Mathematical Programming

, Volume 163, Issue 1–2, pp 359–368 | Cite as

Worst-case evaluation complexity for unconstrained nonlinear optimization using high-order regularized models

  • E. G. Birgin
  • J. L. Gardenghi
  • J. M. Martínez
  • S. A. Santos
  • Ph. L. TointEmail author
Full Length Paper Series A


The worst-case evaluation complexity for smooth (possibly nonconvex) unconstrained optimization is considered. It is shown that, if one is willing to use derivatives of the objective function up to order p (for \(p\ge 1\)) and to assume Lipschitz continuity of the p-th derivative, then an \(\epsilon \)-approximate first-order critical point can be computed in at most \(O(\epsilon ^{-(p+1)/p})\) evaluations of the problem’s objective function and its derivatives. This generalizes and subsumes results known for \(p=1\) and \(p=2\).


Nonlinear optimization Unconstrained optimization Evaluation complexity High-order models Regularization 

Mathematics Subject Classification

90C30 65K05 49M37 90C60 68Q25 



The authors are pleased to thank Coralia Cartis and Nick Gould for valuable comments, in particular on the definition of the tensor Lipschitz condition and associated material. Two anonymous referees also helped to improve the final manuscript.


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Copyright information

© Springer-Verlag Berlin Heidelberg and Mathematical Optimization Society 2016

Authors and Affiliations

  • E. G. Birgin
    • 1
  • J. L. Gardenghi
    • 1
  • J. M. Martínez
    • 2
  • S. A. Santos
    • 2
  • Ph. L. Toint
    • 3
    Email author
  1. 1.Department of Computer Science, Institute of Mathematics and StatisticsUniversity of São PauloSão PauloBrazil
  2. 2.Department of Applied Mathematics, Institute of Mathematics, Statistics, and Scientific ComputingUniversity of CampinasCampinasBrazil
  3. 3.Namur Center for Complex Systems (naXys) and Department of MathematicsUniversity of NamurNamurBelgium

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