On the computation of the cosine measure in high dimensions

TL;DR

This paper addresses the challenge of computing the cosine measure in high-dimensional derivative-free optimization, proposing a new formulation and heuristic to improve computation and benchmarking.

Contribution

It introduces a novel formulation and heuristic for computing the cosine measure in high dimensions, along with methods to construct test sets for benchmarking algorithms.

Findings

The heuristic outperforms existing algorithms in high-dimensional settings.

New methods enable the construction of test sets with specific cosine measures.

The problem remains NP-hard, but practical solutions are proposed.

Abstract

In derivative-free optimization, the cosine measure is a value that often arises in the convergence analysis of direct search methods. Given the increasing interest in high-dimensional derivative-free optimization problems, it is valuable to compute the cosine measure in this setting; however, it has recently been shown to be NP-hard. We propose a new formulation of the problem and heuristic to tackle this problem in higher dimensions and compare it with existing algorithms in the literature. In addition, new results are presented to facilitate the construction of sets with specific cosine measures, allowing for the creation of a test-set to benchmark the algorithms with.