  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.
