We establish a connection between multiplication operators and shift operators. Moreover, we derive positive semidefinite conditions of finite rank moment sequences and use these conditions to strengthen Lasserre's hierarchy for real and complex polynomial optimization. Integration of the strengthening technique with sparsity is considered. Extensive numerical experiments show that our strengthening technique can significantly improve the bound (especially for complex polynomial optimization) and allows to achieve global optimality at lower relaxation orders, thus providing substantial computational savings.
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