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回忆一下关于$\dpi{120}&space;\fn_jvn&space;\large&space;n$元实值函数的$\dpi{120}&space;\fn_jvn&space;\large&space;f:R^{n}\rightarrow&space;R$的求导问题，函数$\dpi{120}&space;\fn_jvn&space;\large&space;f$的一阶导数$\dpi{120}&space;\fn_jvn&space;\large&space;Df$

$\dpi{120}&space;\fn_jvn&space;\large&space;Df\triangleq&space;\begin{bmatrix}&space;\frac{\partial&space;f}{\partial&space;x_{1}},&\frac{\partial&space;f}{\partial&space;x_{2}},&space;&\cdots,&space;&&space;\frac{\partial&space;f}{\partial&space;x_{n}}&space;\end{bmatrix}$

$\dpi{120}&space;\fn_jvn&space;\large&space;H(x)\triangleq&space;D^{2}f(x)=\begin{bmatrix}&space;\frac{\partial^2&space;f}{\partial&space;x_{1}^2}&&space;\cdots&space;&&space;\frac{\partial^2&space;f}{\partial&space;x_{n}\partial&space;x_{1}}&space;\\&space;\vdots&space;&&space;&&space;\vdots&space;\\&space;\frac{\partial^2&space;f}{\partial&space;x_{1}\partial&space;x_{n}}&\cdots&space;&\frac{\partial^2&space;f}{\partial&space;x_{n}^2}&space;\end{bmatrix}$

$\dpi{120}&space;\fn_jvn&space;\large&space;d$$\dpi{120}&space;\fn_jvn&space;\large&space;n$元实值函数$\dpi{120}&space;\fn_jvn&space;\large&space;f:R^{n}\rightarrow&space;R$$\dpi{120}&space;\fn_jvn&space;\large&space;x\in&space;\Omega$处的可行方向，则函数$\dpi{120}&space;\fn_jvn&space;\large&space;f$沿方向$\dpi{120}&space;\fn_jvn&space;\large&space;d$的方向导数$\dpi{120}&space;\fn_jvn&space;\large&space;\partial&space;f/\partial&space;d$可表示为

$\dpi{120}&space;\fn_jvn&space;\large&space;\frac{\partial&space;f}{\partial&space;d}(x)=\lim_{a\rightarrow&space;0}\frac{f(x+ad)-f(x)}{a}$

$\dpi{120}&space;\fn_jvn&space;\large&space;\frac{\partial&space;f}{\partial&space;d}(x)&space;=\left.\dfrac{d}{da}f(x+ad)\right|_{a=0}$

$\dpi{120}&space;\fn_jvn&space;\large&space;\frac{\partial&space;f}{\partial&space;d}(x)&space;=\left.\dfrac{d}{da}f(x+ad)\right|_{a=0}=\triangledown&space;f(x)^{T}d=\left&space;\langle&space;\triangledown&space;f(x),d&space;\right&space;\rangle=d^{T}\triangledown&space;f(x)$

$\dpi{120}&space;\fn_jvn&space;\large&space;d^{T}\triangledown&space;f(x^{*})\geqslant&space;0$

$\dpi{120}&space;\fn_jvn&space;\large&space;\triangledown&space;f(x^{*})=0$

$\dpi{120}&space;\fn_jvn&space;\large&space;d^{T}H(x^{*})d\geqslant&space;0$

$\dpi{120}&space;\fn_jvn&space;\large&space;\triangledown&space;f(x^{*})=0$

hessian矩阵$\dpi{120}&space;\fn_jvn&space;\large&space;H(x^{*})$半正定，也就是说，对于所有的向量$\dpi{120}&space;\fn_jvn&space;\large&space;d\in&space;R^{n}$，都有

$\dpi{120}&space;\fn_jvn&space;\large&space;d^{T}H(x^{*})d\geqslant&space;0$

1   $\dpi{120}&space;\fn_jvn&space;\large&space;\triangledown&space;f(x^{*})=0$

2   $\dpi{120} \fn_jvn \large H(x^{*})>0$

$\dpi{120}&space;\fn_jvn&space;\large&space;x^{*}$是函数$\dpi{120}&space;\fn_jvn&space;\large&space;f$的一个严格局部极小点

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posted on 2015-12-28 04:57 刺猬的温驯 阅读(...) 评论(...) 编辑 收藏