Geodesic
Notes on Extended Real- and Set-Valued Functions
Journal of convex analysis, Tome 19 (2012) no. 2, pp. 355-384
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An order theoretic and algebraic framework for the extended real numbers is established which includes extensions of the usual difference to expressions involving -∞ and/or +∞, so-called residuations. New definitions and results for directional derivatives, subdifferentials and Legendre--Fenchel conjugates for extended real-valued functions are given which admit to include the proper as well as the improper case. For set-valued functions, scalar representation theorems and a new conjugation theory are established. The common denominator is that the appropriate image spaces for set-valued functions share fundamental structures with the extended real numbers: They are order complete, residuated monoids with a multiplication by non-negative real numbers.
Classification : 49N15, 54C60, 90C46
Mots-clés : Extended real-valued functions, directional derivative, subdifferential, Fenchel conjugate, set-valued function, conlinear space, infimal convolution 
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     author = {A. H. Hamel and C. Schrage},
     title = {Notes on {Extended} {Real-} and {Set-Valued} {Functions}},
     journal = {Journal of convex analysis},
     pages = {355--384},
     year = {2012},
     volume = {19},
     number = {2},
     url = {http://geodesic.mathdoc.fr/item/JCA_2012_19_2_JCA_2012_19_2_a2/}
}
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A. H. Hamel; C. Schrage. Notes on Extended Real- and Set-Valued Functions. Journal of convex analysis, Tome 19 (2012) no. 2, pp. 355-384. http://geodesic.mathdoc.fr/item/JCA_2012_19_2_JCA_2012_19_2_a2/