A positivstellensatz for sums of nonnegative circuit polynomials

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Abstract

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.

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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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