A Quadratically Convergent Alternating Projection Method for Nonconvex Sets

TL;DR

This paper introduces a modified alternating projection method with quadratic convergence for solving feasibility problems involving nonconvex sets, combining Newton steps and projections under mild conditions.

Contribution

It proposes a novel alternating projection algorithm with local quadratic convergence for nonconvex feasibility problems, extending existing methods.

Findings

Method achieves quadratic convergence under mild conditions.

Numerical experiments show high efficiency of the proposed method.

Applicable to non-regular, nonconvex constraint 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.