一类向量函数的拉格朗日乘子

青青子衿

\begin{align*}
\mathcal{L}(p,\lambda)&=\left\Vert\,\!p-h\right\Vert^2+\lambda\left\Vert\,\!p-w_{k}\right\Vert^{2}\left\Vert\,\!c_{k}\right\Vert^{2}-\lambda\langle\,\!p-v_{k},c_{k}\rangle^2\\
\frac{1}{2}\cdot\dfrac{\partial}{\partial\,\!p}\mathcal{L}(p,\lambda)&=(p-h)+\lambda\left\Vert\,\!c_{k}\right\Vert^{2}(p-w_k)-\lambda\langle\,\!p-v_{k},c_{k}\rangle\,\!c_{k}\\
\\
(1+\lambda\left\Vert\,\!c_{k}\right\Vert^{2})p&=h+\lambda\left\Vert\,\!c_{k}\right\Vert^{2}w_{k}+\lambda\langle\,\!p-v_{k},c_{k}\rangle\,\!c_{k}\\
(1+\lambda\left\Vert\,\!c_{k}\right\Vert^{2})p&=h+\lambda\left\Vert\,\!c_{k}\right\Vert^{2}w_{k}+\lambda\xi\,\!c_{k}\\
(1+\lambda\left\Vert\,\!c_{k}\right\Vert^{2})p&=h+\lambda\left\Vert\,\!c_{k}\right\Vert^{2}v_{k}+\lambda(\left\Vert\,\!c_{k}\right\Vert^{2}+\xi)c_{k}\\
p&=\frac{1}{1+\lambda\left\Vert\,\!c_{k}\right\Vert^{2}}h+\frac{\lambda\left\Vert\,\!c_{k}\right\Vert^{2}}{1+\lambda\left\Vert\,\!c_{k}\right\Vert^{2}}v_{k}+\frac{\lambda(\left\Vert\,\!c_{k}\right\Vert^{2}+\xi)}{1+\lambda\left\Vert\,\!c_{k}\right\Vert^{2}}c_{k}\\
p-v_{k}&=\frac{1}{1+\lambda\left\Vert\,\!c_{k}\right\Vert^{2}}h-\frac{1}{1+\lambda\left\Vert\,\!c_{k}\right\Vert^{2}}v_{k}+\frac{\lambda(\left\Vert\,\!c_{k}\right\Vert^{2}+\xi)}{1+\lambda\left\Vert\,\!c_{k}\right\Vert^{2}}c_{k}\\
p-v_{k}&=\frac{1}{1+\lambda\left\Vert\,\!c_k\right\Vert^{2}}(h-v_{k})+\frac{\lambda(\xi+\left\Vert\,\!c_{k}\right\Vert^2)}{1+\lambda\left\Vert\,\!c_k\right\Vert^{2}}c_{k}\\

\xi&=\frac{1}{1+\lambda\left\Vert\,\!c_k\right\Vert^{2}}\langle\,\!h-v_{k},c_{k}\rangle+\frac{\lambda(\xi+\left\Vert\,\!c_{k}\right\Vert^2)}{1+\lambda\left\Vert\,\!c_k\right\Vert^{2}}\left\Vert\,\!c_k\right\Vert^{2}\\
\xi&=\langle\,\!p-v_{k},c_{k}\rangle=\langle\,\!h-v_{k},c_{k}\rangle+\lambda\left\Vert\,\!c_k\right\Vert^{4}\\
\\
&p=\frac{h+\lambda\left\Vert\,\!c_{k}\right\Vert^{2}w_{k}+\lambda\langle\,\!h-v_{k},c_{k}\rangle\,\!c_{k}+\lambda^2\left\Vert\,\!c_{k}\right\Vert^{4}c_{k}}{1+\lambda\left\Vert\,\!c_{k}\right\Vert^{2}}\\
\\
&c_k=w_{k}-v_{k}\\
\\
&\left\Vert\,\!p-w_{k}\right\Vert^{2}\left\Vert\,\!c_{k}\right\Vert^{2}=\langle\,\!p-v_{k},c_{k}\rangle^2\\
\\
\end{align*}

