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.

摘要

不精确的非凸和非光滑算法的序列收敛性是在噪声的不现实假设下证明的。在本文中，我们着重于消除假设。没有假设，该算法因此无法用先前的框架和技巧来证明。因此，我们建立了一个新的证明框架，该框架采用了伪充分的下降条件和伪相对误差条件两者都与辅助序列有关；并且假定连续性条件成立。实际上，许多经典的不精确的非凸和非光滑算法都允许这三个条件。在辅助序列的假设下，我们证明了一般算法生成的序列在假定为半代数性质的情况下收敛于目标函数的临界点。证明的核心在于构建一个新的Lyapunov函数，其连续差为算法生成的点的连续差提供了界限。然后，将我们的发现应用于不精确的非凸近端惯性梯度算法，并得出相应的收敛结果。