  An adaptive regularization algorithm for unconstrained nonconvex optimization
is presented in which the objective function is never evaluated, but only
derivatives are used. This algorithm belongs to the class of adaptive
regularization methods, for which optimal worst-case complexity results are
known for the standard framework where the objective function is evaluated. It
is shown in this paper that these excellent complexity bounds are also valid
for the new algorithm, despite the fact that significantly less information is
used. In particular, it is shown that, if derivatives of degree one to $p$ are
used, the algorithm will find a $\epsilon_1$-approximate first-order minimizer
in at most $O(\epsilon_1^{-(p+1)/p})$ iterations, and an
$(\epsilon_1,\epsilon_2)$-approximate second-order minimizer in at most
$O(\max[\epsilon^{-(p+1)/p},\epsilon_2^{-(p+1)/(p-1)}])$ iterations. As a
special case, the new algorithm using first and second derivatives, when
applied to functions with Lipschitz continuous Hessian, will find an iterate
$x_k$ at which the gradient's norm is less than $\epsilon_1$ in at most
$O(\epsilon_1^{-3/2})$ iterations.

Optimization and Control