Automatic Control (AC)

Question 3: Optimization

August 2016 Problem 5

### Solution

The problem equal to
Minimize $(x_1)^2+(x_2)^2-14x_1-6x_2-7$
Subject to $x_1+x_2-2<=0$ and $x_1+2x_2-3<=0$
Form the lagrangian function
$l(x,\mu)=(x_1)^2+(x_2)^2-14x_1-6x_2-7+\mu_1(x_1+x_2-2)+\mu_2(x_1+2x_2-3)$
The KKT condition takes the form
$\nabla_xl(x,\mu)=\begin{bmatrix}2x_1-14+\mu_1+\mu_2 \\ 2x_2-6+\mu_1+2\mu_2\end{bmatrix}=\begin{bmatrix}0 \\ 0\end{bmatrix}$
$\mu_1(x_1+x_2-2)=0$
$\mu_2(x_1+2x_2-3)=0$
$\mu_1>=0$, $\mu_2>=0$
$\Rightarrow \begin{cases} \mu_1=0 & \mu_2=0 & x_1=7 & x_2=3 & wrong \\ \mu_1=0 & \mu_2=4 & x_1=5 & x_2=-1 & wrong \\ \mu_1=8 & \mu_2=4 & x_1=3 & x_2=-1 & f(x)=-33 \\ \mu_1=20 & \mu_2=-8 & x_1=1 & x_2=1 & wrong \end{cases}$
In all $x^T=[3 -1]$ is the maximizer of original function.

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Abstract algebra continues the conceptual developments of linear algebra, on an even grander scale.

Dr. Paul Garrett