min $3x_1^2+x_1x_2+2x_2^2$
s.t $3x_1^2+x_1x_2+2x_2^2+x_1-x_2\geq1$
$ x_1\geq 2x_2$
Find the dual problem and determine if it's convex.
I've set the lagrangian to be $L(x,\lambda,\eta)=3x_1^2+x_1x_2+2x_2^2-\lambda(3x_1^2+x_1x_2+2x_2^2+x_1-x_2-1)+\eta(2x_2-x_1)=3x_1^2(1-\lambda)+x_1x_2(1-\lambda)+2x_2^2(1-\lambda)+\lambda(-x_1+x_2+1)+\eta(2x_2-x_1)$
I know that I will get a finite solution if the matrix of the quad part is PSD which is the same as $0\leq \lambda\leq1$ but I got stuck on finding the dual problem from here. I've tried to minimize w.r.t to $x_1,x_2$ same time here.
Edit I've tried without $\eta(2x_2-x_1)$ and got that the dual problem is $q(\lambda)=\frac{-29\lambda^2+23\lambda}{23(1-\lambda)}$ but not sure if it's the right some help would be nice