Consider the problem 
$$
\min p(x) \text{ subject to } g\_j(x)\le 0
\quad
p,g\_j\in\text{SOS},
\qquad
(\*)
$$


i.e. $p,g\_j$ ($j=1,\ldots,m$) are *sum of squares* (SOS) polynomials. Can this problem be solved efficiently?


The paper [1] shows that unconstrained minimization of SOS polynomials can be reduced to a convex program. Surprisingly, the paper goes onto consider general (non-SOS) constrained polynomial optimization to derive the Lasserre hierarchy, but never explicitly discusses the special case $(\*)$ above. 


[1] *Lasserre, Jean B.*, [**Global optimization with polynomials and the problem of moments**](http://dx.doi.org/10.1137/S1052623400366802), SIAM J. Optim. 11, No. 3, 796-817 (2001). [ZBL1010.90061](https://zbmath.org/?q=an:1010.90061).



Constrained optimization of sum of squares polynomials
If $g\_j$ is SOS, i.e. $g\_j=\sum\_k h\_k^2$, then $g\_j(x)\leq 0$ iff $g\_j(x)=0$, i.e. $h\_1(x)=h\_2(x)=\dots =0$. So this is a general case, although with equality constraints only.


A more interesting question would be to allow inhomogenous constraints $g\_j(x)\leq b\_j$, with $b\_j\in\mathbb{R}$. IIRC, this would be solvable with just one step of the Lassere's hierarchy, as the whole point of it is approximating arbitrary nonnegative polynomials by SOS.
(I believe the latest Lasserre's [book](https://doi.org/10.1017/CBO9781107447226) is discussing this in Chapter 11).


