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DOI 10.5286/raltr.2011011
Persistent URL http://purl.org/net/epubs/work/56238
Record Status Checked
Record Id 56238
Title Optimal Newton-type methods for nonconvex smooth optimization problems
Abstract We consider a general class of second-order iterations for unconstrained optimization that includes regularization and trust-region variants of Newton’s method. For each method in this class, we exhibit a smooth, bounded-below objective function, whose gradient is globally Lipschitz continuous within an open convex set containing any iterates encountered and whose Hessian is _?H¨older continuous (for given _ 2 [0, 1]) on the path of the iterates, for which the method in question takes at least b_?(2+_)/(1+_)c function-evaluations to generate a first iterate whose gradient is smaller than _ in norm. This provides a lower bound on the evaluation complexity of second-order methods in our class when applied to smooth problems satisfying our assumptions. Furthermore, for _ = 1, this lower bound is of the same order in _ as the upper bound on the evaluation complexity of cubic regularization, thus implying cubic regularization has optimal worst-case evaluation complexity within our class of second-order methods.
Organisation CSE , CSE-NAG , STFC
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Language English (EN)
Type Details URI(s) Local file(s) Year
Report RAL Technical Reports RAL-TR-2011-011. 2011. RAL-TR-2011-011.pdf 2011