On the Cardinality of Positively Linearly Independent Sets

TL;DR

This paper investigates the maximum size of positively linearly independent sets in Euclidean spaces, showing that for dimensions 1 and 2 the size is bounded by 2n, but for higher dimensions, the size can be arbitrarily large.

Contribution

It provides a detailed proof that positively linearly independent sets in ^n are bounded by 2n for n=1,2, but unbounded for n263; this clarifies the structure of positive bases.

Findings

For n=1,2, positively linearly independent sets have at most 2n elements.

For n263, such sets can have arbitrarily many elements.

The bounds are tight for n=1,2, but not for n263.

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 $\R^n$ 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 {\em both} positive spanning and positively linearly independent. In this note, we provide details proving that a positively linearly independent set in $\R^n$ for $n \in \{1, 2\}$ has at most $2n$ elements, but a positively linearly independent set in $\R^n$ for $n\geq 3$ can have an arbitrary number of elements.