# Worst-case evaluation complexity and optimality of second-order methods for nonconvex smooth optimization

Coralia Cartis, Nicholas I M Gould, Philippe Toint

Research output: Working paper

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### Abstract

We establish or refute the optimality of inexact second-order methods for
unconstrained nonconvex optimization from the point of view of worst-case evaluation complexity, improving and generalizing the results of Cartis, Gould
and Toint (2010,2011). To this aim, we consider a new general class of inexact second-order algorithms for unconstrained optimization that includes regularization and trust-region variations of Newton's method as well as of
their linesearch variants. For each method in this class and arbitrary
accuracy threshold $\epsilon \in (0,1)$, we exhibit a smooth objective
function with bounded range, whose gradient is globally Lipschitz continuous
and whose Hessian is $\alpha-$H\"older continuous (for given $\alpha\in [0,1]$), for which the method in question takes at least $\lfloor\epsilon^{-(2+\alpha)/(1+\alpha)}\rfloor$ function evaluations to generate a first iterate whose gradient is smaller than $\epsilon$ in norm. Moreover, we also construct another function on which Newton's takes $\lfloor\epsilon^{-2}\rfloor$ evaluations, but whose Hessian is Lipschitz continuous on the path of iterates. These examples provide lower bounds on the worst-case evaluation complexity of methods in our class when applied to smooth problems satisfying the relevant assumptions. Furthermore, for $\alpha=1$, this lower bound is of the same order in $\epsilon$ as the upper bound on the worst-case evaluation complexity of the cubic %(and more generally $(2+\alpha)$) regularization method and other methods in a class of methods proposed in Curtis, Robinson and Samadi (2017) or in Royer and Wright (2017), thus implying that these methods have optimal worst-case evaluation complexity within a wider class of second-order methods, and that Newton's method is suboptimal.
Original language English Arxiv 35 1709.07180 Published - 22 Sep 2017

### Keywords

• evaluation complexity
• nonlinear optimization
• second-order methods