### 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.

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 |
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Publisher | Arxiv |

Number of pages | 35 |

Volume | 1709.07180 |

Publication status | Published - 22 Sep 2017 |

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### Keywords

- evaluation complexity
- nonlinear optimization
- second-order methods

### Cite this

Cartis, C., Gould, N. I. M., & Toint, P. (2017).

*Worst-case evaluation complexity and optimality of second-order methods for nonconvex smooth optimization*. Arxiv.