  This paper studies distributionally robust optimization (DRO) when the
ambiguity set is given by moments for the distributions. The objective and
constraints are given by polynomials in decision variables. We reformulate the
DRO with equivalent moment conic constraints. Under some general assumptions,
we prove the DRO is equivalent to a linear optimization problem with moment and
psd polynomial cones. A moment-SOS relaxation method is proposed to solve it.
Its asymptotic and finite convergence are shown under certain assumptions.
Numerical examples are presented to show how to solve DRO problems.
