A Quadratically Convergent Alternating Projection Method for Nonconvex Sets

Abstract

In this paper, we consider the feasibility problem, which aims to find a feasible point for the constraint set $\{x \in \mathbb{R}^n: c(x) = 0\}$ over a possibly non-regular subset $\mathcal{X} \subset \mathbb{R}^n$. Under the constraint nondegeneracy condition, we propose a modified alternating projection method. In our proposed method, based on the concept of projective mapping for $\mathcal{X}$, we alternate a Newton step for finding an inexact solution within the limiting tangent cone of $\mathcal{X}$ and a projection to $\mathcal{X}$. Under mild conditions, we prove the local quadratic convergence of our proposed method. Preliminary numerical experiments demonstrate the high efficiency of our proposed alternating projection method.