Projection onto hyperbolicity cones and beyond: a dual Frank-Wolfe approach

TL;DR

This paper introduces a dual Frank-Wolfe algorithm for projecting points onto hyperbolicity cones, providing a scalable and closed-form solution approach for complex convex cones, with extensive numerical validation.

Contribution

It proposes a novel dual Frank-Wolfe method for hyperbolicity cone projection, enabling closed-form subproblems and handling large-scale cases efficiently.

Findings

The method outperforms interior point and accelerated gradient methods in experiments.

It efficiently handles hyperbolic polynomials with millions of monomials.

Applicable to p-cones beyond hyperbolicity cones.

Abstract

We discuss the problem of projecting a point onto an arbitrary hyperbolicity cone from both theoretical and numerical perspectives. While hyperbolicity cones are furnished with a generalization of the notion of eigenvalues, obtaining closed form expressions for the projection operator as in the case of semidefinite matrices is an elusive endeavour. To address that we propose a Frank-Wolfe method to handle this task and, more generally, strongly convex optimization over closed convex cones. One of our innovations is that the Frank-Wolfe method is actually applied to the dual problem and, by doing so, subproblems can be solved in closed-form using minimum eigenvalue functions and conjugate vectors. To test the validity of our proposed approach, we present numerical experiments where we check the performance of alternative approaches including interior point methods and an earlier accelerated gradient method proposed by Renegar. We also show numerical examples where the hyperbolic polynomial has millions of monomials. Finally, we also discuss the problem of projecting onto p-cones which, although not hyperbolicity cones in general, are still amenable to our techniques.