A Positivstellensatz for Sums of Nonnegative Circuit Polynomials

TL;DR

This paper introduces a Positivstellensatz for SONC polynomials, providing a new certificate of nonnegativity that enables efficient computation of polynomial bounds on compact sets, independent of sums of squares.

Contribution

The paper proves the SONC cone is full-dimensional, establishes a Positivstellensatz for SONC polynomials, and develops a hierarchy of efficiently computable lower bounds.

Findings

SONC cone is full-dimensional in nonnegative polynomials

Positivstellensatz guarantees representation of positive polynomials on semi-algebraic sets

Hierarchy of lower bounds converges to polynomial minimum

Abstract

Recently, the second and the third author developed sums of nonnegative circuit polynomials (SONC) as a new certificate of nonnegativity for real polynomials, which is independent of sums of squares. In this article we show that the SONC cone is full-dimensional in the cone of nonnegative polynomials. We establish a Positivstellensatz which guarantees that every polynomial which is positive on a given compact, semi-algebraic set can be represented by the constraints of the set and SONC polynomials. Based on this Positivstellensatz we provide a hierarchy of lower bounds converging against the minimum of a polynomial on a given compact set $K$. Moreover, we show that these new bounds can be computed efficiently via interior point methods using results about relative entropy functions.