- Title
- Essential smoothness, essential strict convexity, and Legendre functions in Banach spaces
- Creator
- Bauschke, Heinz H.; Borwein, Jonathan M.; Combettes, Patrick L.
- Relation
- Communications in Contemporary Mathematics Vol. 3, Issue 4, p. 615-647
- Publisher Link
- http://dx.doi.org/10.1142/S0219199701000524
- Publisher
- World Scientific Publishing
- Resource Type
- journal article
- Date
- 2001
- Description
- 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.
- Subject
- convex function of Legendre type; cofinite function; subdifferential; supercoercive; weak Asplund space; zone consistent; coercive; Bregman projection; Bregman distance; essentially smooth; essentially strictly convex; Legendre function; Schur property; Schur space
- Identifier
- http://hdl.handle.net/1959.13/940658
- Identifier
- uon:13061
- Identifier
- ISSN:0219-1997
- Language
- eng
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