This paper examines worst-case evaluation bounds for finding weak minimizers in unconstrained optimization. For the cubic regularization algorithm, Nesterov and Polyak (2006) and Cartis, Gould and Toint (2010a) show that at most O(ε-3) iterations may have to be performed for finding an iterate which is within ε of satisfying second-order optimality conditions. We first show that this bound can be derived for a version of the algorithm which only uses one-dimensional global optimization of the cubic model and that it is sharp. We next consider the standard trust-region method and show that a bound of the same type may also be derived for this method, and that it is also sharp in some cases. We conclude by showing that a comparison of the worst-case behaviour of the ARC and trust-region algorithms favours the first of these methods.
evaluation complexity, worst-case analysis, nonconvex optimization, second-order optimality conditions
Written: 17 December 2010
Published in Journal on Complexity.