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

A Positivstellensatz for Sums of Nonnegative Circuit Polynomials