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Positive bases with maximal cosine measure. (English) Zbl 1429.90099
Summary: Positive spanning sets and positive bases are important in the construction of derivative-free optimization algorithms. The convergence properties of the algorithms might be tied to the cosine measure of the positive basis that is used, and having higher cosine measure might in general be preferable. In this paper, the upper bound of the cosine measure for certain positive bases in $$\mathbb{R}^n$$ are found. In particular, if the size of the positive basis is $$n+1$$ (the minimal positive bases), the maximal value of the cosine measure is $$1 / n$$. A straightforward corollary is that the maximal cosine measure for any positive spanning set of size $$n+1$$ is $$1 / n$$. If the size of a positive basis is $$2n$$ (the maximal positive bases) the maximal cosine measure is $$1/\sqrt{n}$$. In all the cases described, the positive bases achieving these upper bounds are characterized.

##### MSC:
 90C56 Derivative-free methods and methods using generalized derivatives
##### Keywords:
positive basis; positive spanning set; cosine measure
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##### References:
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