Set Optimization Problems with Set Orderings
Nächstes Kapitel A Survey of Set Optimization Problems with Set Solutions
Fußnoten
1
In contrast to many duality results in vector optimization, this can bee seen as a realization of one of the many ‘duality principles in optimization theory that relate a problem expressed in terms of vectors in a space to a problem expressed in terms of hyperplanes in the space,’ see [160, p. 8].
 
2
For apparent reasons, we would like to call this just “set optimization,” but this term is currently used for just too many other purposes.
 
3
R. T. Rockafellar and R.-B. Wets also remark on p. 15 of [197] that the second distributivity law does not extend to all of \(\overline{\mathrm {I\negthinspace R}}\) which is another motivation for the concept of “conlinear” spaces. Finally, it is interesting to note that the authors of [197] consider it a matter of cause to associate minimization with inf-addition (see p. 15). In the set optimization community, there is no clear consensus yet about which relation to use in what context and for what purpose. However, this note makes a clear point towards [197]: associate \(\preccurlyeq _C\) with minimization and \(\curlyeqprec _C\) with maximization because the theory works for these cases. One should have a very strong reason for doing otherwise and be advised that in this case many standard mathematical tools just don’t work.
 
4
Sophie Qingzhen Wang provided the hint to this observation.
 
5
‘In fact the great watershed in optimization isn’t between linearity and nonlinearity, but convexity and nonconvexity’.
 
6
‘Theoretically, what modern optimization can solve well are convex optimization problems’.
 
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Metadaten
Titel
Set Optimization—A Rather Short Introduction
verfasst von
Andreas H. Hamel
Frank Heyde
Andreas Löhne
Birgit Rudloff
Carola Schrage
Copyright-Jahr
2015
Verlag
Springer Berlin Heidelberg
Buch
Set Optimization and Applications - The State of the Art

Print ISBN: 978-3-662-48668-9

Electronic ISBN: 978-3-662-48670-2

Copyright-Jahr: 2015

DOI
https://doi.org/10.1007/978-3-662-48670-2_3