An algorithm for the minimization of nonsmooth nonconvex functions using inexact evaluations and its worst-case complexity

Serge Gratton (1, 2) , Ehouarn Simon (1, 2) , Philippe Toint (3)

1 IRIT-APO - Algorithmes Parallèles et Optimisation
2 Toulouse INP - Institut National Polytechnique (Toulouse)
3 NaXys - Namur Center for Complex Systems [Namur]

Open Archive Toulouse Archive Ouverte OATAO is an open access repository that collects the work of Toulouse researchers and makes it freely available over the web where possible To cite this version: Gratton, Serge and Simon, Ehouarn and Toint, Philippe An algorithm for the minimization of nonsmooth and nonconvex functions using inexact evaluations and its worst-case complexity. (2020) Mathematical Programming, Series A. 1-23. 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(ǫ)| ǫ −2) evaluations of the problem's functions and their derivatives for finding an ǫ-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(ǫ)| 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(ǫ)| + ǫ −2) is also presented.

Mots clés: evaluation complexity; nonsmooth problems; nonconvex optimization; composite functions; inexact evaluations

Domaines: Analyse numérique [math.NA]

Serge Gratton, Ehouarn Simon, Philippe Toint. An algorithm for the minimization of nonsmooth nonconvex functions using inexact evaluations and its worst-case complexity. Mathematical Programming, Series A, 2020, ⟨10.1007/s10107-020-01466-5⟩. ⟨hal-02943960⟩