  Completely positive (CP) tensors, which correspond to a generalization of CP
matrices, allow to reformulate or approximate a general polynomial optimization
problem (POP) with a conic optimization problem over the cone of CP tensors.
Similarly, completely positive semidefinite (CPSD) tensors, which correspond to
a generalization of positive semidefinite (PSD) matrices, can be used to
approximate general POPs with a conic optimization problem over the cone of
CPSD tensors. In this paper, we study CP and CPSD tensor relaxations for
general POPs and compare them with the bounds obtained via a Lagrangian
relaxation of the POPs. This shows that existing results in this direction for
quadratic POPs extend to general POPs. Also, we provide some tractable
approximation strategies for CP and CPSD tensor relaxations. These
approximation strategies show that, with a similar computational effort, bounds
obtained from them for general POPs can be tighter than bounds for these
problems obtained by reformulating the POP as a quadratic POP, which
subsequently can be approximated using CP and PSD matrices. To illustrate our
results, we numerically compare the bounds obtained from these relaxation
approaches on small scale fourth-order degree POPs.
