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Essential smoothness, essential strict convexity, and Legendre functions in Banach spaces

Bauschke, Heinz H. and Borwein, Jonathan M. and Combettes, Patrick L. (2001) Essential smoothness, essential strict convexity, and Legendre functions in Banach spaces. Communications in Contemporary Mathematics, 3 . pp. 615-648.

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      Abstract

      The classical notions of essential smoothness, essential strict convexity, and Legendreness for convex functions are extended from Euclidean to Banach spaces. A pertinent duality theory is developed and several useful characterizations are given.The proofs rely on new results on the more subtle behavior of subdifferentials and directional derivatives at boundary points of the domain. In weak Asplund spaces, a new formula allows the recovery of the subdifferential from nearby gradients.Finally, it is shown that every Legendre function on a reflexive Banach space is zone consistent, a fundamental property in the analysis of optimization algorithms based on Bregman distances. Numerous illustrating examples are provided.

      Item Type: Article
      Additional Information: pubdom FALSE
      Uncontrolled Keywords: Bregman distance, Bregman projection, coercive, cofinite function, convex function of Legendre type, essentially smooth, essentially strictly convex, Legendre function, Schur property, Schur space, subdifferential, supercoercive, weak Asplund space, zone consistent
      Subjects: 90-xx Economics, operations research, programming, games > 90Cxx Mathematical programming
      49-xx Calculus of variations and optimal control; optimization > 49Jxx Existence theories
      46-xx Functional analysis > 46Nxx Miscellaneous applications of functional analysis
      46-xx Functional analysis > 46Gxx Measures, integration, derivative, holomorphy (all involving infinite-dimensional spaces)
      52-xx Convex and discrete geometry > 52Axx General convexity
      Faculty: UNSPECIFIED
      Depositing User: Users 1 not found.
      Date Deposited: 28 Nov 2003
      Last Modified: 28 Sep 2014 14:32
      URI: https://docserver.carma.newcastle.edu.au/id/eprint/232

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