SIAM Journal on Optimization, Volume 30, Issue 1, Page 853-884, January 2020.
Simplex gradients, essentially the gradient of a linear approximation, are a popular tool in derivative-free optimization (DFO). In 2015, a product rule, a quotient rule, and a sum rule for simplex gradients were introduced by Regis [Optim. Lett., 9 (2015), pp. 845--865]. Unfortunately, those calculus rules only work under a restrictive set of assumptions. The purpose of this paper is to provide new calculus rules that work in a wider setting. The rules place minimal assumptions on the functions involved and the interpolation sets. The rules further lead to an alternative approach to gradient approximation in situations where the rules could be applied. We analyze the new approach, provide error bounds, and include some preliminary testing on numerical stability and accuracy.

中文翻译：

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广义单纯形梯度的微积分恒等式：规则和应用

SIAM优化杂志，第30卷，第1期，第853-884页，2020年1月。
单纯形渐变（本质上是线性逼近的渐变）是无导数优化（DFO）中的一种流行工具。2015年，Regis引入了单形梯度的乘积规则，商规则和求和规则[Optim。Lett。，9（2015），第845--865页。不幸的是，这些演算规则仅在一组限制性假设下起作用。本文的目的是提供适用于更广泛环境的新演算规则。该规则对所涉及的函数和插值集进行了最小假设。在可以应用规则的情况下，规则还导致了梯度近似的替代方法。我们分析了这种新方法，提供了误差范围，并包括了一些关于数值稳定性和准确性的初步测试。