The (optimal) function/gradient evaluations worst-case complexity analysis available for the adaptive regularization algorithms with cubics (ARC) for nonconvex smooth unconstrained optimization is extended to finite-difference versions of this algorithm, yielding complexity bounds for first-order and derivative-free methods applied on the same problem class. A comparison with the results obtained for derivative-free methods by Vicente [Worst Case Complexity of Direct Search, Technical report, Preprint 10-17, Department of Mathematics, University of Coimbra, Coimbra, Portugal, 2010] is also discussed, giving some theoretical insight into the relative merits of various methods in this popular class of algorithms. © 2012 Society for Industrial and Applied Mathematics.
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Cartis, C., Gould, N. I. M., & Toint, P. (2012). On the oracle complexity of first-order and derivative-free algorithms for smooth nonconvex minimization. SIAM Journal on Optimization, 22(1), 66-86. https://doi.org/10.1137/100812276