Let K be the unit-cube in Rn and f\,: K $\rightarrow$ R^n be a Morse function. We assume that the function f is given by an evaluation program $Γ$ in the noisy model, i.e., the evaluation program $Γ$ takes an extra parameter $η$ as input and returns an approximation that is $η$-close to the true value of f . In this article, we design an algorithm able to compute all local minimizers of f on K . Our algorithm takes as input $Γ$, $η$, a numerical accuracy parameter $ε$ as well as some extra regularity parameters which are made explicit. Under assumptions of probabilistic nature -- related to the choice of the evaluation points used to feed $Γ$ --, it returns finitely many rational points of K , such that the set of balls of radius $ε$ centered at these points contains and separates the set of all local minimizers of f . Our method is based on approximation theory, yielding polynomial approximants for f , combined with computer algebra techniques for solving systems of polynomial equations. We provide bit complexity estimates for our algorithm when all regularity parameters are known. Practical experiments show that our implementation of this algorithm in the Julia package Globtim can tackle examples that were not reachable until now.

翻译：设K为R^n中的单位立方体，f: K → R^n为莫尔斯函数。我们假设函数f由噪声模型下的评估程序Γ给出，即评估程序Γ将额外参数η作为输入，并返回与f真实值η-接近的近似值。本文设计了一种能够计算K上f所有局部极小值的算法。该算法以Γ、η、数值精度参数ε以及若干明确指定的额外正则性参数为输入。在概率性假设（与用于输入Γ的评估点选择相关）条件下，算法返回K上的有限个有理点，使得以这些点为中心、半径为ε的球体集合包含并分离f的所有局部极小值集合。我们的方法基于逼近理论（构建f的多项式逼近），结合用于求解多项式方程组的计算机代数技术。当所有正则性参数已知时，我们提供了算法位复杂度估计。实际实验表明，我们在Julia软件包Globtim中实现的该算法能够处理此前无法解决的算例。