Positive spanning sets span a given vector space by nonnegative linear combinations of their elements. These have attracted significant attention in recent years, owing to their extensive use in derivative-free optimization. In this setting, the quality of a positive spanning set is assessed through its cosine measure, a geometric quantity that expresses how well such a set covers the space of interest. In this paper, we investigate the construction of positive $k$-spanning sets with geometrical guarantees. Our results build on recently identified positive spanning sets, called orthogonally structured positive bases. We first describe how to identify such sets and compute their cosine measures efficiently. We then focus our study on positive $k$-spanning sets, for which we provide a complete description, as well as a new notion of cosine measure that accounts for the resilient nature of such sets. By combining our results, we are able to use orthogonally structured positive bases to create positive $k$-spanning sets with guarantees on the value of their cosine measures.

翻译：正生成集通过其元素的非负线性组合生成给定向量空间。近年来，由于在无导数优化中的广泛应用，这类集合引起了极大关注。在此背景下，正生成集的质量通过其余弦度量进行评估，这是一个几何量，用于表征该集合对目标空间的覆盖程度。本文研究了具有几何保证的正$k$生成集的构造方法。我们的结果建立在近期发现的一类称为正交结构正基的正生成集之上。我们首先描述了如何识别此类集合并高效计算其余弦度量。随后将研究重点集中于正$k$生成集，为此我们提供了完整的数学描述，并针对此类集合的容错特性提出了一种新的余弦度量概念。通过整合研究结果，我们能够利用正交结构正基来构造具有余弦度量值保证的正$k$生成集。