  We prove that every semidefinite moment relaxation of a polynomial
optimization problem (POP) with a ball constraint can be reformulated as a
semidefinite program involving a matrix with constant trace property (CTP). As
a result such moment relaxations can be solved efficiently by first-order
methods that exploit CTP, e.g., the conditional gradient-based augmented
Lagrangian method. We also extend this CTP-exploiting framework to large-scale
POPs with different sparsity structures. The efficiency and scalability of our
framework are illustrated on second-order moment relaxations for various
randomly generated quadratically constrained quadratic programs.
