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

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, such as the running intersection property. 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 and generalize recent related results in the literature regarding the use of non-SOS polynomials and sparse NNCs to derive Ps\"atze over compact and unbounded sets. Moreover, in contrast to most related literature, our approach is based on basic real analysis rather than algebraic geometry tools.