结构张量的特征值分析与互补问题研究

The structured analysis and eigenvalue theory of higher order tensor can be used to solve the nonlinear complementarity problem, so by means of them, the multivariate polynomial optimization under the spherical constraints may be solved, which gives a new idea for solving the homogeneous polynomial optimization. In this project, we will employ the structured analysis of tensor together with nonlinear spectral theory and metric fixed point theory as well as its iteration convergence to systematically study the tensor complementarity problem and eigenvalue problems and develop the effective numerical methods, and then employ them to analyze and solve the corresponding polynomial optimization. We will mainly study the following four major topics: we will (1) make out the correspondence between the feasible solution of tensor complementary problem and some class of structured tensor; (2) study the H- (Z-) eigenvalue problem of the special structured tensor and the tensor complementary problem, and then to apply to the corresponding homogeneous polynomial optimization problems; (3) research the Pareto H- (Z-) eigenvalues problem of weakly symmetry structured tensor and tensor complementarity problem, and then develop error bounds of tensor complementarity problem; (4) explore the internal relations between the structured properties of the non-Q-tensor such as the copositive tensor and semi-positive tensor and the feasible solution of the corresponding complementarity problems

高阶张量的结构分析与特征值理论能被用于分析求解非线性互补问题，进而可被用于分析求解相应齐次多项式优化问题，这为非线性互补问题与齐次多项式优化问题的求解提供了一个新思路。本项目将采用张量结构分析与非线性谱理论、度量不动点理论及其迭代收敛特性等理论相结合的方法，对高阶张量互补问题与特征值理论进行系统的研究，进而用于齐次多项式优化问题的分析求解。主要包括四个方面内容：(1)揭示张量互补问题可行解与某类张量结构特性之间的结构关系；(2)研究具有特殊结构张量的H-(Z-)特征值与互补问题，探索将其用于相应齐次多项式优化问题；（3）研究弱对称结构张量的Pareto H-（Z-）特征值与互补问题，研究互补问题解的误差界；（4）探索协正张量、半正张量等非Q-张量结构特性与相应互补问题可行解之间的内在联系。

高阶张量的结构分析与特征值理论能被用于分析求解非线性互补问题，进而可被用于分析求解相应齐次多项式优化问题，这为非线性互补问题与齐次多项式优化问题的求解提供了一个新思路。本项目将采用张量结构分析与非线性谱理论、度量不动点理论及其迭代 收敛特性等理论相结合的方法，对高阶张量互补问题与特征值理论进行系统的研究，进而用于齐次多项式优化问题的分析求解。主要包括四个方面内容:(1)揭示张量互补问题解集非空有界性与某类张量结构特性之间的结构关系;(2)研究具有特殊结构张量的H-(Z-)特征值与张量互补问题，探索将其用于相应齐次多项式优化问题;(3)研究弱对称结构张量的Pareto H-(Z-)特征值与互补问题，研究无限维Hilbert张量的谱与算子范数等特性;(4)探索协正张量、半正张量等非Q-张量结构特性与相应互补问题可行解之间的内在联系。经过4年深入系统地研究，取得了一些重要成果，主要有: (1) 确立了张量互补问题解集的有界性等价于相应张量是R_0-张量，获得了严格半正张量与B-张量等的特殊结构确定了相应张量互补问题解集的上下界; (2) 建立了B-张量、Hilbert张量、半正张量与列充分张量等的特殊谱性质与算子范数特性; (3) 首次把m阶无限维Hilbert张量到解析函数空间，获得了Bergman空间上Hilbert张量算子的有界性; (4) 验证了张量特征值互补问题的谱投影梯度算法全局收敛到对称张量对的某个Pareto特征值，给出了求解强M-张量多重线性方程组的预条件张量分离迭代算法.