A Positivstellensatz for Sums of Nonnegative Circuit Polynomials

Mareike Dressler; Sadik Iliman; Timo de Wolff

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arXiv · 1607.06010

A Positivstellensatz for Sums of Nonnegative Circuit Polynomials

Mareike Dressler·Sadik Iliman·Timo de Wolff

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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.

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BibTeX RIS

Mareike Dressler, Sadik Iliman, Timo de Wolff. 2017-03-17. A Positivstellensatz for Sums of Nonnegative Circuit Polynomials. https://arxiv.org/abs/1607.06010

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