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How to project onto extended second order cones
The extended second order cones were introduced by S. Z. N\'emeth and G.
Zhang in [S. Z. N\'emeth and G. Zhang. Extended Lorentz cones and variational
inequalities on cylinders. J. Optim. Theory Appl., 168(3):756-768, 2016] for
solving mixed complementarity problems and variational inequalities on
cylinders. R. Sznajder in [R. Sznajder. The Lyapunov rank of extended second
order cones. Journal of Global Optimization, 66(3):585-593, 2016] determined
the automorphism groups and the Lyapunov or bilinearity ranks of these cones.
S. Z. N\'emeth and G. Zhang in [S.Z. N\'emeth and G. Zhang. Positive operators
of Extended Lorentz cones. arXiv:1608.07455v2, 2016] found both necessary
conditions and sufficient conditions for a linear operator to be a positive
operator of an extended second order cone. This note will give formulas for
projecting onto the extended second order cones. In the most general case the
formula will depend on a piecewise linear equation for one real variable which
will be solved by using numerical methods
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