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 ≥ 1) and to assume Lipschitz continuity of the p-th derivative, then an -approximate first-order critical point can be computed in at most O( −(p+1)/p) evaluations of the problem’s objective function and its derivatives. This generalizes and subsumes results known for p = 1 and p = 2.