Tensor Train分解是表示高阶数据有效的分解形式之一，其可分离的结构特点对大规模张量优化问题的求解带来极大便利。在本项目中，我们将基于Tensor Train分解，围绕张量优化中特征值互补和张量完备化这两类重要的问题展开研究，采用张量分析(如分解、谱理论等)与优化方法(如对偶、松弛等)相结合的技术，分别从交替优化和Riemannian优化的角度对所研究的问题进行理论分析和算法设计。内容包括：1)建立张量特征值互补问题解的性质，采用交替方向法和投影技术相结合思路，设计有效算法并进行收敛性分析；2)针对张量完备化问题，建立相应的张量优化模型，分析目标函数在固定张量TT秩所构成的Riemannian流形上的性质(包括半光滑、强半光滑等)，并设计有效的Riemannian投影梯度型算法；3)将所提出的算法应用到MIMO系统仿真、机器学习等实际问题，进一步检验算法的可行性和有效性。

Tensor Train decomposition is one of the most efficient form to express high-order data. Due to the separable structure, this kind of decomposition brings great convenience for solving large-scale tensor optimization problems. In this project, based on Tensor Train decomposition, we consider two kinds of tensor optimization problems: tensor eigenvalue complementarity problem and tensor completion problem. By integrating tensor analysis (e.g. decomposition, spectral theory) and optimization techniques (e.g. duality, relaxation), we carry out the theoretical analysis and algorithm design from the perspective of alternating optimization and Riemannian optimization, respectively. The main research content includes: 1) For tensor eigenvalue complementarity problem, we analyze the properties of the solutions and propose efficient algorithms with convergence analysis by combining the alternating direction method and projection technique. 2) For tensor completion problem, we establish corresponding tensor optimization model, analyze the properties (e.g. semismooth, strongly semismooth) of objective function on the Riemannian manifold of tensors with fixed TT-ranks, and propose efficient Riemannian gradient method. 3) We apply our proposed methods to practical problems, such as MIMO system simulation and machine learning, in order to check the feasibility and efficiency of our methods.