论文标题
三阶对称张量和T-米边编程的T阳性半铁质
T-positive semidefiniteness of third-order symmetric tensors and T-semidefinite programming
论文作者
论文摘要
三阶张量的T产品已在文献中广泛使用。在本文中，我们首先引入了使用张量t-product的多向量实用值函数的一阶和二阶T衍生物；并受到两次连续t差异的多矢量实价函数的等效表征的启发，我们介绍了三阶对称张量的t阳性半足迹的定义。之后，我们将阳性半金属矩阵的许多特性扩展到三阶对称张量的情况。特别是，类似于广泛使用的半决赛编程（简称SDP），我们在三阶对称张量空间（T-Semidefinite编程或简短的TSDP）上介绍了半决赛编程，并提供了一种通过将其转换为复合域中的SDP问题来解决TSDP问题的方法。此外，我们提供了几个示例，可以作为TSDP问题提出（或放松），并报告两个无约束的多项式优化问题的初步数值结果。实验表明，通过TSDP松弛找到多项式的全局最小值优于传统的SDP松弛，以实现测试示例。
The T-product for third-order tensors has been used extensively in the literature. In this paper, we first introduce the first-order and second-order T-derivatives for the multi-vector real-valued function with the tensor T-product; and inspired by an equivalent characterization of a twice continuously T-differentiable multi-vector 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 third-order symmetric tensor space (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 examples which can be formulated (or relaxed) as TSDP problems, and report 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.
