On the Differentiability of Saddle and Biconvex Functions and Operators

Abstract

We strengthen and generalize results of J. M.

Borwein [Partially monotone operators and the generic differentiability of convex-concave and biconvex mappings, Israel J. Math. 54 (1986) 42-50] and of A.

Ioffe and R. E.

Lucchetti [Typical convex program is very well posed, Math. Program. 104 (2005) 483-499] on Frechet and Gateaux 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 Frechet differentiable except for a G-null, s-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