  We study T-semidefinite programming (SDP) relaxation for constrained
polynomial optimization problems (POPs). T-SDP relaxation for unconstrained
POPs was introduced by Zheng, Huang and Hu in 2022. In this work, we propose a
T-SDP relaxation for POPs with polynomial inequality constraints and show that
the resulting T-SDP relaxation formulated with third-order tensors can be
transformed into the standard SDP relaxation with block-diagonal structures.
The convergence of the T-SDP relaxation to the optimal value of a given
constrained POP is established under moderate assumptions as the relaxation
level increases. Additionally, the feasibility and optimality of the T-SDP
relaxation are discussed. Numerical results illustrate that the proposed T-SDP
relaxation enhances numerical efficiency.
