Cubic-regularization counterpart of a variable-norm trust-region method for unconstrained minimization ∗ J. M. Mart´ ınez † M. Raydan ‡ November 15, 2015 Abstract In a recent paper we introduced a trust-region method with variable norms for uncon- strained minimization and we proved standard asymptotic convergence results. Here we will show that, with a simple modification with respect to the sufficient descent condition and replacing the trust-region approach with a suitable cubic regularization, the complexity of this method for finding approximate first-order stationary points is O(ε -3/2 ). Some numer- ical experiments are also presented to illustrate the impact of the modification on practical performance. Keywords: Smooth unconstrained minimization, cubic modeling, regularization, Newton- type methods. 1 Introduction We consider the unconstrained minimization problem min x∈R n f (x), (1) where f : R n → R is a sufficiently smooth function. In recent years, new ideas for solving (1) have been developed which are based on the minimization of a cubic regularization model, defined as the standard quadratic model plus a regularization term that penalizes the cubic power of the step length. These ideas have been shown to have global convergence properties to second-order minimizers. Moreover, some of them possess a better worst-case evaluation-complexity bound than their quadratic modeling trust-region competitors; see e. g., [1, 2, 5, 6, 9, 12, 14, 15, 16, 19, 20]. Recently [18], an alternative separable cubic model combined with a variable-norm trust- region strategy was proposed and analyzed to solve (1), for which standard asymptotic conver- gence results were established. * This work was supported by PRONEX-CNPq/FAPERJ (E-26/111.449/2010-APQ1), CEPID–Industrial Mathematics/FAPESP (Grant 2011/51305-02), FAPESP (projects 2013/05475-7 and 2013/07375-0), and CNPq (project 400926/2013-0). † Department of Applied Mathematics, IMECC-UNICAMP, University of Campinas, Rua S´ ergio Buarque de Holanda, 651 Cidade Universit´ aria “Zeferino Vaz”, Distrito Bar˜ ao Geraldo, 13083-859 Campinas SP, Brazil. (martinez@ime.unicamp.br). ‡ Departamento de C´ omputo Cient´ ıfico y Estad´ ıstica, Universidad Sim´ on Bol´ ıvar, Ap. 89000, Caracas 1080-A, Venezuela. (mraydan@usb.ve). 1