Minimization of nonsmooth nonconvex functions using inexact evaluations and its worst-case complexity

Abstract: An adaptive regularization algorithm using inexact function and derivatives evaluations is proposed for the solution of composite nonsmooth nonconvex optimization. It is shown that this algorithm needs at most $O(|\log(\epsilon)|\,\epsilon^{-2})$ evaluations of the problem's functions and their derivatives for finding an $\epsilon$-approximate first-order stationary point. This complexity bound therefore generalizes that provided by [Bellavia, Gurioli, Morini and Toint, 2018] for inexact methods for smooth nonconvex problems, and is within a factor $|\log(\epsilon)|$ of the optimal bound known for smooth and nonsmooth nonconvex minimization with exact evaluations. A practically more restrictive variant of the algorithm with worst-case complexity $O(|\log(\epsilon)|+\epsilon^{-2})$ is also presented.