Mathematical Programming

, Volume 112, Issue 1, pp 159–181

Accelerating the cubic regularization of Newton’s method on convex problems

FULL LENGTH PAPER

DOI: 10.1007/s10107-006-0089-x

Cite this article as:
Nesterov, Y. Math. Program. (2008) 112: 159. doi:10.1007/s10107-006-0089-x

Abstract

In this paper we propose an accelerated version of the cubic regularization of Newton’s method (Nesterov and Polyak, in Math Program 108(1): 177–205, 2006). The original version, used for minimizing a convex function with Lipschitz-continuous Hessian, guarantees a global rate of convergence of order \(O\big({1 \over k^2}\big)\), where k is the iteration counter. Our modified version converges for the same problem class with order \(O\big({1 \over k^3}\big)\), keeping the complexity of each iteration unchanged. We study the complexity of both schemes on different classes of convex problems. In particular, we argue that for the second-order schemes, the class of non-degenerate problems is different from the standard class.

Keywords

Convex optimization Unconstrained minimization Newton’s method Cubic regularization Worst-case complexity Global complexity bounds Non-degenerate problems Condition number 

Mathematics Subject Classification (2000)

49M15 49M37 58C15 90C25 90C30 

Copyright information

© Springer-Verlag 2007

Authors and Affiliations

  1. 1.Center for Operations Research and Econometrics (CORE)Catholic University of Louvain (UCL)Louvain-la-NeuveBelgium

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