- Title
- Necessary and sufficient conditions for stable conjugate duality
- Creator
- Burachik, Regina; Jeyakumar, Vaithilingam; Wu, Zhiyou
- Date
- 2006
- Type
- Text; Journal article
- Identifier
- http://researchonline.federation.edu.au/vital/access/HandleResolver/1959.17/33778
- Identifier
- vital:152
- Identifier
-
https://doi.org/10.1016/j.na.2005.07.034
- Identifier
- ISSN:0362-546X
- Abstract
- The conjugate duality, which states that infx∈X φ(x, 0) = maxv∈Y ' −φ∗(0,v), whenever a regularity condition on φ is satisfied, is a key result in convex anal¬ysis and optimization, where φ : X × Y → IR ∪{+∞} is a convex function, X and Y are Banach spaces, Y ' is the continuous dual space of Y and φ∗ is the Fenchel-Moreau conjugate of φ. In this paper, we establish a necessary and sufficient condition for the stable conjugate duality, ∗ ∗ ∈ X' inf {φ(x, 0) + x ∗(x)} = max {−φ ∗(−x ,v)}, ∀x, x∈Xv∈Y ' and obtain a new global dual regularity condition, which is much more general than the popularly known interior-point type conditions, for the conjugate duality. As a consequence we present an epigraph closure condition which is necessary and sufficient for a stable Fenchel-Rockafellar duality theorem. In the case where one of the functions involved in the duality is a polyhedral convex function, we also provide generalized interior-point conditions for the epigraph closure condition. Moreover, we show that a stable Fenchel’s duality for sublinear functions holds whenever a subdifferential sum formula for the functions holds. As applications, we give general sufficient conditions for a minimax theorem, a subdifferential composition formula and for duality results of convex programming problems.; C1
- Publisher
- Elsevier
- Relation
- Journal of Nonlinear Analysis Vol. 64, no. 9 (2006), p. 1998-2005
- Rights
- Copyright Elsevier
- Rights
- Open Access
- Rights
- This metadata is freely available under a CCO license
- Subject
- 0101 Pure Mathematics; Conjugate duality; Constraint qualification; Convex programming; Polyhedral functions; Sublinear functions
- Full Text
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