用尺度KKT条件、高阶模型和临界性测度$χ$计算光滑约束非线性优化的复杂性界
Evaluation complexity bounds for smooth constrained nonlinear optimisation using scaled KKT conditions, high-order models and the criticality measure $χ$
摘要：
考虑凸约束优化的评估复杂性，并且首先示出由Cartis、Gould和Toint（IMAJNA 32（4）2012，pp.1662-1695）证明的用于计算$ε$近似一阶临界点的$O(ε^{-3/2})$的复杂性界可以在显著较弱的假设下获得。此外，该结果被推广到高阶导数的情况下，导致在一个界的$O(ε^{-(p+1)/p})$评价时，阶$p$的衍生物。还表明，由Cartis，Gould和Toint（SINUM，2015）提出的涉及等式和不等式约束的一般非凸情况下的$O(ε_P^{-1/2}ε_D^{-3/2})$评估（$ε_P$和$ε_D$是原始和对偶精度阈值）的界限可以推广到类似减弱假设下的$O(ε_P^{-1/p}ε_D^{-(p+1)/p})$评估的界限。本文是配套报告（NTR-11-2015，比利时那慕尔大学）的变体，该报告使用不同的一阶临界性度量来获得相同的复杂性界限。
英文摘要：
Evaluation complexity for convexly constrained optimization is considered and it is shown first that the complexity bound of $O(ε^{-3/2})$ proved by Cartis, Gould and Toint (IMAJNA 32(4) 2012, pp.1662-1695) for computing an $ε$-approximate first-order critical point can be obtained under significantly weaker assumptions. Moreover, the result is generalized to the case where high-order derivatives are used, resulting in a bound of $O(ε^{-(p+1)/p})$ evaluations whenever derivatives of order $p$ are available. It is also shown that the bound of $O(ε_P^{-1/2}ε_D^{-3/2})$ evaluations ($ε_P$ and $ε_D$ being primal and dual accuracy thresholds) suggested by Cartis, Gould and Toint (SINUM, 2015) for the general nonconvex case involving both equality and inequality constraints can be generalized to a bound of $O(ε_P^{-1/p}ε_D^{-(p+1)/p})$ evaluations under similarly weakened assumptions. This paper is variant of a companion report (NTR-11-2015, University of Namur, Belgium) which uses a different first-order criticality measure to obtain the same complexity bounds.
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