Limiting ε-subgradient characterizations of constrained best approximation

In this paper, we show that the strong conical hull intersection property (CHIP) completely characterizes the best approximation to any x in a Hilbert space X from the setK:=C∩x∈X:-g(x)∈S,by a perturbation x-l of x from the set C for some l in a convex cone of X, where C is a closed convex subset of X, S is a closed convex cone which does not necessarily have non-empty interior, Y is a Banach space and g:X→Y is a continuous S-convex function. The point l is chosen as the weak*-limit of a net of ɛ-subgradients. We also establish limiting dual conditions characterizing the best approximation to any x in a Hilbert space X from the set K without the strong CHIP. The ε-subdifferential calculus plays the key role in deriving the results.