Reducing Nonnegativity over General Semialgebraic Sets to Nonnegativity over Simple Sets

Abstract

SIAM Journal on Optimization, Volume 34, Issue 2, Page 1970-2006, June 2024.
Abstract. A nonnegativity certificate (NNC) is a way to write a polynomial so that its nonnegativity on a semialgebraic set becomes evident. Positivstellensätze (Psätze) guarantee the existence of NNCs. Both NNCs and Psätze underlie powerful algorithmic techniques for optimization. This paper proposes a universal approach to derive new Psätze for general semialgebraic sets from ones developed for simpler sets, such as a box, a simplex, or the nonnegative orthant. We provide several results illustrating the approach. First, by considering Handelman’s Positivstellensatz (Psatz) over a box, we construct non-SOS Schmüdgen-type Psätze over any compact semialgebraic set, that is, a family of Psätze that follow the structure of the fundamental Schmüdgen’s Psatz but where instead of SOS polynomials, any class of polynomials containing the nonnegative constants can be used, such as SONC, DSOS/SDSOS, hyperbolic, or sums of AM/GM polynomials. Second, by considering the simplex as the simple set, we derive a sparse Psatz over general compact sets which does not rely on any structural assumptions of the set. Finally, by considering Pólya’s Psatz over the nonnegative orthant, we derive a new non-SOS Psatz over unbounded sets which satisfy some generic conditions. All these results contribute to the literature regarding the use of non-SOS polynomials and sparse NNCs to derive Psätze over compact and unbounded sets. Throughout the article, we illustrate our results with relevant examples and numerical experiments.