How to project onto extended second order cones

The extended second order cones were introduced by Nemeth and Zhang (J Optim Theory Appl 168(3):756–768, 2016) for solving mixed complementarity problems and variational inequalities on cylinders. Sznajder (J Glob Optim 66(3):585–593, 2016) determined the automorphism groups and the Lyapunov or bilinearity ranks of these cones. Nemeth and Zhang (Positive operators of extended Lorentz cones, 2016. arXiv:1608.07455v2) found both necessary conditions and sufficient conditions for a linear operator to be a positive operator of an extended second order cone. In this note we give formulas for projecting onto the extended second order cones. In the most general case the formula depends on a piecewise linear equation for one real variable which is solved by using numerical methods.