Gradient approximations are a class of numerical approximation techniques that are of central importance in numerical optimization. In derivative-free optimization, most of the gradient approximations, including the simplex gradient, centred simplex gradient, and adapted centred simplex gradient, are in the form of simplex derivatives. Owing to machine precision, the approximation accuracy of any numerical approximation technique is subject to the influence of floating point errors. In this paper, we provide a general framework for floating point error analysis of simplex derivatives. Our framework is independent of the choice of the simplex derivative as long as it satisfies a general form. We review the definition and approximation accuracy of the generalized simplex gradient and generalized centred simplex gradient. We define and analyze the accuracy of a generalized version of the adapted centred simplex gradient. As examples, we apply our framework to the generalized simplex gradient, generalized centred simplex gradient, and generalized adapted centred simplex gradient. Based on the results, we give suggestions on the minimal choice of approximate diameter of the sample set.

翻译：梯度近似是一类在数值优化中至关重要的数值近似技术。在无导数优化中，大多数梯度近似方法，包括单纯形梯度、中心单纯形梯度和自适应中心单纯形梯度，均属于单纯形导数的形式。受限于机器精度，任何数值近似技术的近似精度都会受到浮点误差的影响。本文提出了一个用于单纯形导数浮点误差分析的通用框架。只要满足一般形式，该框架独立于具体单纯形导数的选择。我们回顾了广义单纯形梯度和广义中心单纯形梯度的定义及其近似精度。我们定义并分析了广义自适应中心单纯形梯度的精度。作为示例，我们将该框架应用于广义单纯形梯度、广义中心单纯形梯度和广义自适应中心单纯形梯度。基于分析结果，我们针对样本集近似直径的最小选择给出了建议。