Projecting onto a Capped Rotated Second-Order Cone

We derive a closed-form expression for the projection onto a capped rotated second-order cone – a convex set that arises in perspective relaxations of nonlinear programs with binary indicator variables. The closed-form solution involves three distinct cases, one of which reduces to the classical projection onto a second-order cone. The remaining two cases yield nontrivial projections, for which we provide necessary and sufficient conditions under which the solution lies on the intersection of the cone and a facet of a box.