I am trying to find Lagrange Multipliers in QP problem:
$$\begin{aligned}\min_x~~&\frac{1}{2}\begin{bmatrix}x_1 & x_2\end{bmatrix}\begin{bmatrix}1 & 1\\1 & 2\end{bmatrix}\begin{bmatrix}x1\\x2\end{bmatrix} + \begin{bmatrix}8\\9\end{bmatrix}^T\begin{bmatrix}x1\\x2\end{bmatrix} \\
\mathrm{s.t}~~ &x_1 + x_2 = 0 \\
&2x_1 + x_2 = 0 \\
&3x_1 + 2x_2 = 0
\end{aligned}
$$
Lagrangian of this problem is
$$L(x,\lambda) = \frac{1}{2}x_1^2 +x_1x_2 + x_2^2 + 8x_1 + 9x_2 + \lambda_1(x_1 + x_2) + \lambda_2(2x_1 + x_2) + \lambda_3(3x_1+2x_2)$$
$\nabla_xL(x,\lambda)$ set to zero gives (for feasible solution $x_1=0,~x_2=0)$
$$\begin{aligned}\lambda_1 + 2\lambda_2 + 3\lambda_3 &= -8\\
\lambda_1 + \lambda_2 + 2\lambda_3 &= -9\end{aligned}$$
And I ended up with two equations with three unknowns, although solving above QP problem with quadprog in MATLAB gives:
$$\lambda_1 = 0,~~\lambda_2 =11 ,~~\lambda_3 = -10$$
Am I missing some equation or how can I figure out that the first multiplier is zero?
Best Answer
The third constraint is a consequence ( a sum) of the other two, throw it away, and be happy.