Projecting onto the intersection of a cone and a sphere

TL;DR

This paper derives closed-form projection formulas onto the intersection of a cone with a sphere or ball, facilitating efficient computations in optimization problems involving such sets.

Contribution

It provides systematic analysis and explicit formulas for projecting onto the intersection of cones with spheres or balls, including practical examples and numerical experiments.

Findings

Closed-form projection formulas derived for specific cone intersections

Examples include finitely generated cones, Lorentz cone, and positive semidefinite cone

Numerical experiments demonstrate the practical utility of the formulas

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.