Duality of sum of nonnegative circuit polynomials and optimal SONC bounds

TL;DR

This paper introduces an efficient iterative method to compute the optimal SONC lower bounds for polynomials by identifying the best circuits, improving bounds for sparse polynomial optimization problems.

Contribution

It provides a new proof removing the nondegeneracy assumption and develops an algorithm to find the optimal set of circuits for SONC bounds, enhancing computational efficiency.

Findings

Method efficiently computes optimal SONC bounds for large sparse polynomials.

Algorithm successfully applied to problems with up to 40 variables and 3000 monomials.

Results demonstrate practical efficiency and effectiveness of the proposed approach.

Abstract

Circuit polynomials are polynomials satisfying a number of conditions that make it easy to compute sharp and certifiable global lower bounds for them. Consequently, one may use them to find certifiable lower bounds for any polynomial by writing it as a sum of circuit polynomials with known lower bounds (if possible), in a fashion similar to the better-known sum-of-squares polynomials. Seidler and de Wolff recently showed that sums of nonnegative circuit (SONC) polynomials can be used to compute global lower bounds (called SONC bounds) for polynomials in this manner in polynomial time, as long as the polynomial is bounded from below and its support satisfies a nondegeneracy assumption. The quality of the SONC bound depends on the circuits used in the computation, but finding the set of circuits that yield the best attainable SONC bound among the astronomical number of candidate circuits is a non-trivial task that has not been addressed so far. In this paper we propose an efficient method to compute the optimal SONC lower bound by iteratively identifying the optimal circuits to use in the SONC bounding process. The method is based on a new proof of a recent result by Wang which states that (under the same nondegeneracy assumption) every SONC polynomial decomposes into SONC polynomials on the same support. Our proof, based on convex programming duality, removes the nondegeneracy assumption and motivates an algorithm that generates an optimal set of circuits and computes the corresponding SONC bound in a manner that is particularly attractive for sparse polynomials. The method is implemented and tested on a large set of sparse polynomial optimization problems with up to 40 unknowns, of degree up to 60, and up to 3000 monomials in the support. The results indicate that the method is efficient in practice. [abstract truncated for arXiv character limits]