# Constrained Best Approximation with Nonsmooth Nonconvex Constraints

## TL;DR

This paper investigates the best approximation problem under nonsmooth nonconvex constraints, characterizing the perturbation property via strong CHIP and extending classical results with new dual and Lagrange multiplier conditions.

## Contribution

It introduces a dual cone characterization and sufficient conditions for strong CHIP in nonsmooth nonconvex constraints, extending existing approximation theory.

## Key findings

- Characterization of perturbation property via strong CHIP.
- Dual cone representation of the constraint set.
- Lagrange multiplier conditions under non-smooth Abadie's qualification.

## Abstract

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.