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Subdifferentials of Finite Convex Functions

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Fundamentals of Convex Analysis

Part of the book series: Grundlehren Text Editions ((TEXTEDITIONS))

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We have mentioned in our preamble to Chap. C that sublinearity permits the approximation of convex functions to first order around a given point. In fact, we will show here that, if f : ℝn → ℝ is convex and x ∈ ℝn is fixed, then the function

$$ f(x + h) = f(x) + f'(x,h) + o(||h||). $$

exists and is finite sublinear. Furthermore, fapproximates f around x in the sense that

$$ d \mapsto f'(x, d): = \mathop {\lim }\limits_{t \downarrow 0} \frac{{f(x + td) - f(x)}} {t}$$

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© 2001 Springer-Verlag Berlin Heidelberg

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Hiriart-Urruty, JB., Lemaréchal, C. (2001). Subdifferentials of Finite Convex Functions. In: Fundamentals of Convex Analysis. Grundlehren Text Editions. Springer, Berlin, Heidelberg.

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  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-42205-1

  • Online ISBN: 978-3-642-56468-0

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