Positive bases with maximal cosine measure

Abstract

Positive spanning sets and positive bases are important in the construction of derivativefree 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 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/ √ n. In all the cases described, the positive bases achieving these upper bounds are characterized.

Convergence of derivative-free nonmonotone Direct Search Methods for unconstrained and box-constrained mixed-integer optimization

“…By Lemma 4, orthogonal directions implicitly enforce (12) with α = 1/ √ n. It is also known that the cosine value α = 1/ √ n cannot be improved by any D that spans R n positively, with either n + 1 or 2n search directions [35]. It has been argued that performance of a DFO algorithm may improve with n + 1 search directions [4].…”