非负共轭多项式：张量表达，最优化算法及应用

There are some recent problems arising from Signal Processing, Quantum Entanglement, Smart Grid etc., which can be casted as polynomial optimization in complex variables with conjugate terms. However, the current literature on complex optimization that considered the conjugate polynomial or tensor is quite limited. In this project, we shall conduct a systematic study on conjugate polynomials including the tensor representation of nonnegative conjugate polynomials and their relations; conjugate partial symmetric rank-one-tensor optimization problems; the conjugate numerical range of complex tensors; optimization algorithms for complex polynomial optimization. We expect to find some interesting results that exclusively hold for conjugate polynomials comparing to the polynomials in real variable. Finally, we shall use the algorithms proposed in this project to solve some concrete problems in Signal Processing, which can in term justify the capability of our methods.

近年来，信号处理，量子纠缠，智能电网等领域出现了一些新的问题，它们用带共轭项的复数多项式优化问题来描述。而当前研究复数优化问题的文献很少考虑将复数的共轭性融入到多项式与张量中。本项目将系统地研究带共轭项的复数多项，研究内容包括：非负共轭多项式的张量表达及关系、共轭半对称张量的秩一优化问题、复数张量的共轭数值域、复数多项式的优化算法。相对于实数多项式而言，我们期望找到一些共轭多项式所特有的结果。最后，我们还会用本项目提出的算法去求解一些信号处理中的具体问题，来验证算法的有效性。

近年来，信号处理，量子纠缠，智能电网等领域出现了一些新的问题，它们用带共轭项的复数多项式优化问题来描述。而当前研究复数优化问题的文献很少考虑将复数的共轭性融入到多项式与张量中。本项目将系统地研究带共轭项的复数多项、张量及相关的张量优化问题。研究内容包括：共轭半对称张量的秩一优化问题；对基于张量分解的非凸问题提出了多个一阶算法，并在很弱的条件下得到收敛率的结果；带非凸分布约束的高阶矩问题；对基于张量的高阶优化算法进行研究，提出了最优高阶算法和一种可实现的高阶算法，并通过数值实验验证了算法的有效性。