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 [15, 19]. 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 <i>ϵ</i> ∈ (0, 1), we exhibit a smooth objective function with bounded range, whose gradient is globally Lipschitz continuous and whose Hessian is <i>α</i>−Hölder continuous (for given <i>α</i> ∈ [0, 1]), for which the method in question takes at least [<i>ϵ</i>^{−(2+<i>α</i>)/(1+<i>α</i>)}] function evaluations to generate a first iterate whose gradient is smaller than <i>ϵ</i> in norm. Moreover, we also construct another function on which Newton’s takes [<i>ϵ</i>⁻²] 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 <i>α</i> = 1, this lower bound is of the same order in <i>ϵ</i> as the upper bound on the worst-case evaluation complexity of the cubic regularization method and other methods in a class of methods proposed in [36] or in [65], thus implying that these methods have optimal worstcase evaluation complexity within a wider class of second-order methods, and that Newton’s method is suboptimal.