Subdifferentials of Finite Convex Functions

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


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


Convex Function Normal Cone Support Function Directional Derivative Cluster Point 
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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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