A Global Approach to Nonlinearly Constrained Best Approximation
V. Jeyakumar, H. Mohebi · Numerical Functional Analysis and Optimization · 2005
In this paper, we study the problem of whether the best approximation to any x in a Hilbert space X from the set where C is a closed convex subset of X, S is a closed convex cone that does not necessarily have nonempty interior, Y is a Banach space, and g : X → Y is a continuous S-convex function, can be characterized by the best approximation to a perturbation x − l of x from the set C for some l ∈ X. We provide a global approach to this problem by presenting a dual global constraint qualification, which is less restrictive than the Slater type (or interior-point) conditions, guaranteeing the strong conical hull intersection property (CHIP). We then show that the strong CHIP characterizes the perturbation property under a mild closure condition. The closure condition, for instance, holds whenever the explicit constraint set is described by finitely many linear inequality constraints. We also establish easily verifiable dual conditions that are equivalent to the best approximation from the set K. We finally demonstrate that our results recapture the corresponding known results in the particular case where Y is a finite dimensional space.