The T-product for third-order tensors has been used extensively in the literature. In this paper, we first introduce first-order and second-order T-derivatives for the multi-variable real-valued function with the tensor T-product. Inspired by an equivalent characterization of a twice continuously T-differentiable multi-variable real-valued function being convex, we present a definition of the T-positive semidefiniteness of third-order symmetric tensors. After that, we extend many properties of positive semidefinite matrices to the case of third-order symmetric tensors. In particular, analogue to the widely used semidefinite programming (SDP for short), we introduce the semidefinite programming over the space of third-order symmetric tensors (T-semidefinite programming or TSDP for short), and provide a way to solve the TSDP problem by converting it into an SDP problem in the complex domain. Furthermore, we give several TSDP examples and especially some preliminary numerical results for two unconstrained polynomial optimization problems. Experiments show that finding the global minimums of polynomials via the TSDP relaxation outperforms the traditional SDP relaxation for the test examples.

三阶张量的T乘积已在文献中广泛使用。在本文中，我们首先介绍具有张量T积的多变量实值函数的一阶和二阶T导数。受到两次连续T可微分多元实值函数为凸的等效特征的启发，我们给出了三阶对称张量的T正半定性的定义。之后，我们将正半定矩阵的许多性质扩展到三阶对称张量的情况。特别是，类似于广泛使用的半定规划（简称SDP），我们介绍了三阶对称张量空间上的半定规划（简称T-半定规划或TSDP），并提供一种将TSDP问题转换为复杂域中的SDP问题的方法。此外，我们给出了几个TSDP示例，尤其是针对两个无约束多项式优化问题的一些初步数值结果。实验表明，通过TSDP弛豫找到多项式的全局最小值优于测试例的传统SDP弛豫。