Complexity of Geometric programming in the Turing model and application to nonnegative tensors

We consider a version of geometric programming problem consisting in minimizing a function given by the maximum of finitely many log-Laplace transforms of discrete nonnegative measures on a Euclidean space. Under a coerciveness assumption, we show that an $\varepsilon$-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 an $\varepsilon$-error in polynomial time. For strongly irreducible tensors, we show in addition that the logarithm of the positive eigenvector is polynomial time approximable. Our results also yield that the the maximum of a nonnegative homogeneous $d$-form in the $\ell_d$ unit ball can be approximated in polynomial time. In particular, the spectral radius of uniform weighted hypergraphs and some known upper bounds for the clique number of uniform hypergraphs are polynomial time computable. In contrast, we provide an example showing that the Phase I approach needs exponentially many bits to solve the feasibility problem in geometric programming.