Scalable nonconvex inexact proximal splitting

We study large-scale, nonsmooth, nonconconvex optimization problems. In particular, we focus on nonconvex problems with \emph{composite} objectives. This class of problems includes the extensively studied convex, composite objective problems as a special case. To tackle composite nonconvex problems, we introduce a powerful new framework based on asymptotically \emph{nonvanishing} errors, avoiding the common convenient assumption of eventually vanishing errors. Within our framework we derive both batch and incremental nonconvex proximal splitting algorithms. To our knowledge, our framework is first to develop and analyze incremental \emph{nonconvex} proximal-splitting algorithms, even if we disregard the ability to handle nonvanishing errors. We illustrate our theoretical framework by showing how it applies to difficult large-scale, nonsmooth, and nonconvex problems.