正交张量优化问题的理论与算法

The class of optimization problems on orthogonally decomposable tensors (ODTs) is a class of mathematical programming problems based on decompositions and approximations of ODTs, which plays key roles in diverse applications, serves as the most important foundation of tensor optimization problems, and is a both theoretically significant and practically useful research topic. This project will carry out a systematic study on theory, algorithms and efficient implementations of optimization problems associated to both decompositions and approximations of ODTs as well as their important subclasses of completely orthogonally decomposable tensors and partially orthogonally decomposable tensors; and establish the framework of optimization problems on ODTs based on their decompositions and approximations. Especially, this project will put emphasis on the existence, uniqueness, robustness, certifications of decompositions, as well as uniqueness, error bound analysis of the solution sets of the corresponding optimization problems of approximations, etc. for ODTs; investigations on algorithms for finding decompositions and approximations of ODTs in large scale and their perturbation analysis and convergent properties, etc.; and highly efficient software implementations and their practical applications.

正交张量优化问题是指以正交张量分解和正交张量逼近为核心的数学规划问题，此类问题在众多实际应用问题的求解中起着关键作用，是张量优化问题的重要组成部分，其研究具有重要的理论意义和实际应用价值。本项目将系统地研究由完全正交张量和部分正交张量等重要子类构成的正交张量的分解和逼近等优化问题的理论、算法和高效数值实现；建立以正交张量分解和逼近为核心的正交张量优化问题的系统框架。特别地，本项目将重点讨论正交张量分解的存在性，唯一性，鲁棒性，判别准则，及对应正交张量逼近优化问题解集的唯一性，误差界分析等理论性质；研究求解大规模正交张量分解和逼近的高效算法，建立其扰动分析，收敛性分析等理论基础；设计出高效的数值程序实现并应用于实际问题。

项目以张量的正交分解和逼近问题为中心进行了一系列的相关研究。研究的主要内容为张量正交分解的基本性质和高效算法、张量低秩正交逼近问题解集的基本性质和相应算法的全局收敛性和局部收敛率。项目建立了完全正交可分解张量集合的几何性质，包括局部微分同胚等与高效算法设计相应的重要性质；刻画了完全正交可分解张量的奇异向量对的非退化性。特别地，项目系统地研究了张量低秩正交逼近问题解的非退化性，为后续的研究奠定了坚实的基础；据此，完全解决了张量秩一逼近问题的交替极小化方法的线性收敛性问题。同时，项目还研究了与张量低秩逼近问题相关的系列其他基础性质，包括谱的对称性、不变量、子问题的高效算法求解、相应投影函数的次微分等。