张量计算中的半定松弛算法研究

In the era of big data, tensor are often used to express the large scale multimensional data of applied science. Tensor computations become more and more important in data science and mathematics. This project aims to establish some semidefinite relaxation methods for tensor computations. The following are included: (1) model multilinear Pagerank and other practical problems as tensor eigenvalue problems, reformulate them as polynomial optimization and design semidefinite relaxation methods with global convergence and finite convergence; (2) discuss the structure properties of structured tensors, establish them as optimization problem and propose the semidefinite relaxation methods to detect structured tensor; (3) for tensor decomposable detection problems in quantum physics, establish semidefinite relaxation methods to detect the decomposable tensors. (4) establish the semidefinite relaxation methods for higher order and higher dimensional tensor or complex tensor computation problems by use of symmetrical or sparse structure of tensor. This project has scientific significance and practical values for not only providing numerical methods for tensor computation, but also applications in applied sciences such as quantum physics.

随着大数据时代的到来，张量已成为众多应用学科中数据表示的有效手段，而张量计算也成为数据科学与数学的一个重要问题。本项目旨在对应用学科中的张量计算问题建立快速有效的半定松弛算法，主要内容包括：(1)将多线性Pagerank等实际问题建模为张量特征值问题，构造合适的多项式优化模型并设计具有全局收敛性质与有限收敛性质的半定松弛算法；(2)研究P张量，SOS张量等结构张量的理论与性质，建立合适的优化模型并建立半定松弛算法有效判定张量的结构性质；(3)将量子物理的经典性与可分离性等判定问题转化为具有某种性质的张量分解判定问题，对相应问题建立合适的优化模型并设计快速易算的半定松弛算法；(4)利用对称性、稀疏性等结构性质建立适用于高阶、高维张量与复张量计算的半定松弛算法。该项目的实施不仅能为张量计算问题本身提供新的数值方法，还可应用到实际问题中，因而具有重要的理论意义与实用价值。

张量是众多应用学科中数据表示的有效手段，因而张量计算成为数据科学与数学研究中的一个重要课题。本项目主要研究各应用学科中张量计算问题的建模与计算，主要研究内容包括：建立了P张量与P_0张量判定的半定松弛算法；建立了二阶锥上张量互补问题与张量特征值互补问题以及带有二阶锥互补约束的多项式优化问题的半定松弛算法；建立了非负张量Perron特征对求解的半定松弛算法；建立了低秩张量完成问题的各种优化模型与相应的算法设计。项目负责人共发表学术论文19篇（含在线发表4篇），其中在SIAM Journal on Control and Optimization上1篇，在SIAM Journal on Matrix Analysis and Applications上1篇；在Computational Optimization and Applications上4篇；在Journal of Global Optimization上1篇，在Journal of Scientific Computing上1篇等。

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