Computing critical angles between two convex cones

TL;DR

This paper introduces an efficient numerical method for computing critical angles between convex cones by transforming the problem into a fractional programming task and solving it with a novel algorithm, validated by experiments in high-dimensional spaces.

Contribution

The paper presents a new approach to compute critical angles between convex cones using fractional programming and a partial linearization algorithm, offering high efficiency and reliability.

Findings

The proposed algorithm asymptotically computes critical angles.

Numerical experiments show the method is fast in high-dimensional spaces.

Only a few seconds are needed for large-dimensional problems.

Abstract

This paper addresses the numerical computation of critical angles between two convex cones in finite-dimensional Euclidean spaces. We present a novel approach to computing these critical angles by reducing the problem to finding stationary points of a fractional programming problem. To efficiently compute these stationary points, we introduce a partial linearization-like algorithm that offers significant computational advantages. Solving a sequence of strictly convex subproblems with straightforward solutions in several settings gives the proposed algorithm high computational efficiency while delivering reliable results: our theoretical analysis demonstrates that the proposed algorithm asymptotically computes critical angles. Numerical experiments validate the efficiency of our approach, even when dealing with problems of relatively large dimensions: only a few seconds are necessary to compute critical angles between different types of cones in spaces of dimension 1000.