A Quadratically Convergent Alternating Projection Method for Nonconvex Sets
Nachuan Xiao, Shiwei Wang, Tianyun Tang, Kim-Chuan Toh
TL;DR
This work tackles the nonconvex feasibility problem K = X ∩ M by introducing a constraint-dissolving alternating projection powered by a projective mapping Q and Clarke tangent cones. The authors prove local quadratic convergence under a nondegeneracy condition and develop a globally convergent algorithm Adap_AP2 that adaptively blends feasibility updates with projected-gradient steps. They extend the framework to a quadratically convergent Bregman proximal method for convex X and demonstrate strong empirical performance across diverse nonconvex scenarios, often outperforming NewtonSLRA and standard alternating projection. The results offer a practical, theory-backed approach for fast convergence to feasible points in challenging nonconvex settings, with broad applicability to low-rank, sparsity, and norm-constrained problems.
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.