Fenchel Duality and the Strong Conical Hull Intersection Property

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.