subdifferentiable mapping

Let $X$ be a Banach space, and let $X^{*}$ be the dual space of $X$ . For a function $f\colon X\rightarrow\mathbb{R}$ , and $x\in X$ , let us define

$\partial f(x)=\{r^{*}\in X^{*}\;:f(x)-f(y)\leq r^{\ast}(x-y)\;\ \mbox{for all}% \ y\in X\}.$

If $\partial f(x)$ is non-empty, then $f$ is subdifferentiable at $x\in X$ , and if $\partial f(x)$ is non-empty for all $x$ , then $f$ is subdifferentiable [1, 2].

References

1 C. Zalinescu, Convex Analysis in General Vector Spaces, World Scientific Publishing Company, 2002.
2 R.T. Rockafellar, Convex Analysis, Princeton University Press, 1996.

Title	subdifferentiable mapping
Canonical name	SubdifferentiableMapping
Date of creation	2013-03-22 14:31:19
Last modified on	2013-03-22 14:31:19
Owner	matte (1858)
Last modified by	matte (1858)
Numerical id	13
Author	matte (1858)
Entry type	Definition
Classification	msc 39B62
Classification	msc 52-00