报告题目:  T-Positive Semidefiniteness of Third-Order Symmetric Tensors and T-Semidefinite Programming

报  告  人:  黄正海教授（天津大学）

报告时间：2019年11月02日，10:30--

报告地点：数计学院4号楼229报告厅

报告摘要: 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.

报告人简介：黄正海，天津大学数学学院教授、博士生导师。主要从事最优化理论、算法及其应用方面的研究工作，在求解互补与变分不等式问题、对称锥优化与对称锥互补问题、稀疏优化、张量优化、核磁共振医学成像、人脸识别等方面取得了一些有意义的成果。已发表论文百余篇、连续获得多项国家自然科学基金资助。曾获得中科院优秀博士后奖和教育部高等学校自然科学奖二等奖。目前为中国运筹学会常务理事，中国运筹学会数学规划分会副理事长；国际期刊《Pacific Journal of Optimization》、《Applied Mathematics and Computation》和《Optimization,Statistics & Information Computing》的编委、中国核心期刊《运筹学学报》的编委。