On the use of third-order models with fourth-order regularization for unconstrained optimization

  • E. G. BirginEmail author
  • J. L. Gardenghi
  • J. M. Martínez
  • S. A. Santos
Original paper


In a recent paper (Birgin et al. in Math Program 163(1):359–368, 2017), it was shown that, for the smooth unconstrained optimization problem, worst-case evaluation complexity \(O(\epsilon ^{-(p+1)/p})\) may be obtained by means of algorithms that employ sequential approximate minimizations of p-th order Taylor models plus \((p+1)\)-th order regularization terms. The aforementioned result, which assumes Lipschitz continuity of the p-th partial derivatives, generalizes the case \(p=2\), known since 2006, which has already motivated efficient implementations. The present paper addresses the issue of defining a reliable algorithm for the case \(p=3\). With that purpose, we propose a specific algorithm and we show numerical experiments.


Unconstrained minimization Third-order models Regularization Complexity 



We are thankful to the anonymous reviewers, and to the Associate Editor, whose comments and suggestions improved the presentation of our work. This work has been partially supported by FAPESP (Grants 2013/03447-6, 2013/05475-7, 2013/07375-0, 2013/23494-9, 2016/01860-1, and 2017/03504-0) and CNPq (Grants 309517/2014-1, 303750/2014-6, and 302915/2016-8)


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Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2019

Authors and Affiliations

  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

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