# 二次规划笔记

$\displaystyle \min \frac{1}{2} x^\top G x - c^\top x$ s.t. $A^\top x = b, C^\top x \geq d$

$\displaystyle L(x, \alpha, \beta) = \frac{1}{2} x^\top G x - c^\top x - \alpha^\top (A^\top x - b) - \beta^\top (C^\top x - d)$

$\displaystyle \begin{pmatrix} G & -A \\ A^\top & 0 \end{pmatrix} \begin{pmatrix} x \\ \alpha\end{pmatrix} = \begin{pmatrix} c \\ b\end{pmatrix}$

$\displaystyle \begin{pmatrix} G & A \\ A^\top & 0 \end{pmatrix} \begin{pmatrix} -p \\ \alpha\end{pmatrix} = \begin{pmatrix} Gx + c \\ A^\top x - b\end{pmatrix} \stackrel{\triangle}{=} \begin{pmatrix} g \\ h \end{pmatrix}$

$\displaystyle F(x, y, \lambda; \sigma, \mu) = \begin{pmatrix} Gx - C \lambda + c \\ C^\top x - y - b \\ Y\Lambda e - \sigma\mu\end{pmatrix} = 0$

$\displaystyle \begin{pmatrix} G & 0 & -C \\ C^\top & - I & 0 \\ 0 & \Lambda & Y\end{pmatrix} \begin{pmatrix} \delta x \\ \delta y \\ \delta \lambda\end{pmatrix} = \begin{pmatrix} Gx - A\lambda + c \\ A^\top x - y - b \\ -\Lambda Y e + \sigma\mu e\end{pmatrix} \stackrel{\triangle}{=} \begin{pmatrix} r_d \\ r_P \\ -\Lambda Y e + \sigma\mu e\end{pmatrix}$

$\displaystyle \begin{pmatrix} G & 0 & C \\ 0 & -Y\Lambda & -I \\ C^\top & I & 0\end{pmatrix} \begin{pmatrix} \delta x \\ -\delta y \\ -\delta \lambda \end{pmatrix} = \begin{pmatrix} -r_d \\ \Lambda e - \sigma\mu Y^{-1}e \\ -r_P\end{pmatrix}$

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And in your seed shall all the nations of the earth be blessed; because you have obeyed my voice.