Exact Optimization via Sums of Nonnegative Circuits and Sums of AM/GM Exponentials

Abstract: 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.