The adaptive cubic regularization algorithms described in Cartis, Gould & Toint (2009, 2010) for unconstrained (nonconvex) optimization are shown to have improved worst-case efficiency in terms of the function- and gradient-evaluation count when applied to convex and strongly convex objectives. In particular, our complexity upper bounds match in order (as a function of the accuracy of approximation), and sometimes even improve, those obtained by Nesterov (2004, 2008) and Nesterov & Polyak (2006) for these same problem classes, without employing the Hessian’s Lipschitz constant explicitly in the algorithms or requiring exact or global solution of the subproblem. An additional outcome of our approximate approach is that our complexity results can naturally capture the advantages of both first- and second-order methods.
Citation
ERGO Technical Report 10-006, 2010, School of Mathematics, University of Edinburgh, United Kingdom