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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: $$x < y \leq 1$$

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    What is KKT? Karush-Kuhn-Tucke? – 2017-02-14
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    Yes. Karush-Kuhn-Tucker conditions. – 2017-02-14
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    I think there is no solution. function is convex and you want to maximize it which leads to unbounded above solution. for example consider the case $x=-n \quad n\in \Bbb N$ and $y=1$. – 2017-02-14
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    Furthermore, the maximization of a convex function is not a convex optimization problem, so it shouldn't be tagged as such. – 2017-02-14

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Hint: Try solving $\max_{x\leq1,y\leq1}x^2+xy+y^2$ and pick solutions that satisfy $x0$ and send $y\rightarrow-\infty$).

If the constraint is $0\leq x

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 $x-y\leq0$.

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    I picked a bad example. What if the constraint is $0 \leq x < y \leq 1$ – 2017-02-14
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    This answer should have been a comment. – 2017-02-14
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    @Anon123 I added couple of lines for the alternative constraint. – 2017-02-14