青青子衿

\begin{align*}
p-w_{k}&=\frac{h+\lambda\left\Vert\,\!c_{k}\right\Vert^{2}w_{k}+\lambda\langle\,\!h-v_{k},c_{k}\rangle\,\!c_{k}+\lambda^2\left\Vert\,\!c_{k}\right\Vert^{4}c_{k}}{1+\lambda\left\Vert\,\!c_{k}\right\Vert^{2}}-w_{k}\\
&=\frac{h-w_{k}+\lambda\langle\,\!h-v_{k},c_{k}\rangle\,\!c_{k}+\lambda^2\left\Vert\,\!c_{k}\right\Vert^{4}c_{k}}{1+\lambda\left\Vert\,\!c_{k}\right\Vert^{2}}\\
&=\frac{s_{k}+(\lambda\langle\,\!t_{k},c_{k}\rangle+\lambda^2\left\Vert\,\!c_{k}\right\Vert^{4})c_{k}}{1+\lambda\left\Vert\,\!c_{k}\right\Vert^{2}}\\

\\
M&=1+\lambda\left\Vert\,\!c_{k}\right\Vert^{2}\\
M^2\left\Vert\,\!p-w_{k}\right\Vert^2&=\left\Vert\,\!s_{k}\right\Vert^{2}+(\lambda\langle\,\!t_{k},c_{k}\rangle+\lambda^2\left\Vert\,\!c_{k}\right\Vert^{4})^2\left\Vert\,\!c_{k}\right\Vert^{2}\\
&\qquad\>\>+2(\lambda\langle\,\!t_{k},c_{k}\rangle+\lambda^2\left\Vert\,\!c_{k}\right\Vert^{4})\langle\,\!s_{k},c_{k}\rangle\\
M^2\left\Vert\,\!p-w_{k}\right\Vert^2\left\Vert\,\!c_{k}\right\Vert^2&=\color{blue}{c^{12} \lambda ^4}+2 c^8 T\lambda ^3\\
&\qquad+c^4 \left(2 c^2 S+T^2\right)\lambda ^2\\
&\qquad\quad+2 c^2  S T\lambda+c^2 s^2\\
\\
p-v_{k}&=\frac{h+\lambda\left\Vert\,\!c_{k}\right\Vert^{2}w_{k}+\lambda\langle\,\!h-v_{k},c_{k}\rangle\,\!c_{k}+\lambda^2\left\Vert\,\!c_{k}\right\Vert^{4}c_{k}}{1+\lambda\left\Vert\,\!c_{k}\right\Vert^{2}}-v_{k}\\
&=\frac{h-v_{k}+\lambda\left\Vert\,\!c_{k}\right\Vert^{2}c_{k}+\lambda\langle\,\!h-v_{k},c_{k}\rangle\,\!c_{k}+\lambda^2\left\Vert\,\!c_{k}\right\Vert^{4}c_{k}}{1+\lambda\left\Vert\,\!c_{k}\right\Vert^{2}}\\

&=\frac{t_{k}+\lambda(\left\Vert\,\!c_{k}\right\Vert^{2}+\langle\,\!t_{k},c_{k}\rangle)c_{k}+\lambda^2\left\Vert\,\!c_{k}\right\Vert^{4}c_{k}}{1+\lambda\left\Vert\,\!c_{k}\right\Vert^{2}}\\
\\

M\langle\,\!p-v_{k},c_{k}\rangle&=\langle\,\!t_{k},c_{k}\rangle+\lambda(\left\Vert\,\!c_{k}\right\Vert^{2}+\langle\,\!t_{k},c_{k}\rangle)\left\Vert\,\!c_{k}\right\Vert^{2}+\lambda^2\left\Vert\,\!c_{k}\right\Vert^{6}\\
&=(1+\lambda\left\Vert\,\!c_{k}\right\Vert^{2})(\langle\,\!t_{k},c_{k}\rangle+\lambda\left\Vert\,\!c_{k}\right\Vert^{4})\\
M^2\langle\,\!p-v_{k},c_{k}\rangle^2
&=\color{blue}{c^{12} \lambda ^4}+2 c^8\left(c^2+T\right)\lambda ^3\\
&\qquad+c^4 \left(c^4+4 c^2 T+T^2\right)\lambda ^2\\
&\qquad\quad+2 c^2 T \left(c^2+T\right)\lambda+T^2\\
\\
\\
\Delta&\colon=M^2(\left\Vert\,\!p-w_{k}\right\Vert^{2}\left\Vert\,\!c_{k}\right\Vert^{2}-\langle\,\!p-v_{k},c_{k}\rangle^2)\\
&=2 c^{10} \lambda ^3+2 c^2 T \left(c^2-S+T\right)\lambda\\
&\qquad+c^6 \left(c^2-2 S+4 T\right)\lambda ^2\\
&\qquad\quad+T^2-c^2 s^2
\\
c&=\left\Vert\,\!w_{k}-v_{k}\right\Vert\\
s&=\left\Vert\,\!h-w_{k}\right\Vert\\
S&=\langle\,\!h-w_{k},w_{k}-v_{k}\rangle\\
T&=\langle\,\!h-v_{k},w_{k}-v_{k}\rangle\\

