On the properties of positive spanning sets and positive bases

Abstract

The concepts of positive span and positive basis are important in derivative-free optimization. In fact, a well-known result is that if the gradient of a continuously differentiable objective function on $\mathbb{R}^n$ is nonzero at a point, then one of the vectors in any positive basis (or any positive spanning set) of $\mathbb{R}^n$ is a descent direction for the objective function from that point. This article summarizes the basic results and explores additional properties of positive spanning sets, positively independent sets and positive bases that are potentially useful in the design of derivative-free optimization algorithms. In particular, it provides construction procedures for these special sets of vectors that were not previously mentioned in the literature. It also proves that invertible linear transformations preserve positive independence and the positive spanning property. Moreover, this article introduces the notion of linear equivalence between positive spanning sets and between positively independent sets to simplify the analysis of their structures. Linear equivalence turns out to be a generalization of the concept of structural equivalence between positive bases that was introduced by Coope and Price (SIAM J Optim 11:859–869, 2001). Furthermore, this article clarifies which properties of linearly independent sets, spanning sets and ordinary bases carry over to positively independent sets, positive spanning sets, and positive bases. For example, a linearly independent set can always be extended to a basis of a linear space but a positively independent set cannot always be extended to a positive basis. Also, the maximum size of a linearly independent set in $R^n$ is n but there is no limit to the size of a positively independent set in $\mathbb{R}^n$ when $n \ge 3$. Whenever possible, the results are proved for the more general case of frames of convex cones instead of focusing only on positive bases of linear spaces. In addition, this article discusses some algorithms for determining whether a given set of vectors is positively independent or whether it positively spans a linear subspace of $\mathbb{R}^n$. Finally, it provides an algorithm for extending any finite set of vectors to a positive spanning set of $\mathbb{R}^n$ using only a relatively small number of additional vectors.