Projecting onto a Capped Rotated Second-Order Cone

TL;DR

This paper derives a closed-form expression for projecting onto a capped rotated second-order cone, facilitating convex relaxations in nonlinear programs with binary variables, with three distinct cases including classical and nontrivial projections.

Contribution

It introduces a novel closed-form projection formula for a complex convex set relevant in nonlinear optimization relaxations.

Findings

Closed-form projection formula derived for the capped rotated second-order cone.

Three cases identified, including classical and nontrivial projections.

Conditions established for solutions on the intersection of the cone and a box facet.

Abstract

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.