Projecting onto the intersection of a cone and a sphere

Abstract: The projection onto the intersection of sets generally does not allow for a closed form even when the individual projection operators have explicit descriptions. In this work, we systematically analyze the projection onto the intersection of a cone with either a ball or a sphere. Several cases are provided where the projector is available in closed form. Various examples based on finitely generated cones, the Lorentz cone, and the the cone of positive semidefinite matrices are presented. The usefulness of our formulae is illustrated by numerical experiments for determining copositivity of real symmetric matrices.