Sequence Convergence of Inexact Nonconvex and Nonsmooth Algorithms with More Realistic Assumptions

Abstract

The sequence convergence of inexact nonconvex and nonsmooth algorithms is proved with an unrealistic assumption on the noise. In this paper, we focus on removing the assumption. Without the assumption, the algorithm consequently cannot be proved with previous framework and tricks. Thus, we build a new proof framework which employs a pseudo sufficient descent condition and a pseudo relative error condition both related to an auxiliary sequence; and a continuity condition is assumed to hold. In fact, a lot of classical inexact nonconvex and nonsmooth algorithms allow these three conditions. Under an 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 semi-algebraic 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 the inexact nonconvex proximal inertial gradient algorithm and derive the corresponding convergence results.