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 ${\mathbb{R}}^n$ from a convex set $\text{tK}:=C\cap K$ 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 ${\mathbb{R}}^n$ and the set $K$ is represented by $K:=\{x\in {\mathbb{R}}^n:g_j(x)\le 0,\mathrm{\forall }j=1,2,\dots ,m\}$ with $g_j:{\mathbb{R}}^n\text{lrar}\mathbb{R}$ $(j=1,2,\cdots ,m)$ is a tangentially convex function at a given point $\bar{x}\in K\mathrm{.}$ 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 $K\mathrm{.}$ Moreover, we present sufficient conditions for which the strong CHIP property holds. In particular, when the set $\text{tK}$ 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.