Fenchel Duality and the Strong Conical Hull Intersection Property

F. Deutsch, W. Li and J. Swetits
Additional contact information
F. Deutsch: Pennsylvania State University
W. Li: Old Dominion University
J. Swetits: Old Dominion University

Journal of Optimization Theory and Applications, 1999, vol. 102, issue 3, No 10, 695 pages

Abstract: Abstract We study a special dual form of a convex minimization problem in a Hilbert space, which is formally suggested by Fenchel dualityand is useful for the Dykstra algorithm. For this special duality problem, we prove that strong duality holds if and only if the collection of underlying constraint sets {C 1,...,C m} has the strong conical hull intersection property. That is, $$\left( {\mathop \cap \limits_1^m C_i - x)^ \circ = \sum\limits_1^m {(C_1 - x} } \right)^ \circ {\text{, for each }}x \in \mathop \cap \limits_1^m C_1$$ where D° denotes the dual cone of D. In general, we can establish weak duality for a convex minimization problem in a Hilbert space by perturbing the constraint sets so that the perturbed sets have the strong conical hull intersection property. This generalizes a result of Gaffke and Mathar.

Keywords: Convex optimization; Fenchel duality; best approximation in Hilbert space; strong conical hull intersection property (search for similar items in EconPapers)
Date: 1999
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Citations: View citations in EconPapers (2)

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DOI: 10.1023/A:1022658308898

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