A convergence framework for inexact nonconvex and nonsmooth algorithms and its applications to several iterations
This provides a theoretical foundation for analyzing convergence in nonconvex optimization, which is incremental as it extends existing frameworks to more general conditions.
The paper tackles the convergence analysis of inexact nonconvex and nonsmooth algorithms by establishing a framework with pseudo sufficient descent, pseudo relative error, and continuity conditions, proving convergence to critical points under Kurdyka-Lojasiewicz property and a summable assumption, and applies it to classical iterative algorithms.
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