Worst-case evaluation complexity for unconstrained nonlinear optimization using high-order regularized models - Birgin, E. G.; Gardenghi, J. L.; Martínez, J. M.; Santos, S. A.; Toint, Ph. L. (Mathematical programming, 2017)
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 ϵ -approximate first-order critical point can be computed in at most O ( ϵ - ( 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 .