In this paper, we consider the constraint set K := {x ? Rn : gj(x)? 0,? j =
1,2,...,m} of inequalities with nonsmooth nonconvex constraint
functions gj : 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.
Lagrange multiplier characterizations of constrained best approximation with nonsmooth nonconvex constraints
