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 a recently identified type of positive spanning sets called orthogonally structured positive bases. We first describe how to identify such sets and compute their cosine measure 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：正张成集通过其元素的非负线性组合张成给定向量空间。由于其广泛用于无导数优化，这些集合近年来引起了极大关注。在此背景下，正张成集的质量通过其余弦度量来评估，这是一种几何量，用于衡量该集合覆盖目标空间的程度。本文研究了具有几何保证的正 $k$-张成集的构造。我们的结果基于近期发现的一类称为正交结构化正基的正张成集。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，我们聚焦于正 $k$-张成集的研究，为其提供了完整描述，并引入了一种考虑此类集合弹性性质的新余弦度量概念。通过整合我们的结果，我们能够利用正交结构化正基构造出余弦度量值具有保证的正 $k$-张成集。