Operator polynomials L(λ) = λ l I - λ l - 1 A l - 1 - ... - λA 1 - A 0 are considered, where A 0 ,..., A l - 1 are nonnegative operators in a Banach space X with normal cone X + . For x X + we define the local spectral radius r L (x) and the lower and upper Collatz-Wielandt numbers r L (x) and r L (x), respectively, of x with respect to L. We characterize these quantities with the help of corresponding quantities with respect to the first companion operator belonging to L and the operator function S(λ) = A l - 1 + λ - 1 A l - 2 + ... + λ - l + 1 A 0 . Many properties known in the linear case l = 1 have generalizations to the case l > 1; e.g., r L (x) =< r L (x) =< r L (x) is true for all x X + . From these local results we obtain results for the global spectral radius r(L), which were proved earlier under more restrictive conditions.