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

Serge Gratton(serge.gratton***at***enseeiht.fr)
Ehouarn Simon(ehouarn.simon***at***enseeiht.fr)
Philippe L. Toint(philippe.toint***at***unamur.be)

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.

Keywords: evaluation complexity, nonsmooth problems, nonconvex optimization, composite functions, inexact evaluations

Category 1: Nonlinear Optimization (Unconstrained Optimization )

Category 2: Nonlinear Optimization (Constrained Nonlinear Optimization )

Category 3: Convex and Nonsmooth Optimization (Nonsmooth Optimization )

Citation:

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Entry Submitted: 02/27/2019
Entry Accepted: 02/27/2019
Entry Last Modified: 02/27/2019

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