A regularization algorithm allowing random noise in derivatives and inexact function values is proposed for computing approximate local critical points of any order for smooth unconstrained optimization problems. For an objective function with Lipschitz continuous p-th derivative and given an arbitrary optimality order $q\le p$, an upper bound on the number of function and derivatives evaluations is established for this algorithm. This bound is in expectation, and in terms of a power of the required tolerances, this power depending on whether $q\le 2$ or $q>2$. Moreover these bounds are sharp in the order of the accuracy tolerances. An extension to convexly constrained problems is also outlined.

提出了一种允许导数中的随机噪声和不精确函数值的正则化算法，用于计算平滑无约束优化问题的任何阶的近似局部临界点。对于具有 Lipschitz 连续p阶导数并给定任意最优性阶的目标函数$q\le p$，为此算法建立了函数和导数评估数量的上限。这个界限是预期的，就所需容差的幂而言，这个幂取决于$q\le 2$ 或者 $q>2$. 此外，这些界限在精度容差的顺序上是尖锐的。还概述了凸约束问题的扩展。