An adaptive regularization algorithm is proposed that uses Taylor models of the objective of order p, $p\ge 2$, of the unconstrained objective function, and that is guaranteed to find a first- and second-order critical point in at most $O(max\{{ϵ}_1^{-\frac{p+1}{p}},{ϵ}_2^{-\frac{p+1}{p-1}}\})$ function and derivatives evaluations, where ${ϵ}_1$ and ${ϵ}_2$ 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.

提出了一种自适应正则化算法，该算法使用目标p的泰勒模型，$p\ge 2$不受约束的目标函数，并且可以保证最多找到一个一阶和二阶临界点 $Ø（最高\{{ϵ}_{\mathrm{1个}}^{-\frac{p+\mathrm{1个}}{p}}，{ϵ}_2^{-\frac{p+\mathrm{1个}}{p-\mathrm{1个}}}\}）$ 功能和派生评估，其中 ${ϵ}_{\mathrm{1个}}$ 和 ${ϵ}_2$规定了一阶和二阶最优容差。与Cartis等人中更为通用的方法相比，这是一种简单的算法和相关分析。[适用于具有廉价约束的任意阶非凸优化的尖锐最坏情况评估复杂度界限，arXiv：1811.01220，2018年]解决了大于2的临界复杂度；在这里，我们使用标准的最优性条件和实际的子问题来解决，以显示出针对二阶关键性的同阶尖锐复杂性。我们的方法还扩展了Birgin等人的方法。[使用高阶正则化模型进行无约束非线性优化的最坏情况评估复杂度， 数学。编 163（1）（2017），第359–368页]，在与一阶复杂度相同的问题平滑度假设下，找到二阶临界点。