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