Projecting onto the intersection of a cone and a sphere
For researchers in optimization and matrix analysis, it offers new computational tools for problems involving cone-sphere intersections, though the results are incremental extensions of known projection techniques.
The paper provides closed-form expressions for projecting onto the intersection of a cone and a sphere or ball, which previously lacked such formulas. Numerical experiments demonstrate the utility for determining copositivity of matrices.
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.