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Set Optimization—A Rather Short Introduction

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Set Optimization and Applications - The State of the Art

Part of the book series: Springer Proceedings in Mathematics & Statistics ((PROMS,volume 151))

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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Notes

  1. 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. 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. 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. 4.

    Sophie Qingzhen Wang provided the hint to this observation.

  5. 5.

    ‘In fact the great watershed in optimization isn’t between linearity and nonlinearity, but convexity and nonconvexity’.

  6. 6.

    ‘Theoretically, what modern optimization can solve well are convex optimization problems’.

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Acknowledgments

Andreas H. Hamel gratefully acknowledges the support of Free University of Bozen-Bolzano via a generous start-up grant through the academic years 2013/14 and 2014/15.

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  1. Free University Bolzano-Bozen, Bolzano, Italy

    Andreas H. Hamel

  2. TU Bergakademie Freiberg, Freiberg, Germany

    Frank Heyde

  3. Friedrich Schiller University Jena, Jena, Germany

    Andreas Löhne

  4. Vienna University of Economics and Business, Vienna, Austria

    Birgit Rudloff

  5. Free University of Bolzano-Bozen, Bolzano, Italy

    Carola Schrage

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    Andreas H Hamel

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    Frank Heyde

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    Andreas Löhne

  4. and Business, Vienna University of Economics, Vienna, Austria

    Birgit Rudloff

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    Carola Schrage

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Hamel, A.H., Heyde, F., Löhne, A., Rudloff, B., Schrage, C. (2015). Set Optimization—A Rather Short Introduction. In: Hamel, A., Heyde, F., Löhne, A., Rudloff, B., Schrage, C. (eds) Set Optimization and Applications - The State of the Art. Springer Proceedings in Mathematics & Statistics, vol 151. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-662-48670-2_3

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