SONC Optimization and Exact Nonnegativity Certificates via Second-Order Cone Programming

TL;DR

This paper demonstrates that the SONC cone can be represented using second-order cones, enabling efficient polynomial optimization and exact nonnegativity certificates through SOC programming.

Contribution

It proves the SOC representability of the SONC cone and introduces a new SOC-based algorithm for polynomial optimization with exact certificates.

Findings

Efficient algorithm for large-degree polynomials

Constructive proof of SONC cone SOC representation

Hybrid scheme for exact nonnegativity certificates

Abstract

The second-order cone (SOC) is a class of simple convex cones and optimizing over them can be done more efficiently than with semidefinite programming. It is interesting both in theory and in practice to investigate which convex cones admit a representation using SOCs, given that they have a strong expressive ability. In this paper, we prove constructively that the cone of sums of nonnegative circuits (SONC) admits a SOC representation. Based on this, we give a new algorithm for unconstrained polynomial optimization via SOC programming. We also provide a hybrid numeric-symbolic scheme which combines the numerical procedure with a rounding-projection algorithm to obtain exact nonnegativity certificates. Numerical experiments demonstrate the efficiency of our algorithm for polynomials with fairly large degree and number of variables.