Constrained Best Approximation with Nonsmooth Nonconvex Constraints

In this paper, we consider the constraint set $K$ of inequalities with nonsmooth nonconvex constraint functions. We show that under Abadie's constraint qualification the "perturbation property" of the best approximation to any $x$ in $\R^n$ from a convex set \tK:=C \cap K is characterized by the strong conical hull intersection property (strong CHIP) of $C$ and $K,$ where $C$ is a non-empty closed convex subset of $\R^n$ and the set $K$ is represented by $K:=\{x\in \R^n : g_j(x) \le 0, \ \forall \ j=1,2,\ldots,m \}$ with g_j : \R^n \lrar \R $(j=1,2, \cdots,m)$ is a tangentially convex function at a given point $\bar x \in K.$ 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 \tK is closed and convex, we show that the Lagrange multiplier characterization of 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.Comment: 20 pages; MS# 19-041 (2019