| Line 20: | Line 20: | ||
We form the lagrangian:<br> | We form the lagrangian:<br> | ||
<math>l(x,\lambda)=x_1x_2+\lambda_1(x_1+x_2+x_3-1)+\lambda_2(x_1+x_2-x_3)</math><br> | <math>l(x,\lambda)=x_1x_2+\lambda_1(x_1+x_2+x_3-1)+\lambda_2(x_1+x_2-x_3)</math><br> | ||
| − | <math>\begin{cases}\ | + | <math>\begin{cases} |
| + | \nabla_xl=\begin{bmatrix} x_2+\lambda_1+\lambda_2 \\ x_1+\lambda_1+\lambda_2 \\ \lambda_1+\lambda_2\end{bmatrix}=\begin{bmatrix} 0 \\ 0 \\ 0 \end{bmatrix} \\ | ||
x_1+x_2+x_3-1=0 \\ | x_1+x_2+x_3-1=0 \\ | ||
x_1+x_2-x_3=0 | x_1+x_2-x_3=0 | ||
Revision as of 22:24, 18 February 2019
Automatic Control (AC)
Question 3: Optimization
August 2016 Problem 4
Solution
We form the lagrangian:
$ l(x,\lambda)=x_1x_2+\lambda_1(x_1+x_2+x_3-1)+\lambda_2(x_1+x_2-x_3) $
$ \begin{cases} \nabla_xl=\begin{bmatrix} x_2+\lambda_1+\lambda_2 \\ x_1+\lambda_1+\lambda_2 \\ \lambda_1+\lambda_2\end{bmatrix}=\begin{bmatrix} 0 \\ 0 \\ 0 \end{bmatrix} \\ x_1+x_2+x_3-1=0 \\ x_1+x_2-x_3=0 \end{cases} $
No valid solution for lagrangian condition
Such that the problem can not be optimized
