3 research outputs found
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
Generalized power cones: optimal error bounds and automorphisms
Error bounds are a requisite for trusting or distrusting solutions in an
informed way. Until recently, provable error bounds in the absence of
constraint qualifications were unattainable for many classes of cones that do
not admit projections with known succinct expressions. We build such error
bounds for the generalized power cones, using the recently developed framework
of one-step facial residual functions. We also show that our error bounds are
tight in the sense of that framework. Besides their utility for understanding
solution reliability, the error bounds we discover have additional applications
to the algebraic structure of the underlying cone, which we describe. In
particular we use the error bounds to compute the dimension of the automorphism
group for the generalized power cones, and to identify a set of generalized
power cones that are self-dual, irreducible, nonhomogeneous, and perfectComment: 24 pages, title change, some minor fixes throughout the paper and
removed the appendix. Comments welcom