论文标题
回溯新Q-Newton的方法：一种用于优化和求解方程系统的良好算法
Backtracking New Q-Newton's method: a good algorithm for optimization and solving systems of equations
论文作者
论文摘要
在本文中，通过将新Q -Newton的算法结合在作者以前的联合工作中开发的新Q -Newton的方法 - 与Armijo的回溯线搜索相结合，我们解决了牛顿方法遇到的融合问题（例如，融合到鞍点或吸引超过1分的支持称），同时吸引了快速的纽顿融合速度的速率。我们还为一般的二阶方法开发了这样一种方法的家庭，其中一些人对准牛顿的方法有利。开发的算法非常易于实现。从动态系统的角度来看，新的迭代方法具有一个有趣的功能：虽然它是确定性的，但它对Armijo的回溯线搜索的依赖性使其表现就像一个随机过程，因此可以帮助其具有良好的性能。在实验方面，我们将算法的性能与牛顿方法对某些方程式（实际变量和复杂变量）上的众所周知的变化进行了比较。我们还探索了回溯到新Q-Newton的方法引起的一些吸引力的盆地，这些方法似乎很规律，并且没有标准牛顿方法观察到的分形结构。回溯梯度下降的吸引力盆地似乎不是那么规律。
In this paper, by combining the algorithm New Q-Newton's method - developed in previous joint work of the author - with Armijo's Backtracking line search, we resolve convergence issues encountered by Newton's method (e.g. convergence to a saddle point or having attracting cycles of more than 1 point) while retaining the quick rate of convergence for Newton's method. We also develop a family of such methods, for general second order methods, some of them having the favour of quasi-Newton's methods. The developed algorithms are very easy to implement. From a Dynamical Systems' viewpoint, the new iterative method has an interesting feature: while it is deterministic, its dependence on Armijo's Backtracking line search makes its behave like a random process, and thus helps it to have good performance. On the experimental aspect, we compare the performance of our algorithms with well known variations of Newton's method on some systems of equations (both real and complex variables). We also explore some basins of attraction arising from Backtracking New Q-Newton's method, which seem to be quite regular and do not have the fractal structures as observed for the standard Newton's method. Basins of attraction for Backtracking Gradient Descent seem not be that regular.
