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

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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(ϵ)|ϵ-2) evaluations of the problem’s functions and their derivatives for finding an ϵ-approximate first-order stationary point. This complexity bound therefore generalizes that provided by Bellavia et al. (Theoretical study of an adaptive cubic regularization method with dynamic inexact Hessian information. arXiv:1808.06239, 2018) for inexact methods for smooth nonconvex problems, and is within a factor | log (ϵ) | 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 (ϵ) | + ϵ - 2) is also presented.

Original languageEnglish
Number of pages19
JournalMathematical Programming
Publication statusAccepted/In press - 1 Jan 2020



  • evaluation complexity
  • nonsmooth problems
  • nonconvex optimization
  • inexact evaluations
  • composite functions
  • Evaluation complexity
  • Nonconvex optimization
  • Composite functions
  • Nonsmooth problems
  • Inexact evaluations

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