Complexity bounds for second-order optimality in unconstrained optimization

Coralia Cartis, N.I.M. Gould, Philippe Toint

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This paper examines worst-case evaluation bounds for finding weak minimizers in unconstrained optimization. For the cubic regularization algorithm, Nesterov and Polyak (2006) [15] and Cartis et al. (2010) [3] show that at most O(ε ) iterations may have to be performed for finding an iterate which is within of satisfying second-order optimality conditions. We first show that this bound can be derived for a version of the algorithm, which only uses one-dimensional global optimization of the cubic model and that it is sharp. We next consider the standard trust-region method and show that a bound of the same type may also be derived for this method, and that it is also sharp in some cases. We conclude by showing that a comparison of the bounds on the worst-case behaviour of the cubic regularization and trust-region algorithms favours the first of these methods. © 2011 Elsevier Inc. All rights reserved.
Original languageEnglish
Pages (from-to)93-108
Number of pages16
JournalJournal of Complexity
Publication statusPublished - 1 Feb 2012


  • evaluation complexity
  • second-order optimality conditions
  • nonconvex optimization
  • worst-case analysis


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