Recently, the second and third authors 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 paper 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, semialgebraic 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 to 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.
CITATION STYLE
Dressler, M., Iliman, S., & de Wolff, T. (2017). A positivstellensatz for sums of nonnegative circuit polynomials. SIAM Journal on Applied Algebra and Geometry, 1(1), 536–555. https://doi.org/10.1137/16M1086303
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