Abstract
Consider a nonempty finite set of nonzero vectors \(S \subset \mathbb {R}^n\). The angle between a nonzero vector \(v \in \mathbb {R}^n\) and S is the smallest angle between v and an element of S. The cosine measure of S is the cosine of the largest possible angle between a nonzero vector \(v \in \mathbb {R}^n\) and S. The cosine measure provides a way of quantifying the positive spanning property of a set of vectors, which is important in the area of derivative-free optimization. This paper proves some of the properties of the cosine measure for a nonempty finite set of nonzero vectors. It also introduces the notion of the uniform angle subspace and some cones associated with it and proves some of their properties. Moreover, this paper proves some results that characterize the Karush–Kuhn–Tucker (KKT) points for the optimization problem of calculating the cosine measure. These characterizations of the KKT points involve the uniform angle subspace and its associated cones. Finally, this paper provides an outline for calculating the cosine measure of any nonempty finite set of nonzero vectors.
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Acknowledgements
The author wishes to thank the reviewers for their thorough review of this paper and their excellent comments. Also, special thanks to Geir Nævdal for suggesting the result on the maximal cosine measure of a set in \(\mathbb {R}^n\) with at most n elements.
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Regis, R.G. On the properties of the cosine measure and the uniform angle subspace. Comput Optim Appl 78, 915–952 (2021). https://doi.org/10.1007/s10589-020-00253-4
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DOI: https://doi.org/10.1007/s10589-020-00253-4