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Revisiting some Rules of Convex Analysis

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Abstract

Convex analysis is devoted to the study and the use of four notions: conjugate functions, normal cones, subdifferentials, support functions under convexity assumptions. These notions are closely related and the calculus rules for one of them imply calculus rules for the other ones. But for none of these notions such rules are always valid without additional assumptions. Thus, much effort has been devoted to these “qualification conditions” (Attouch-Brézis, Boţ and his co-authors, Burachik-Jeyakumar, Rockafellar...). We introduce a new condition that requires a pointwise outer semicontinuity (or closedness) property of an appropriate multifunction. For subdifferentials (resp. normal cones), the multifunction is defined in terms of subdifferentials (resp. normal cones). We relate the new condition with previous ones from the literature.

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Correspondence to Jean-Paul Penot.

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Dedicated to Michel Théra on the occasion of his seventieth birthday

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Penot, J. Revisiting some Rules of Convex Analysis. Set-Valued Var. Anal 25, 773–788 (2017). https://doi.org/10.1007/s11228-017-0462-3

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Keywords

  • Composition rule
  • Convexity
  • Normal cone
  • Qualification conditions
  • Regularity
  • Sum rule
  • Subdifferential

Mathematics Subject Classification (2010)

  • 26B25
  • 52A05
  • 26B40
  • 49K99
  • 90C25