\end{align*}

青青子衿

青青子衿 发表于 2024-1-5 17:54
\begin{align*}
p-w_{k}&=\frac{h+\lambda\left\Vert\,\!c_{k}\right\Vert^{2}w_{k}+\lambda\langle\,\!h-v_{k},c_{k}\rangle\,\!c_{k}+\lambda^2\left\Vert\,\!c_{k}\right\Vert^{4}c_{k}}{1+\lambda\left\Vert\,\!c_{k}\right\Vert^{2}}-w_{k}\\
&=\frac{h-w_{k}+\lambda\langle\,\!h-v_{k},c_{k}\rangle\,\!c_{k}+\lambda^2\left\Vert\,\!c_{k}\right\Vert^{4}c_{k}}{1+\lambda\left\Vert\,\!c_{k}\right\Vert^{2}}\\
&=\frac{s_{k}+(\lambda\langle\,\!t_{k},c_{k}\rangle+\lambda^2\left\Vert\,\!c_{k}\right\Vert^{4})c_{k}}{1+\lambda\left\Vert\,\!c_{k}\right\Vert^{2}}\\

\\
\end{align*}

\begin{align*}
\mathcal{L}(p,\lambda_1,\lambda_2)&=\left\Vert\,\!p-h\right\Vert^2+\lambda_1\left\Vert\,\!p-w_{k}\right\Vert^{2}\left\Vert\,\!c_{k}\right\Vert^{2}-\lambda_1\langle\,\!p-v_{k},c_{k}\rangle^2+\lambda_2\langle\,\!p-v_{k},g_{k}\rangle\\

\dfrac{1}{2}\cdot\dfrac{\partial}{\partial\,\!p}\mathcal{L}(p,\lambda_1,\lambda_2)&=(p-h)+\lambda_1\left\Vert\,\!c_{k}\right\Vert^{2}(p-w_k)-\lambda_1\langle\,\!p-v_{k},c_{k}\rangle\,\!c_{k}+\dfrac{\lambda_2}{2}g_{k}\\
\\
(1+\lambda_1\left\Vert\,\!c_{k}\right\Vert^{2})p&=h+\lambda_1\left\Vert\,\!c_{k}\right\Vert^{2}w_{k}+\lambda_1\langle\,\!p-v_{k},c_{k}\rangle\,\!c_{k}
-\dfrac{\lambda_2}{2}g_{k}\\

(1+\lambda_1\left\Vert\,\!c_{k}\right\Vert^{2})p&=h+\lambda_1\left\Vert\,\!c_{k}\right\Vert^{2}w_{k}+\lambda_1\xi\,\!c_{k}-\dfrac{\lambda_2}{2}g_{k}\\

