  This paper establishes a closed-form expression for projecting onto a capped
rotated second-order cone. This convex set arises in the perspective relaxation
of mixed-integer nonlinear programs (MINLP) with binary indicator variables.
The rapid computation of the projection onto this set is expected to enable the
development of effective methods for solving the continuous relaxation of
MINLPs whose feasible region may involve a Cartesian product of a large number
of such sets. The closed-form established herein consists of seven cases, one
of which is a solution of a cubic equation and another is a solution of a
quartic equation. Although quartic equations possess closed-form solutions,
numerical solutions are typically used in practice. Based on bounds that we
prove using additional case analysis, we develop a specialized bisection-based
method to solve the resulting quartic equation. In experiments we first
demonstrate that the projection problem is solved faster and more accurately
with our closed-form, together with a standard polynomial equation solver,
compared with a general state-of-the-art interior-point solver and compared
with a state-of-the art conic first-order method solver. We also demonstrate
the efficacy of our bisection-based specialized numerical method for solving
the quartic equation.
