A convergence framework for inexact nonconvex and nonsmooth algorithms and its applications to several iterations

In this paper, we consider the convergence of an abstract inexact nonconvex and nonsmooth algorithm. We promise a pseudo sufficient descent condition and a pseudo relative error condition, which are both related to an auxiliary sequence, for the algorithm; and a continuity condition is assumed to hold. In fact, a lot of classical inexact nonconvex and nonsmooth algorithms allow these three conditions. Under a special kind of summable assumption on the auxiliary sequence, we prove the sequence generated by the general algorithm converges to a critical point of the objective function if being assumed Kurdyka- Lojasiewicz property. The core of the proofs lies in building a new Lyapunov function, whose successive difference provides a bound for the successive difference of the points generated by the algorithm. And then, we apply our findings to several classical nonconvex iterative algorithms and derive the corresponding convergence results