# Are those two Sum-Of-Squares approach for unconstrained polynomial optimization related? [closed]

This is a crosspost of mathoverflow/345282

I found 2 approaches to solve an unconstrained polynomial optimization problem using the Lasserre / SOS hierarchy: $$\inf_{x\in\mathbb{R}^n}\quad p(x)$$ where $$p$$ is a polynomial of even degree degree, $$d=2\ell$$.

1. The approach of Lasserre, which relies on Putinar's Nullstellensatz, requires to restrict to an archimedean domain, which can be done by adding the constraint $$\|x\|^2\leq a$$ for a sufficiently large $$a$$. This approach is described e.g. in Theorem 5.5. of this lecture, and yields the hierarchy:

\begin{align}\tag{P_r} \sup &\quad \lambda\\ s.t. &\quad p(x)-\lambda = q(x) + \sigma(x) (a-\sum_i x_i^2)\\ &\quad \text{for some SOS polynomials p,\sigma}\\ &\quad \text{of degree at most 2(\ell+r) and 2(\ell+r-1), respectively.} \end{align}

1. The approach of Reznick, which is described e.g. in this lecture, and consists in expressing $$p(x)-p^*$$ as a rational function (the ratio between a SOS polynomial and $$(1+\|x\|^2)^r$$ for increasing values of $$r$$):

\begin{align}\tag{Q_r} \sup &\quad \lambda\\ s.t. &\quad (1+\sum_i x_i^2)^r \ (p(x)-\lambda) = q(x)\\ &\quad \text{for some SOS polynomials p of degree 2(\ell+r).} \end{align}

So my question is: Are these two approaches related / equivalent? Is it known if one approach converges faster than the other one?