KKT Conditions for thresholds? My main question is that when I use Lagrange Multipliers/KKT conditions to perform optimization with threshold constraints, I seem to get contradictory FOC.
Here is a characteristic example:
take an optimization problem like the following:
$$
\begin{align*}
&\max_{x,y} 2x+y-(x-y)^2\\
&s.t.\\
&x,y \geq 0\\
&x+y \leq I
\end{align*}
$$
For some value $I>0$. If I translate this directly to a lagrangian I believe the KKT conditions should give that I need to optimize:
$$
\mathcal{L}(x,y,\mu_1, \mu_2, \mu_3)=2x+y-(x-y)^2 + \mu_1(-x) + \mu_2(-y)+\mu_3(x+y-I)
$$
However partial derivatives w.r.t $\mu_1, \mu_2,\mu_3$ gives:
$$\begin{align*}
&-x=0\\
&-y=0\\
&x+y-I=0\\
\end{align*}$$
which gives a contradiction for $I>0$.
Could someone help me understand why I seem to get this contradiction?

Instead of differentiating w.r.t. the multipliers you should properly write the complementary slackness condition (defined in your link). In your case this is
$$-\mu_1 x=0$$
$$-\mu_2 y=0$$
$$\mu_3 (x+y-I)=0$$
This says that either the constraint binds with equality or the multiplier is zero. If you just differentiate with respect to $\mu_1$ for instance you would incorrectly conclude that always $x=0$, while this can be strictly positive.
