基于复数神经网络的极小极大分式规划理论研究

Optimization problem has plenty of applications in scientific research and manufacturing production. Neurodynamic optimization algorithm plays an important role in the calculation of lagre-scale optimization problems. Recently, minimax fractional optimization problems in complex space,which are applied in various fileds such as filter thoery and statistic signal processing, have drawn interests of more and more researchers . However, few researchers have studied the minimax fractional programming problems in complex space by using complex-valued neural networks. Although the optimization problems in complex space can be transformed to real space by splitting the complex variables to their real and imaginary parts, in some cases such an approch may be quite complicated or even impossible. Besides, in some optimization problems in complex space, the variables and constraint functions may be represented in moduls and argument, which makes the analysis of the optimization conditions more difficult. In this project, we extend the optimization theories, such as KKT condition, dual theory and projection theory from real space to complex space and obtain the optimal conditions represented in complex gradient and constrcut the corresponding complex-valued neural networks to solve the minimax fractional programming problems. Since the complex-valued neural networks that we construct may be discontinuous, we take use of the approach of differential inclusion and Fillippov’s theory to analyze the dynmamical behaviors of the complex-valued neural networks, including the stability and convergency. It is the first time that this project takes use of complex-valued neural networks to deal with the algorithm of minimax fractioanl problems in complex-space, which is quite creative. The results of the project will be greatly helpful to the algorithm and applications to the optimization problems in complex space.

神经动态优化算法在处理大规模优化问题的计算中起着关键的作用。近年来，复空间上的极小极大分式规划问题因其在滤波理论，信号处理等方面的应用引起了人们的重视，但利用复数神经网络研究该问题却还是空白。虽然复空间上的优化问题可以通过将实部和虚部拆开的方式利用实优化算法求解，但是往往相当复杂，有些情况甚至无法拆开；此外，在复空间的优化问题中，目标函数和约束条件往往用模和幅角表示，这也使得对优化条件的分析变得更加困难。本课题将建立复空间上的KKT条件，对偶理论和投影定理，得到极小极大分式规划以复梯度形式表示的最优解条件，并据此构造相应的复数神经网络，利用微分包含及Fillippov理论，对网络的动力学性质包括稳定性和收敛性进行分析，从而求得问题的最优解。本项目将首次用复数神经网络的方法研究极小极大分式规划问题的算法，填补在该问题研究上的空白，项目的结论对于复空间上优化问题的算法与应用有一定的指导意义。

神经动力学优化算法是处理大规模优化问题的重要算法。复空间上极小极大分式规划问题在滤波理论和信号处理方面有着重要的应用，但利用复数神经网络来研究该问题还是空白。本项目利用松弛CR微积分理论得到复空间上的KKT条件，对偶理论和投影定理，结合广义Charnes-Cooper变换，将分式规划问题转化为非分式规划问题，得到用复梯度表示的复优化问题最优解存在的充分条件；进而构造复数投影神经网络，求解相应的复优化问题；计算机仿真验证了所得算法的有效性。该方法为复值分式规划以及复值极小极大分式规划问题提供了一个有效的解决方案和框架，对今后复优化问题的进一步研究奠定了坚实的基础。此外，本项目还对复数神经网络的动力学进行了深入的分析。利用半离散化，Poincare映射，M-矩阵，矩阵不等式，自适应控制，压缩映射原理和同胚映射原理等方法，我们对离散和连续复数神经网络，复数Cohn-Grossberg神经网络及四元数神经网络的全局指数稳定性，周期性，μ稳定性，鲁棒稳定性以及多稳定性等动力学性质进行了深入的探讨，得到了有意义的结果。这些结果不仅对研究复数神经网络的动力学性质大有帮助，而且对研究基于复数神经网络的神经动力学优化算法也很有价值。本项目在研期间发表SCI论文8篇，EI会议论文2篇。