Abstract
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.
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This work is partially supported by the National Natural Science Foundation of China (Grant No. 11871051).
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Zheng, MM., Huang, ZH. & Wang, Y. T-positive semidefiniteness of third-order symmetric tensors and T-semidefinite programming. Comput Optim Appl 78, 239–272 (2021). https://doi.org/10.1007/s10589-020-00231-w
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DOI: https://doi.org/10.1007/s10589-020-00231-w
Keywords
- T-product
- T-positive semidefiniteness
- T-semidefinite cone
- T-semidefinite programming
- Polynomial optimization