In this paper, we consider the constraint set K := {x ∈ Rn : 1 j(x) ≤ 0, ∀ j = 1, 2, . . . ,m} of inequalities with nonsmooth nonconvex constraint functions 1 j : Rn −→ R ( j = 1, 2, · · · ,m). We show that under Abadie’s constraint qualification the “perturbation property“ of the best approximation to any x in Rn from a convex set K˜ := C ∩ K is characterized by the strong conical hull intersection property (strong CHIP) of C and K, where C is an arbitrary non-empty closed convex subset of Rn. By using the idea of tangential subdifferential and a non-smooth version of Abadie’s constraint qualification, we do this by first proving a dual cone characterization of the constraint set K. Moreover, we present sufficient conditions for which the strong CHIP property holds. In particular, when the set K˜ is closed and convex, we show that the Lagrange multiplier characterizations of constrained best approximation holds under a non-smooth version of Abadie’s constraint qualification. The obtained results extend many corresponding results in the context of constrained best approximation. Several examples are provided to clarify the results