Consider the problem to minimize:
$z = -2x_1 + 8x_2 + x_1x_2 + x_1^2+6x_2^2$
subject to restrictions:
$\displaystyle ( x_1 + 2x_2 \leq 4 )\\ ( 2x_1 + x_2 \leq 5 )\\ ( x_1, x_2 \geq 0 )$
Derive the Dual problem and solve by the gradient method We first introduce the Lagrange multipliers λ_1 and λ_2 for the first two constraints:
$L(x_1 x_2, λ_1, λ_2) = -2x_1 + 8x_2 + x_1x_2+ x_2^2 + 6x_2^2 + λ_1(x_1 + 2x_2 - 4) + λ_2(2x_1 + x_2 - 5)$
The Lagrangian derived with respect to $x_1$, $x_2, λ_1 and λ_2$ and set to zero:
$∂L/∂x_1 = -2 + x_2 + λ_1 + 2λ_2 = 0\\ ∂L/∂x_2 = 8 + x_1 + 2x_2 + 2λ_1 + λ_2 = 0\\ ∂L/∂λ_1 = x_1 + 2x_2 - 4 = 0\\ ∂L/∂λ_2 = 2x_1 + x_2 - 5 = 0$
The last two equations form a system of linear equations in terms of $x_1$ and $x_2$. Let's solve this system:
$x_1 + 2x_2 = 4\\ 2x_1 + x_2 = 5$
Multiplying the first equation by $-2$ and adding it to the second, we get:
$-3x_2 = -3 $
Therefore, $x_2= 1$. Substituting $x_2 = 1$ in the first equation:
$x_2 + 2(1) = 4$ So, $x_1 = 2$.
Now, replacing $x_1 = 2$ and $x_2 = 1$ in the first two equations of the original system, we can find the values of $λ_1$ and $λ_2$:
$-2 + 1 + λ_1 + 2λ_2 = 0 \\ 8 + 2 + 2 + 2λ_1 + λ_2 = 0 $
Simplifying:
$λ_1+ 2λ_2 = 1\\ 2λ_1 + λ_2 = -12 $ Solution of the system: $λ_1 = -25/3\\ λ_2 = 14/3 $ the value of lambda 1 is negative, but I have the conditions of non-negativity, so I don't know how to state it anymore. I need help step by step
