Abstract

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.

Author information

Contact details are reproduced from the original publication and may be historical.

Libor Veselý

Dip. di Matematica, Università degli Studi, Via C. Saldini 50, 20133 Milano, Italy

libor.vesely@unimi.it

Keywords

  • Saddle function
  • convex-concave operator
  • biconvex function
  • biconvex operator
  • partially convex operator
  • Fréchet differentiability
  • Gâteaux differentiability
  • strict differentiability

Mathematics Subject Classification

46G05; 49J50, 26B25

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Suggested citation

L. Veselý, L. Zajíček. “On the Differentiability of Saddle and Biconvex Functions and Operators.” Journal of Convex Analysis 27 (2020), No. 2, 705–731. https://doi.org/10.68381/jca27036

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