  We exhibit a convex polynomial optimization problem for which the
diagonally-dominant sum-of-squares (DSOS) and the scaled diagonally-dominant
sum-of-squares (SDSOS) hierarchies, based on linear programming and
second-order conic programming respectively, do not converge to the global
infimum. The same goes for the r-DSOS and r-SDSOS hierarchies. This refutes the
claim in the literature according to which the DSOS and SDSOS hierarchies can
solve any polynomial optimization problem to arbitrary accuracy. In contrast,
the Lasserre hierarchy based on semidefinite programming yields the global
infimum and the global minimizer with the first order relaxation. We further
observe that the dual to the SDSOS hierarchy is the moment hierarchy where
every positive semidefinite constraint is relaxed to all necessary second-order
conic constraints. As a result, the number of second-order conic constraints
grows quadratically in function of the size of the positive semidefinite
constraints in the Lasserre hierarchy. Together with the counterexample, this
suggests that DSOS and SDSOS are not necessarily more tractable alternatives to
sum-of-squares.
