  In this article we combine two developments in polynomial optimization. On
the one hand, we consider nonnegativity certificates based on sums of
nonnegative circuit polynomials, which were recently introduced by the second
and the third author. On the other hand, we investigate geometric programming
methods for constrained polynomial optimization problems, which were recently
developed by Ghasemi and Marshall. We show that the combination of both results
yields a new method to solve certain classes of constrained polynomial
optimization problems. We test the new method experimentally and compare it to
semidefinite programming in various examples.
