I have the following optimization problem: \begin{align} \max_{x,y,z} \quad & ax + by + cz\\ \text{subject to} \quad & z \leq xy\\ & z \leq L \\ & x + y = M \end{align} The KKT conditions yield FOC: \begin{align} a+\lambda_1 y - \lambda_3 &= 0\\ b+\lambda_1 x - \lambda_3 &= 0\\ c-\lambda_1-\lambda_2 &=0\\ \lambda_1(xy-z) &=0\\ \lambda_2(L-z) &=0 \end{align} Let $a,b,c>0$, and $a\neq b$.
From Eqs. 1 and 2, $$a+\lambda_1 y = b+\lambda_1 x \implies \lambda_1 \neq 0$$ But then Eq. 4 implies $z=xy$. Yet if we suppose $L=0$, clearly $z=0$, but $xy \neq 0$.
I'm fairly sure I violated some assumption of KKT conditions with my "endogenous" constraint, but two hours of googling got me nowhere. Where did I go wrong?
This question is based on an incorrect assumption. In the event that $L=0$, the maximum value of the objective function would be realized by setting the variable ($x$ or $y$) multiplied by the larger of $a$ or $b$ to equal $M$, and the other to 0, so $xy$ would be 0.