A concise second-order complexity analysis for unconstrained optimization using high-order regularized models

An adaptive regularization algorithm is proposed that uses Taylor models of the objective of order p, (Formula presented.), of the unconstrained objective function, and that is guaranteed to find a first- and second-order critical point in at most (Formula presented.) function and derivatives evaluations, where ε ₁ and ε ₁ are prescribed first- and second-order optimality tolerances. This is a simple algorithm and associated analysis compared to the much more general approach in Cartis et al. [Sharp worst-case evaluation complexity bounds for arbitrary-order nonconvex optimization with inexpensive constraints, arXiv:1811.01220, 2018] that addresses the complexity of criticality higher-than two; here, we use standard optimality conditions and practical subproblem solves to show a same-order sharp complexity bound for second-order criticality. Our approach also extends the method in Birgin et al. [Worst-case evaluation complexity for unconstrained nonlinear optimization using high-order regularized models, Math. Prog. A 163(1) (2017), pp. 359–368] to finding second-order critical points, under the same problem smoothness assumptions as were needed for first-order complexity.