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subdifferentiable mapping

(Definition)

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

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

If $\partial f(x)$ is non-empty, then

is subdifferentiable at $x \in X$ , and if $\partial f(x)$ is non-empty for all

, then

is subdifferentiable [1,2].

Bibliography

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

"subdifferentiable mapping" is owned by matte. [ full author list (2) | owner history (1) ]

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Cross-references: function, dual space, Banach space
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This is version 10 of subdifferentiable mapping, born on 2004-08-03, modified 2006-10-15.
Object id is 6063, canonical name is SubDifferentiableMapping.
Accessed 1239 times total.

Classification:

AMS MSC:	52-00 (Convex and discrete geometry :: General reference works )
	39B62 (Difference and functional equations :: Functional equations and inequalities :: Functional inequalities, including subadditivity, convexity, etc.)

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