Consider the following optimization problem \begin{equation}\notag \begin{split} \max & x^2+y^2 \\ \mathrm{s.t.} & x^2 \leq 1 \\ & 0\leq y\leq 2 \end{split} \end{equation} Obviously, the optimal solution should be $x^*=1, y^*=2$ or $x^*=-1,y^*=2$. I am wondering how to use Lagrange dual method and KKT condition to solve it.
It seems that $$L(x,y,\lambda)=x^2+y^2-\lambda_1 (x^2-1) - \lambda_2 (y-2) + \lambda_3 y.$$ Due to the KKT condition, the gradient of $L$: $$ [2(1-\lambda_1^*)x^*, 2y^*-\lambda_2^*+\lambda_3^*]=0$$ And also, $$x^{*2}\leq 1$$ $$0\leq y^*\leq 2$$ $$\lambda_i^*\geq 0$$ $$\lambda_1^*(x^2-1)=0$$ $$\lambda_2^*(y-2)=0$$ $$-\lambda_3^* y=0$$ However, I still couldn't get the solution. Thanks.