求解绝对值方程的光滑化同伦方法

A smoothing homotopy method for the absolute value equation

先构造绝对值函数光滑化函数,利用此光滑化函数对绝对值方程Ax-|x|=b建立同伦方程,用逆像定理证明同伦路径的存在性,用一维流形分类定理证明了同伦路径的收敛性,得到该光滑同伦路径的极限点就是绝对值方程的解。

First we construct a smoothing function for absolute function.With the smoothing function,we propose a homotopy formula for the absolute equation Ax-｜x｜=b.The inverse theorem is used to prove the existence of the homotopy path and one-dimensional manifold theorem the convergence.We can come to the conclusion that the limit point of the smooth homotopy path is the solution of the absolute equation.