In this paper we consider the cubic regularization (CR) method for minimizing
a twice continuously differentiable function. While the CR method is widely
recognized as a globally convergent variant of Newton's method with superior
iteration complexity, existing results on its local quadratic convergence
require a stringent non-degeneracy condition. We prove that under a local error
bound (EB) condition, which is much weaker a requirement than the existing
non-degeneracy condition, the sequence of iterates generated by the CR method
converges at least Q-quadratically to a second-order critical point. This
indicates that adding a cubic regularization not only equips Newton's method
with remarkable global convergence properties but also enables it to converge
quadratically even in the presence of degenerate solutions. As a byproduct, we
show that without assuming convexity, the proposed EB condition is equivalent
to a quadratic growth condition, which could be of independent interest. To
demonstrate the usefulness and relevance of our convergence analysis, we focus
on two concrete nonconvex optimization problems that arise in phase retrieval
and low-rank matrix recovery, respectively, and prove that with overwhelming
probability, the sequence of iterates generated by the CR method for solving
these two problems converges at least Q-quadratically to a global minimizer. We
also present numerical results of the CR method when applied to solve these two
problems to support and complement our theoretical development.

介绍了三次正则化方法及其改进的全局收敛性和局部二次收敛性。同时，介绍了一种弱化的错误界条件来证明方法的局部二次收敛性，使得该方法能处理出现退化情况的问题，并将其应用于相位恢复和低秩矩阵恢复的非凸优化问题。