Originally developed in 1954, positive bases and positive spanning sets have been found to be a valuable concept in derivative-free optimization (DFO). The quality of a positive basis (or positive spanning set) can be quantified via the cosine measure and convergence properties of certain DFO algorithms are intimately linked to the value of this measure. However, it is unclear how to compute the cosine measure for a positive basis from the definition. In this paper, a deterministic algorithm to compute the cosine measure of any positive basis or finite positive spanning set is provided. The algorithm is proven to return the exact value of the cosine measure in finite time.

中文翻译：

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确定性算法，用于计算有限正跨集的余弦量度

正基和正生成集最初开发于1954年，在无导数优化（DFO）中被发现是一个有价值的概念。可以通过余弦度量来量化正基准（或正扩展集）的质量，并且某些DFO算法的收敛特性与该度量的值紧密相关。但是，尚不清楚如何从定义中算出正弦余弦量。本文提供了一种确定性算法，可以计算任何正基或有限正跨度集的余弦量度。证明该算法可在有限时间内返回余弦量度的精确值。