Reducing non-negativity over general semialgebraic sets to non-negativity over simple sets

TL;DR

This paper introduces a universal method to derive new Positivstellensätze for complex semialgebraic sets from simpler sets, expanding the toolkit for polynomial non-negativity certificates in optimization.

Contribution

It develops a general framework to generate Positivstellensätze for complex sets from simpler ones, including non-SOS and sparse certificates, broadening the scope of polynomial non-negativity proofs.

Findings

Constructed non-SOS Schmüdgen-type Psatz over any compact set.

Derived sparse Psatz over general compact sets without structural assumptions.

Introduced a new non-SOS Psatz for unbounded sets satisfying certain conditions.

Abstract

A non-negativity certificate (NNC) is a way to write a polynomial so that its non-negativity on a semialgebraic set becomes evident. Positivstellens\"atze (Ps\"atze) guarantee the existence of NNCs. Both, NNCs and Ps\"atze underlie powerful algorithmic techniques for optimization. This paper proposes a universal approach to derive new Ps\"atze for general semialgebraic sets from ones developed for simpler sets, such as a box, a simplex, or the non-negative orthant. We provide several results illustrating the approach. First, by considering Handelman's Positivstellensatz (Psatz) over a box, we construct non-SOS Schm\"{u}dgen-type Ps\"atze over any compact semialgebraic set. That is, a family of Ps\"atze that follow the structure of the fundamental Schm\"{u}dgen's Psatz, but where instead of SOS polynomials, any class of polynomials containing the non-negative constants can be used, such as SONC, DSOS/SDSOS, hyperbolic or sums of AM/GM polynomials. Secondly, by considering the simplex as the simple set, we derive a sparse Psatz over general compact sets, which does not require any structural assumptions of the set. Finally, by considering P\'olya's Psatz over the non-negative 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 Ps\"atze over compact and unbounded sets. Throughout the article, we illustrate our results with relevant examples and numerical experiments.