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Journal of Convex Analysis 27 (2020), No. 2, 705--731
Copyright Heldermann Verlag 2020

On the Differentiability of Saddle and Biconvex Functions and Operators

Libor Vesely
Dip. di Matematica, Universitŕ degli Studi, Via C. Saldini 50, 20133 Milano, Italy

Ludek Zajícek
Faculty of Mathematics and Physics, Charles University, 186 75 Praha 8, Czech Republic

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.

Keywords: Saddle function, convex-concave operator, biconvex function, biconvex operator, partially convex operator, Frechet differentiability, Gateaux differentiability, strict differentiability.

MSC: 46G05; 49J50, 26B25.

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