论文标题
用混合牛顿方法在复杂空间中进行优化
Optimization in complex spaces with the Mixed Newton Method
论文作者
论文摘要
我们提出了一种二阶方法，用于将功能的无条件最小化$ f（z）$的复杂参数。我们称其为混合的牛顿方法，这是由于使用混合的Wirtinger衍生物$ \ frac {\ partial^2f} {\ partial \ bar z \ partial z} $来计算搜索方向，而不是完整的hessian $ \ frac {\ partial^partial^2f partial^2f} {\ p p p p p p p p p p p p p p p p p p p p p p p bar z，z，^$ natials^2 $ in。该方法是针对无线网络通信中特定应用程序开发的，但其全局收敛属性在更一般的函数$ f $上是优越性的，即霍尔态函数绝对值的正方形。特别是，对于此类目标函数，最小值被吸引盆地包围，而迭代却是从其他类型的关键点中排斥的。我们为渐近收敛速率提供公式，并表明在标量情况下，该方法还原为众所周知的复杂牛顿方法，用于搜索全体形函数的零。在这种情况下，它表现出普遍的分形全局收敛模式。
We propose a second-order method for unconditional minimization of functions $f(z)$ of complex arguments. We call it the Mixed Newton Method due to the use of the mixed Wirtinger derivative $\frac{\partial^2f}{\partial\bar z\partial z}$ for computation of the search direction, as opposed to the full Hessian $\frac{\partial^2f}{\partial(z,\bar z)^2}$ in the classical Newton method. The method has been developed for specific applications in wireless network communications, but its global convergence properties are shown to be superior on a more general class of functions $f$, namely sums of squares of absolute values of holomorphic functions. In particular, for such objective functions minima are surrounded by attraction basins, while the iterates are repelled from other types of critical points. We provide formulas for the asymptotic convergence rate and show that in the scalar case the method reduces to the well-known complex Newton method for the search of zeros of holomorphic functions. In this case, it exhibits generically fractal global convergence patterns.
