How to solve the following optmization problem using KKT?

Question

I have a simple optmization problem, I just want to see how to do it using Karush-Kuhn-Tucker to understand it better.

$$\max_{x, y} x^2+xy+y^2$$ subject to: $$x < y \leq 1$$

Answer 1

Hint: Try solving $\max_{x\leq1,y\leq1}x^2+xy+y^2$ and pick solutions that satisfy $x<y$ (if any). Why? Since KKT do not deal with strict inequality constraints. Also, you problem has no solution (set $x=y-\epsilon$ for some small $\epsilon>0$ and send $y\rightarrow-\infty$).

If the constraint is $0\leq x<y\leq1$, then my remark about KKT not dealing with strict inequalities still applies.

If the constraint is $0\leq x\leq y\leq1$, then the KKT approach is to split this series of inequalities into constraints that can be written in terms of $g_{i}(x,y)\leq c$ where $g_{i}:\mathbb{R}^{2}\rightarrow\mathbb{R}$. That is, into five constraints $-x\leq0$, $x\leq1$, $-y\leq0$, $y\leq1$ and $x-y\leq0$.