## 简介

\begin{aligned}\min_x \quad f(x), \\ s.t\quad Ax=b,x \in \bar{C}, \end{aligned}\tag{1.1}

\begin{aligned}\min_x f(x)，\end{aligned} \tag{1.2}

$$\dot{x}(t)=-\nabla f(x(t))\tag{1.3}$$

## 海森黎曼梯度流

$$\int_0^T|\dot{\gamma}(t)|dt,\tag{2.1}$$

$$\int_0^T\sqrt{\left<\dot{\gamma}(t),\dot{\gamma}(t)\right>}dt,\tag{2.2}$$

$$g(u,v)=\left<u,v\right>\tag{2.3}$$

$$g(u,v)=\left<H(x)u,v\right>\tag{2.4}$$

$$H(x)^{-1}\nabla f(x). \tag{2.5}$$

$$P_x=I-H(x)^{-1}A^T(AH(x)^{-1}A^T)^{-1}A.$$

$$P_xH(x)^{-1}\nabla f(x).\tag{2.6}$$

\left\{\begin{aligned}\dot{x}(t)&=-P_xH(x)^{-1}\nabla f(x),\\x(0)&\in\mathcal{F}.\end{aligned}\right.\tag{2.7}

## 例子

\left\{\begin{aligned}min_x\quad x,\\s.t. x\geq 0.\end{aligned}\right.\tag{3.1}

$$\dot{x}(t)=-1.\tag{3.2}$$

$$H(x)=\frac{1}{x}.\tag{3.3}$$

$$H(x)^{-1}\nabla x=x.\tag{3.4}$$

$$\dot{x}(t)=-x(t).\tag{3.5}$$

$$x(t)=x_0e^{-t}.\tag{3.6}$$

## 总结与拓展

$$\left<F(x)-F(y),x-y\right>\geq 0.\tag{4.1}$$

$$\dot{x}(t)=-F(x).\tag{4.2}$$

\left\{\begin{aligned}\dot{x}(t)&=-P_xH(x)^{-1}F(x),\\x(0)&\in\mathcal{F}.\end{aligned}\right.