This work extends the iterative framework proposed by Attouch et al. (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, a relative error condition and a subsequence condition, where the first two conditions involve a parameter \(p\!>0\) . We prove that any sequence conforming to this framework is globally convergent when \(\Phi\) is a Kurdyka–Łojasiewicz (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 \(\theta \in (0,\frac{p}{p+1})\) . Then, we demonstrate that the iterate sequence generated by an inexact q-order regularized proximal Newton method with \(q\in [2,3]\) for composite optimization problems falls into this framework, and first achieve the Q-superlinear convergence rate of order 4/3 for an inexact cubic regularization method to solve this class of nonconvex and nonsmooth problems with KL property of exponent 1/2.
