Worst-case evaluation complexity for unconstrained nonlinear optimization using high-order regularized models

The worst-case evaluation complexity for smooth (possibly nonconvex) unconstrained optimization is considered. It is shown that, if one is willing to use derivatives of the objective function up to order p (for $$p\ge 1$$p≥1) and to assume Lipschitz continuity of the p-th derivative, then an $$\epsilon $$ϵ-approximate first-order critical point can be computed in at most $$O(\epsilon ^{-(p+1)/p})$$O(ϵ-(p+1)/p) evaluations of the problem’s objective function and its derivatives. This generalizes and subsumes results known for $$p=1$$p=1 and $$p=2$$p=2.