Nicely structured positive bases with maximal cosine measure

TL;DR

This paper investigates the structure of intermediate positive bases with maximal cosine measure, providing a simple, easily generated structure that enhances understanding and application in derivative-free optimization and mathematics.

Contribution

It characterizes the structure of intermediate positive bases with maximal cosine measure, offering a simple form that facilitates computational generation.

Findings

Identifies the structure of intermediate positive bases with maximal cosine measure.

Provides a simple form for these bases, making them easy to generate.

Enhances understanding of positive bases in optimization and mathematics.

Abstract

The properties of positive bases make them a useful tool in derivative-free optimization (DFO) and an interesting concept in mathematics. The notion of the \emph{cosine measure} helps to quantify the quality of a positive basis. It provides information on how well the vectors in the positive basis uniformly cover the space considered. The number of vectors in a positive basis is known to be between $n+1$ and $2n$ inclusively. When the number of vectors is strictly between $n+1$ and $2n$, we say that it is an intermediate positive basis. In this paper, the structure of intermediate positive bases with maximal cosine measure is investigated. The structure of an intermediate positive basis with maximal cosine measure over a certain subset of positive bases is provided. This type of positive bases has a simple structure that makes them easy to generate with a computer software.