The talk will consider the question of the worst-case evaluation complexity of finding approximate first-order critical points in nonlinear (nonconvex) smooth optimization using p-th order models. A remarkably simple proof, based on a standard regularization algorithm, will be given that at worst O(epsilon^{-(p+1)/p}) evaluations of the objective function and its derivatives are needed to compute an epsilon-approximate critical point for unconstrained and convexly-constrained cases. A two-phases framework will also be described for handling the case where constraints are fully general (equalities and inequalities) and the evaluation complexity shown to be at worst O(epsilon^{-(p+2)/p}) in this case.

Scheduled as part of