  We consider a geometric programming problem consisting in minimizing a
function given by the supremum of finitely many log-Laplace transforms of
discrete nonnegative measures on a Euclidean space. Under a coerciveness
assumption, we show that a $\varepsilon$-minimizer can be computed in a time
that is polynomial in the input size and in $|\log\varepsilon|$. This is
obtained by establishing bit-size estimates on approximate minimizers and by
applying the ellipsoid method. We also derive polynomial iteration complexity
bounds for the interior point method applied to the same class of problems. We
deduce that the spectral radius of a partially symmetric, weakly irreducible
nonnegative tensor can be approximated within $\varepsilon$ error in poly-time.
For strongly irreducible tensors, we also show that the logarithm of the
positive eigenvector is poly-time computable. Our results also yield that the
the maximum of a nonnegative homogeneous $d$-form in the unit ball with respect
to $d$-H\"older norm can be approximated in poly-time. In particular, the
spectral radius of uniform weighted hypergraphs and some known upper bounds for
the clique number of uniform hypergraphs are poly-time computable.
