Q: Specify Karush-Kuhn Tucker (KKT) optimality conditions for the following problem. Find a KKT
point (i.e., a point that satisfies all KKT conditions) by analyzing all possible values of the Lagrange
multipliers. Is this point an optimal solution?
$$
\min{z} = 2(x_1-1)^2 - (x_2)^2 -(4 x_1 x_2)
$$
s.t.
\begin{align}
-x_1 + x_2 & = 1
\\
x_1 + x_2 & \le 3
\\
x_1, x_2 & \ge 0
\end{align}
Thanks in advance!:)
Note that this is not a convex optimization model (check the hessian of the objective function). So in order to find the optimal point you may check all the KKT solutions. I am not writing them one by one since this is a very simple generic problem. Only trick is about proving if one of these points is minimum. For that, you can either take the dual and see if it gives a solution same as your minimum KKT value. However, this will only work if strong duality holds. Thus, I recommend you to check First or Second Order Sufficient Optimality Conditions.