  Several notions of spectral radius arise in the study of nonlinear
order-preserving positively homogeneous self-maps of cones in Banach spaces. We
give conditions that guarantee that all these notions lead to the same value.
In particular, we give a Collatz-Wielandt type formula, which characterizes the
growth rate of the orbits in terms of eigenvectors in the closed cone or
super-eigenvectors in the interior of the cone. This characterization holds
when the cone is normal and when a quasi-compactness condition, involving an
essential spectral radius defined in terms of $k$-set-contractions, is
satisfied. Some fixed point theorems for non-linear maps on cones are derived
as intermediate results. We finally apply these results to show that non-linear
spectral radii commute with respect to suprema and infima of families of order
preserving maps satisfying selection properties.
