KKT Conditions

Solve the following optimization problem \(\begin{alignat*}{2} & \text{minimize: } && xy \\ & \text{subject to: } && \begin{aligned}[t] x+2y &\leq 8\\ \dfrac{x^2}{16}+1 &\leq y \\ x &\geq 0 \\ 0 \leq y &\leq 3 \end{aligned} \end{alignat*}\)

We can see that the constraint $y\geq 0$ is not active. So we don’t have to include this constraint in our solution.

Case I

First we search for critical points in the interior of the region.

$\displaystyle f = x y
g_1 = - y + 3
g_2 = - x - 2 y + 8
g_3 = - \frac{x^{2}}{16} + y - 1
g_4 = x $

The lagrangian $L=\lambda_{1} \left(- y + 3\right) + \lambda_{2} \left(- x - 2 y + 8\right) + \lambda_{3} \left(- \frac{x^{2}}{16} + y - 1\right) + \lambda_{4} x + x y$

$\displaystyle - \lambda_{2} - \frac{\lambda_{3} x}{8} + \lambda_{4} + y= 0 \- \lambda_{1} - 2 \lambda_{2} + \lambda_{3} + x= 0 \\lambda_{1} \left(- y + 3\right)= 0 \\lambda_{2} \left(- x - 2 y + 8\right)= 0 \\lambda_{3} \left(- \frac{x^{2}}{16} + y - 1\right)= 0 \\lambda_{4} x= 0 \$