SIAM Journal on Optimization, Volume 30, Issue 1, Page 513-541, January 2020.
We provide sharp worst-case evaluation complexity bounds for nonconvex minimization problems with general inexpensive constraints, i.e., problems where the cost of evaluating/enforcing of the (possibly nonconvex or even disconnected) constraints, if any, is negligible compared to that of evaluating the objective function. These bounds unify, extend, or improve all known upper and lower complexity bounds for nonconvex unconstrained and convexly constrained problems. It is shown that, given an accuracy level $\epsilon$, a degree of highest available Lipschitz continuous derivatives $p$, and a desired optimality order $q$ between one and $p$, a conceptual regularization algorithm requires no more than $O(\epsilon^{-\frac{p+1}{p-q+1}})$ evaluations of the objective function and its derivatives to compute a suitably approximate $q$th order minimizer. With an appropriate choice of the regularization, a similar result also holds if the $p$th derivative is merely Hölder rather than Lipschitz continuous. We provide an example that shows that the above complexity bound is sharp for unconstrained and a wide class of constrained problems; we also give reasons for the optimality of regularization methods from a worst-case complexity point of view, within a large class of algorithms that use the same derivative information.

中文翻译：

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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

SIAM优化杂志，第30卷，第1期，第513-541页，2020年1月。
对于具有一般廉价约束的非凸最小化问题，我们提供了最尖锐的最坏情况评估复杂度界限，即，与评估非约束最小化问题相比，评估/执行（可能是非凸甚至断开）约束的成本（如果有）可以忽略不计的问题目标函数。这些边界统一，扩展或改善了非凸，无约束和凸约束问题的所有已知的上下复杂度边界。结果表明，给定精度水平$ \ epsilon $，可用Lipschitz连续导数$ p $的最高程度以及在1到$ p $之间的理想最优顺序$ q $，概念上的正则化算法所要求的不超过$对目标函数及其导数的O（\ epsilon ^ {-\ frac {p + 1} {p-q + 1}}）$个求值，以计算适当近似的$ q $阶最小化器。通过选择适当的正则化，如果$ p $ th的导数仅仅是Hölder而不是Lipschitz连续的，则也会得到相似的结果。我们提供了一个示例，表明上述复杂度界限对于不受约束的问题和各种各样的受约束的问题而言是非常明显的；我们还从最坏情况的复杂性角度，在使用相同导数信息的一大类算法中，给出了正则化方法最优的原因。