On the cardinality of positively linearly independent sets

Abstract

Positive bases, which play a key role in understanding derivative free optimization methods that use a direct search framework, are positive spanning sets that are positively linearly independent. The cardinality of a positive basis in $\mathbb {R}^n$ has been established to be between $n+1$ and 2n (with both extremes existing). The lower bound is immediate from being a positive spanning set, while the upper bound uses both positive spanning and positively linearly independent. In this note, we provide details proving that a positively linearly independent set in $\mathbb {R}^n$ for $n \in \{1, 2\}$ has at most 2n elements, but a positively linearly independent set in $\mathbb {R}^n$ for $n\ge 3$ can have an arbitrary number of elements.