Fenchel duality and the strong conical hull intersection property

We study a special dual form of a convex minimization problem in a Hilbert space, which is formally suggested by Fenchel duality and 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, (boolean AND(1)(m) C-i-x)degrees = Sigma(1)(m) (C-i-x)degrees, for each x is an element of boolean AND(1)(m) C-i. where D degrees 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 (see Ref. 1).