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Subdifferentials of Finite Convex Functions

  • Jean-Baptiste Hiriart-Urruty
  • Claude Lemaréchal
Part of the Grundlehren Text Editions book series (TEXTEDITIONS)

Abstract

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
$$ f(x + h) = f(x) + f'(x,h) + o(||h||). $$
(0.1)
exists and is finite sublinear. Furthermore, fapproximates f around x in the sense that
$$ d \mapsto f'(x, d): = \mathop {\lim }\limits_{t \downarrow 0} \frac{{f(x + td) - f(x)}} {t}$$

Keywords

Convex Function Normal Cone Support Function Directional Derivative Cluster Point 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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Copyright information

© Springer-Verlag Berlin Heidelberg 2001

Authors and Affiliations

  • Jean-Baptiste Hiriart-Urruty
    • 1
  • Claude Lemaréchal
    • 2
  1. 1.Département de MathématiquesUniversité Paul SabatierToulouseFrance
  2. 2.INRIA, Rhône AlpesZIRSTMontbonnotFrance

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