A superlinear convergence iterative framework for Kurdyka-{\L}ojasiewicz optimization and application

This work extends the iterative framework proposed by Attouch et al. (in Math. Program. 137: 91-129, 2013) for minimizing a nonconvex and nonsmooth function $\Phi$ so that the generated sequence possesses a Q-superlinear convergence rate. This framework consists of a monotone decrease condition and a relative error condition, both involving a parameter $p\!>0$. We justify that any sequence conforming to this framework is globally convergent when $\Phi$ is a KL function, and the convergence has a Q-superlinear rate of order $\frac{p}{\theta(1+p)}$ when $\Phi$ is a KL function of exponent $\theta\in(0,\frac{p}{p+1})$. In particular, we propose an inexact proximal Newton method with a $q\in[2,3]$-order regularization term for nonconvex and nonsmooth composite problems and show that its iterate sequence belongs to this framework, and consequently first achieve the Q-superlinear convergence rate of order $4/3$ for an inexact cubic regularization method to solve this class of composite problems with KL property of exponent $1/2$.