Limiting [formula omitted]-subgradient characterizations of constrained best approximation

Abstract

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 set K := 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.