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] <cite authors="Lasserre, Jean B.">_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).</cite>