A convergence framework for inexact nonconvex and nonsmooth algorithms and its applications to several iterations
It provides a general convergence theory for a broad class of inexact nonconvex optimization algorithms, but the results are theoretical and incremental over existing analyses.
The paper develops a unified convergence framework for inexact nonconvex and nonsmooth algorithms by introducing pseudo descent and relative error conditions, proving convergence to critical points under the Kurdyka-Łojasiewicz property, and applying it to several classical 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