Abstract
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.
Highlights
The seminal work on positive bases by Davis [6] established many of their properties
The present paper shows that the maximal cosine measure for a positive spanning set for Rn of size n + 1 in 1/n
Both the minimal and maximal positive bases with maximal cosine measure turn out to be given as examples of positive bases in [5]
Summary
The seminal work on positive bases by Davis [6] established many of their properties. Conn et al [5] which gives an introduction to derivative-free optimization introduces positive spanning sets and positive bases and the cosine measure (if necessary, see definitions below) as part of the background before starting on developing any optimization routines. The present paper shows that the maximal cosine measure for a positive spanning set for Rn of size n + 1 in 1/n. The present paper shows that this is the case for the maximal positive bases (which is a subset of the positive spanning sets of size 2n). Both the minimal and maximal positive bases with maximal cosine measure turn out to be given as examples of positive bases in [5].
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