Let us start with the optimization question, \begin{eqnarray*}min \{ c \vert c - f \in SOS_d \}\end{eqnarray*}for some function $f : \{0,1\}^n \rightarrow \mathbb{R}$ and $SOS_d $ being the cone of all polynomials on $\mathbb{R}^n$ which can be written as a sum of squares of polynomials each of degree at most $\frac{d}{2}$. This can be cast as a SDP by doing the gram matrix rewriting of any polynomial in $SOS_d$. Now how do we show that the above is dual with no duality gap to this following optimization question?  \begin{eqnarray*} &max_D \{ \tilde{\mathbb{E}}_D [f] \} \\&s.t \\&D : \{ 0,1\} ^n \rightarrow \mathbb{R} \\&\sum_{x \in \{0,1\}^n} D(x) = 1 \\& \forall u \in SOS_d \\ &\tilde{\mathbb{E}}_D [u] \geq 0 \end{eqnarray*} where for any function $g : \{0,1\}^n \rightarrow \mathbb{R}$ we have $\tilde{\mathbb{E}}_D [g] := \sum_{x \in \{0,1\}^n} D(x)g(x)$I tried writing a conic duality for the first to get it to look like an optimization over Lasserre maps but that approach didn't work...
A question about SOS duality
complexity theory;np complete;np hard;np