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

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 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.