  This paper considers sparse polynomial optimization with unbounded sets. When
the problem possesses correlative sparsity, we propose a sparse homogenized
Moment-SOS hierarchy with perturbations to solve it. The new hierarchy
introduces one extra auxiliary variable for each variable clique according to
the correlative sparsity pattern. Under the running intersection property, we
prove that this hierarchy has asymptotic convergence. Furthermore, we provide
two alternative sparse hierarchies to remove perturbations while preserving
asymptotic convergence. As byproducts, new Positivstellens\"atze are obtained
for sparse positive polynomials on unbounded sets. Extensive numerical
experiments demonstrate the power of our approach in solving sparse polynomial
optimization problems on unbounded sets with up to thousands of variables.
Finally, we apply our approach to tackle two trajectory optimization problems
(block-moving with minimum work and optimal control of Van der Pol).
