In this paper we study the worst-case complexity of an inexact augmented Lagrangian method for nonconvex constrained problems. Assuming that the penalty parameters are bounded we prove a complexity bound of |$\mathcal{O}(|\log (\epsilon )|)$| outer iterations for the referred algorithm to generate an |$\epsilon$|-approximate KKT point for |$\epsilon \in (0,1)$|⁠. When the penalty parameters are unbounded we prove an outer iteration complexity bound of |$\mathcal{O}(\epsilon ^{-2/(\alpha -1)} )$|⁠, where |$\alpha>1$| controls the rate of increase of the penalty parameters. For linearly constrained problems these bounds yield to evaluation complexity bounds of |$\mathcal{O}(|\log (\epsilon )|^{2}\epsilon ^{-2})$| and |$\mathcal{O}(\epsilon ^{- (\frac{2(2+\alpha )}{\alpha -1}+2 )})$|⁠, respectively, when appropriate first-order methods (⁠|$p=1$|⁠) are used to approximately solve the unconstrained subproblems at each iteration. In the case of problems having only linear equality constraints the latter bounds are improved to |$\mathcal{O}(|\log (\epsilon )|^{2}\epsilon ^{-(p+1)/p})$| and |$\mathcal{O}(\epsilon ^{-(\frac{4}{\alpha -1}+\frac{p+1}{p})})$|⁠, respectively, when appropriate |$p$|-order methods (⁠|$p\geq 2$|⁠) are used as inner solvers.

中文翻译：

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关于非凸优化的增强拉格朗日方法的复杂性

在本文中，我们研究了非凸约束问题的不精确增强拉格朗日方法的最坏情况复杂度。假设惩罚参数是有界的，我们证明了| $ \ mathcal {O}（| \ log（\ epsilon）|）$ |的复杂度界 引用算法的外部迭代生成| $ \ epsilon $ | - | $ \ epsilon \ in（0,1）$ |⁠的大约KKT点。当罚分参数无界时，我们证明了| $ \ mathcal {O}（\ epsilon ^ {-2 /（\\ alpha -1）}）$ |⁠的外部迭代复杂度边界，其中| $ \ alpha> 1 $ | 控制惩罚参数的增加率。对于线性约束的问题，这些界限产生评估的复杂度界限| $ \ mathcal {O}（| \ log（\ epsilon）| ^ {2} \ epsilon ^ {-2}）$ | 和| $ \ mathcal {O}（\ epsilon ^ {-（\ frac {2（2+ \ alpha）} {\ alpha -1} +2}}）$ |⁠，分别是适当的一阶方法（⁠| $ p = 1 $ |⁠）用于近似解决每次迭代中不受约束的子问题。在仅具有线性等式约束的问题的情况下，后者的边界将提高为| $ \ mathcal {O}（| \ log（\ epsilon）| ^ {2} \ epsilon ^ {-（p + 1）/ p}） $ | 和| $ \ mathcal {O}（\ epsilon ^ {-（\ frac {4} {\ alpha -1} + \ frac {p + 1} {p}）}）$ |⁠，分别在适当的情况下| $ p $ | -order方法（⁠| $ p \ geq 2 $ |⁠）被用作内部求解器。