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Circuit polynomials are polynomials satisfying a number of conditions thatmake it easy to compute sharp and certifiable global lower bounds for them.Consequently, one may use them to find certifiable lower bounds for anypolynomial by writing it as a sum of circuit polynomials with known lowerbounds (if possible), in a fashion similar to the better-known sum-of-squarespolynomials. Seidler and de Wolff recently showed that sums of nonnegativecircuit (SONC) polynomials can be used to compute global lower bounds (calledSONC bounds) for polynomials in this manner in polynomial time, as long as thepolynomial is bounded from below and its support satisfies a nondegeneracyassumption. The quality of the SONC bound depends on the circuits used in thecomputation, but finding the set of circuits that yield the best attainableSONC bound among the astronomical number of candidate circuits is a non-trivialtask that has not been addressed so far. In this paper we propose an efficientmethod to compute the optimal SONC lower bound by iteratively identifying theoptimal circuits to use in the SONC bounding process. The method is based on anew proof of a recent result by Wang which states that (under the samenondegeneracy assumption) every SONC polynomial decomposes into SONCpolynomials on the same support. Our proof, based on convex programmingduality, removes the nondegeneracy assumption and motivates an algorithm thatgenerates an optimal set of circuits and computes the corresponding SONC boundin a manner that is particularly attractive for sparse polynomials. The methodis implemented and tested on a large set of sparse polynomial optimizationproblems with up to 40 unknowns, of degree up to 60, and up to 3000 monomialsin the support. The results indicate that the method is efficient in practice.[abstract truncated for arXiv character limits]

Duality of sum of nonnegative circuit polynomials and optimal SONC bounds