Geodesic
On the Differentiability of Saddle and Biconvex Functions and Operators
Journal of convex analysis, Tome 27 (2020) no. 2, pp. 705-731
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We strengthen and generalize results of J. M. Borwein [ Generic differentiability of order-bounded convex operators, J. Austral. Math. Soc. Ser. B 28 (1986) 22--29] and of A. Ioffe and R. E. Lucchetti [Typical convex program is very well posed, Math. Program. 104 (2005) 483--499] on Fréchet and Gâteaux differentiability of saddle and biconvex functions (and operators). For example, we prove that in many cases (also in some cases which were not considered before) these functions (and operators) are Fréchet differentiable except for a Γ-null, σ-lower porous set. Moreover, we prove these results for more general "partially convex (up or down)" functions and operators defined on the product of n Banach spaces.
Classification : 46G05, 49J50, 26B25
Mots-clés : Saddle function, convex-concave operator, biconvex function, biconvex operator, partially convex operator, Frechet differentiability, Gateaux differentiability, strict differentiability 
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     author = {L. Vesely and L. Zaj{\'\i}cek},
     title = {On the {Differentiability} of {Saddle} and {Biconvex} {Functions} and {Operators}},
     journal = {Journal of convex analysis},
     pages = {705--731},
     year = {2020},
     volume = {27},
     number = {2},
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L. Vesely; L. Zajícek. On the Differentiability of Saddle and Biconvex Functions and Operators. Journal of convex analysis, Tome 27 (2020) no. 2, pp. 705-731. http://geodesic.mathdoc.fr/item/JCA_2020_27_2_JCA_2020_27_2_a14/