T-semidefinite programming relaxation with third-order tensors for constrained polynomial optimization

We study T-semidefinite programming (SDP) relaxation for constrained polynomial optimization problems (POPs). T-SDP relaxation for unconstrained POPs was introduced by Zheng et al. (JGO 84:415–440, 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.