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A Duality Theory for Set-Valued Functions I: Fenchel Conjugation Theory

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Abstract

It is proven that a proper closed convex function with values in the power set of a preordered, separated locally convex space is the pointwise supremum of its set-valued affine minorants. A new concept of Legendre–Fenchel conjugates for set-valued functions is introduced and a Moreau–Fenchel theorem is proven. Examples and applications are given, among them a dual representation theorem for set-valued convex risk measures.

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Hamel, A.H. A Duality Theory for Set-Valued Functions I: Fenchel Conjugation Theory. Set-Valued Anal 17, 153–182 (2009). https://doi.org/10.1007/s11228-009-0109-0

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