(1+\lambda_1\left\Vert\,\!c_{k}\right\Vert^{2})p&=h-\dfrac{\lambda_2}{2}g_{k}+\lambda_1\left\Vert\,\!c_{k}\right\Vert^{2}v_{k}+\lambda_1(\left\Vert\,\!c_{k}\right\Vert^{2}+\xi)c_{k}\\
p&=\frac{1}{1+\lambda_1\left\Vert\,\!c_{k}\right\Vert^{2}}(h-\dfrac{\lambda_2}{2}g_{k})+\frac{\lambda_1\left\Vert\,\!c_{k}\right\Vert^{2}}{1+\lambda_1\left\Vert\,\!c_{k}\right\Vert^{2}}v_{k}+\frac{\lambda_1(\left\Vert\,\!c_{k}\right\Vert^{2}+\xi)}{1+\lambda_1\left\Vert\,\!c_{k}\right\Vert^{2}}c_{k}\\
p-v_{k}&=\frac{1}{1+\lambda_1\left\Vert\,\!c_{k}\right\Vert^{2}}(h-\dfrac{\lambda_2}{2}g_{k})-\frac{1}{1+\lambda_1\left\Vert\,\!c_{k}\right\Vert^{2}}v_{k}+\frac{\lambda_1(\left\Vert\,\!c_{k}\right\Vert^{2}+\xi)}{1+\lambda_1\left\Vert\,\!c_{k}\right\Vert^{2}}c_{k}\\
p-v_{k}&=\frac{1}{1+\lambda_1\left\Vert\,\!c_k\right\Vert^{2}}(h-\dfrac{\lambda_2}{2}g_{k}-v_{k})+\frac{\lambda_1(\xi+\left\Vert\,\!c_{k}\right\Vert^2)}{1+\lambda_1\left\Vert\,\!c_k\right\Vert^{2}}c_{k}\\

\xi&=\frac{1}{1+\lambda_1\left\Vert\,\!c_k\right\Vert^{2}}\langle\,\!h-\dfrac{\lambda_2}{2}g_{k}-v_{k},c_{k}\rangle+\frac{\lambda_1(\xi+\left\Vert\,\!c_{k}\right\Vert^2)}{1+\lambda_1\left\Vert\,\!c_k\right\Vert^{2}}\left\Vert\,\!c_k\right\Vert^{2}\\
\xi&=\langle\,\!p-v_{k},c_{k}\rangle=\langle\,\!h-\dfrac{\lambda_2}{2}g_{k}-v_{k},c_{k}\rangle+\lambda_1\left\Vert\,\!c_k\right\Vert^{4}\\
\\

\end{align*}

\begin{align*}
&p=\frac{h-\frac{\lambda_2}{2}g_{k}+\lambda_1\left\Vert\,\!c_{k}\right\Vert^{2}w_{k}+\lambda_1\langle\,\!h-v_{k},c_{k}\rangle\,\!c_{k}-\frac{\lambda_1\lambda_2}{2}\langle\,\!g_{k},c_{k}\rangle\,\!c_{k}+\lambda_1^2\left\Vert\,\!c_{k}\right\Vert^{4}c_{k}}{1+\lambda_1\left\Vert\,\!c_{k}\right\Vert^{2}}\\
&p-v_k=\frac{h-\frac{\lambda_2}{2}g_{k}-v_{k}+\lambda_1\left\Vert\,\!c_{k}\right\Vert^{2}c_{k}+\lambda_1\langle\,\!h-v_{k},c_{k}\rangle\,\!c_{k}-\frac{\lambda_1\lambda_2}{2}\langle\,\!g_{k},c_{k}\rangle\,\!c_{k}+\lambda_1^2\left\Vert\,\!c_{k}\right\Vert^{4}c_{k}}{1+\lambda_1\left\Vert\,\!c_{k}\right\Vert^{2}}\\
&p-w_k=\frac{h-\frac{\lambda_2}{2}g_{k}-w_{k}+\lambda_1\langle\,\!h-v_{k},c_{k}\rangle\,\!c_{k}-\frac{\lambda_1\lambda_2}{2}\langle\,\!g_{k},c_{k}\rangle\,\!c_{k}+\lambda_1^2\left\Vert\,\!c_{k}\right\Vert^{4}c_{k}}{1+\lambda_1\left\Vert\,\!c_{k}\right\Vert^{2}}\\
\\
&c_k=w_{k}-v_{k}\\
\\
&\left\Vert\,\!p-w_{k}\right\Vert^{2}\left\Vert\,\!c_{k}\right\Vert^{2}=\langle\,\!p-v_{k},c_{k}\rangle^2\\
\\
\end{align*}

\begin{align*}
&\quad\>J=\frac{\lambda_2}{2}\left(\Vert\,\!g_{k}\Vert^2+\lambda_1\langle\,\!c_{k},g_{k}\rangle^2\right)=\frac{\lambda_2}{2}\left(g^2+\lambda_1R^2\right)=\frac{\lambda_2L}{2}\\

