  We provide two hybrid numeric-symbolic optimization algorithms, computing
exact sums of nonnegative circuits (SONC) and sums of
arithmetic-geometric-exponentials (SAGE) decompositions. Moreover, we provide a
hybrid numeric-symbolic decision algorithm for polynomials lying in the
interior of the SAGE cone. Each framework, inspired by previous contributions
of Parrilo and Peyrl, is a rounding-projection procedure.
  For a polynomial lying in the interior of the SAGE cone, we prove that the
decision algorithm terminates within a number of arithmetic operations, which
is polynomial in the number of terms of the input, and linear in the distance
to the boundary of the cone. We also provide experimental comparisons regarding
the implementation of the two optimization algorithms.
