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Optim."],"published-print":{"date-parts":[[2025,6,30]]},"abstract":"<jats:p>Abstract.<\/jats:p>\n                  <jats:p>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 asymptotically converges to a value close to the optimum. Furthermore, we also provide two sparse homogenized moment-SOS hierarchies without perturbations, each having asymptotic convergence to the exact optimum. As by-products, new Positivstellens\u00e4tze are obtained for sparse positive polynomials on unbounded sets. Extensive numerical experiments demonstrate the power of our approaches in solving sparse polynomial optimization problems on unbounded sets with up to thousands of variables. 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