Search

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 recently identified positive spanning sets, called orthogonally structured positive bases. We first describe how to identify such sets and compute their cosine measures 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. [ABSTRACT FROM AUTHOR]

We consider the question of numerically approximating the derivative of a smooth function using only function evaluations. In particular, we examine the regression gradient , the generalized simplex gradient and the generalized centered simplex gradient , three numerical techniques based on using function values at a collection of sample points to construct 'best-fit' linear models. Under some conditions, these gradient approximations have error bounds dependent on the number of sample points used, the Lipschitz constant of the true gradient, and the geometry of the sample set. Perhaps counter-intuitively, as the number of sample points increases (to infinity) on a fixed domain, the error bounds can increase (to infinity). In this work, we first explore the behavior of the error bound for generalized simplex gradients of a single-variable function (f : R ↦ R). Thereafter, we investigate the behavior of the absolute error for these three gradient approximation techniques as the number of sample points tends to infinity. Under reasonable assumptions, we prove that the absolute errors remain bounded as the number of sample points increases to infinity on a fixed interval. [ABSTRACT FROM AUTHOR]