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Conjugacy in Convex Analysis

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

Abstract

In classical real analysis, the gradient of a differentiable function f : ℝn → ℝ. plays a key role - to say the least. Considering this gradient as a mapping xs(x) = ∇f(x) from (some subset X of) ℝn to (some subset S of) ℝn, an interesting object is then its inverse: to a given sS, associate the xX such that s =f(x). This question may be meaningless: not all mappings are invertible! but could for example be considered locally, taking for X x S a neighborhood of some (x 0, s 0 = ∇f(x 0)), with ∇2 f continuous and invertible at x 0 (use the local inverse theorem).

Keywords

Support Function Convex Analysis Conjugate Function Strict Convexity Closed Convex Cone 
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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