&=\langle\,\!h-v_{k},g_k\rangle+\lambda_1\left(\left\Vert\,\!c_{k}\right\Vert^{2}+\langle\,\!h-v_{k},c_{k}\rangle\right)\langle\,\!c_{k},g_{k}\rangle+\lambda_1^2\left\Vert\,\!c_{k}\right\Vert^{4}\langle\,\!c_{k},g_{k}\rangle\\

&=Q+\lambda_1\left(c^{2}+T\right)R+\lambda_1^2c^{4}R\\
\end{align*}

\begin{align*}
&\quad\>\>LM\left(p-v_k\right)\\
&=Lh-\frac{\lambda_2L}{2}g_{k}-Lv_{k}+\lambda_1\left\Vert\,\!c_{k}\right\Vert^{2}Lc_{k}\\
&\qquad+\lambda_1\langle\,\!h-v_{k},c_{k}\rangle\,\!Lc_{k}-\lambda_1\frac{\lambda_2L}{2}\langle\,\!g_{k},c_{k}\rangle\,\!c_{k}+\lambda_1^2\left\Vert\,\!c_{k}\right\Vert^{4}Lc_{k}\\
&=Lh-\frac{\lambda_2L}{2}g_{k}-Lv_{k}+\lambda_1c^{2}Lc_{k}+\lambda_1(LT-\frac{\lambda_2L}{2}R+\lambda_1c^{4}L)c_{k}\\

&=L(h-v_{k})-Jg_{k}+\lambda_1(H+c^{2}L)c_{k}\\
\\
\\
&\quad\>\>LM\left(p-w_k\right)\\
&=Lh-\frac{\lambda_2L}{2}g_{k}-Lw_{k}\\
&\qquad+\lambda_1\langle\,\!h-v_{k},c_{k}\rangle\,\!Lc_{k}-\frac{\lambda_2L}{2}\lambda_1\langle\,\!g_{k},c_{k}\rangle\,\!c_{k}+\lambda_1^2\left\Vert\,\!c_{k}\right\Vert^{4}Lc_{k}\\
&=L(h-w_{k})-\frac{\lambda_2L}{2}g_{k}+\lambda_1(LT-\frac{\lambda_2L}{2}R+\lambda_1c^{4}L)c_{k}\\
&=L(h-w_{k})-Jg_{k}+\lambda_1Hc_{k}\\
\\
&c_k=w_{k}-v_{k}\\
\\
&\left\Vert\,\!p-w_{k}\right\Vert^{2}\left\Vert\,\!c_{k}\right\Vert^{2}=\langle\,\!p-v_{k},c_{k}\rangle^2\\
\\
\end{align*}


\begin{align*}
&\quad\>\>LM\langle\,\!p-v_{k},c_{k}\rangle\\

&=L\langle\,\!h-v_{k},c_{k}\rangle-J\langle\,\!g_{k},c_{k}\rangle+\lambda_1(H+c^{2}L)\left\Vert\,\!c_{k}\right\Vert^{2}\\
&=LT-JR+\lambda_1c^2(H+c^2L)\\
\\
\\
&\quad\>\>L^2M^2\langle\,\!p-v_{k},c_{k}\rangle^2\\
&=(LT-JR+\lambda_1c^2(H+c^2L))^2\\
\\
\\
&\quad\>\>c^2L^2M^2\left\Vert\,\!p-w_{k}\right\Vert^{2}\\

