求解广义Lyapunov方程的非单调谱投影梯度法

Nonmonotone Spectral Projection Gradient Method for Solving Generalized Lyapunov Equations

本文研究双线性控制系统中的一类广义Lyapunov方程的半正定解.基于凸函数的局部极小解就是全局极小解这一良好性质,首先将广义Lyapunov方程的半正定解问题等价转化为凸优化问题.利用非单调线搜索技术确定步长,构造了非单调谱投影梯度方法求解这一等价问题.最后用数值例子验证了新方法的可行性和有效性.

In this paper,we consider the positive semidefinite solution to a class of generalized Lyapunov matrix equation,which arises in bilinear systems.Based on the good property that the local minimizer of a convex function is also the global minimizer,the positive semidefinite solution of the generalized Lyapunov equation is transformed into a convex optimization problem.By using the nonmonotone line search technique,we develop a nonmonotone spectral projected gradient method to solve this equivalent problem.Finally,numerical examples are presented to illustrate the feasibility and effectiveness of the new method.