&=c^2\left\Vert\,\!L(h-w_{k})-Jg_{k}+\lambda_1Hc_{k}\right\Vert^{2}\\

&=c^2L^2\left\Vert\,\!h-w_{k}\right\Vert^{2}+c^2J^2\left\Vert\,\!g_{k}\right\Vert^{2}+\lambda_1^2c^2H^2\left\Vert\,\!c_{k}\right\Vert^{2}\\
&\qquad-2c^2JL\langle\,\!h-w_{k},g_{k}\rangle+2c^2\lambda_1HL\langle\,\!h-w_{k},c_{k}\rangle\\
&\qquad\quad-2\lambda_1c^2JH\langle\,\!g_{k},c_{k}\rangle\\
&=c^2s^2L^2+c^2g^2J^2+c^4\lambda_1^2H^2\\
&\qquad-2c^2JLP+2c^2\lambda_1HLS-2c^2\lambda_1JHR\\
\\
\\
&J=Q+\lambda_1\left(c^{2}+T\right)R+\lambda_1^2c^{4}R\\
&H=LT-JR+\lambda_1c^{4}L\\
&\quad\,=g^2 T-Q R+\lambda_1c^2 (c^2 g^2  -  R^2)\\
&L=g^2+\lambda_1R^2\\
&g=\left\Vert\,\!g_{k}\right\Vert=\left\Vert\,\!v_{0}-v_{k}\right\Vert\\
&c=\left\Vert\,\!c_{k}\right\Vert=\left\Vert\,\!w_{k}-v_{k}\right\Vert\\
&s=\left\Vert\,\!h-w_{k}\right\Vert\\
&P=\langle\,\!h-w_{k},g_{k}\rangle=\langle\,\!h-w_{k},v_{0}-v_{k}\rangle\\
&Q=\langle\,\!h-v_{k},g_{k}\rangle=\langle\,\!h-v_{k},v_{0}-v_{k}\rangle\\
&R=\langle\,\!g_{k},c_{k}\rangle=\langle\,\!v_{0}-v_{k},w_{k}-v_{k}\rangle\\
&S=\langle\,\!h-w_{k},c_{k}\rangle=\langle\,\!h-w_{k},w_{k}-v_{k}\rangle\\
&T=\langle\,\!h-v_{k},c_{k}\rangle=\langle\,\!h-v_{k},w_{k}-v_{k}\rangle\\
\\
\end{align*}

\begin{align*}
(LT - JR + \lambda_1c^2 (H + c^2 L) )^2=c^2 (s^2 L^2 + g^2 J^2 - 2 JLP+ 2\lambda_1H(LS - JR) +\lambda_1^2 c^2H^2)
\end{align*}

1. (L*T - J*R + \[Lambda]*c^2 (H + c^2 L) )^2 -
2.       c^2 (s^2 L^2 + g^2 J^2 + c^2 \[Lambda]^2 H^2 - 2 J*L*P +
3.          2 \[Lambda]*H*L*S - 2 \[Lambda]*H*J*R) /. {H ->
4.        L*T - J*R + \[Lambda]*c^4 L} /. {
5.      J -> (Q + \[Lambda] (c^2 + T) R + \[Lambda]^2 c^4 R),
6.      L -> (g^2 + \[Lambda]*R^2),
7.      M -> 1 + \[Lambda]*c^2} /. {
8.     s -> Sqrt[(h - w) . (h - w)],
9.     c -> Sqrt[(w - v) . (w - v)],
10.     g -> Sqrt[(V - v) . (V - v)],
11.     P -> ((h - w) . (V - v)),
12.     Q -> ((h - v) . (V - v)),
13.     R -> ((V - v) . (w - v)),
14.     S -> ((h - w) . (w - v)),
15.     T -> ((h - v) . (w - v))} /. {v -> {1, 3, 5}, w -> {-2, 3, -7},
16.    h -> {-6, 4, -2}, V -> {5, 2, 2}} // Factor

复制代码

\begin{gather*}
\frac{1}{2}(\boldsymbol{x}^{T},1)\begin{pmatrix}
\boldsymbol{A}&\boldsymbol{b}\\
\boldsymbol{b}^{T}&c_0
\end{pmatrix}\begin{pmatrix}
\boldsymbol{x}\\
1
\end{pmatrix}=0\\
\\
\left\{
\begin{split}
\boldsymbol{A}&=2\Vert\boldsymbol{w}-\boldsymbol{v}\Vert^2\boldsymbol{E}-2(\boldsymbol{w}-\boldsymbol{v})(\boldsymbol{w}-\boldsymbol{v})^{T}\\

\boldsymbol{b}&=2\langle\boldsymbol{v},\boldsymbol{w}-\boldsymbol{v}\rangle(\boldsymbol{w}-\boldsymbol{v})-2\Vert\boldsymbol{w}-\boldsymbol{v}\Vert^2\boldsymbol{w}\\

c_0&=2\Vert\boldsymbol{w}\Vert^2\Vert\boldsymbol{w}-\boldsymbol{v}\Vert^2-2\langle\boldsymbol{v},\boldsymbol{w}-\boldsymbol{v}\rangle^2\\
\end{split}\right.
\end{gather*}