min $x_1+2x_2^2+x_3^2+\sqrt{4x_1^2+x_1x_3+x_3^2+2}$
s.t $x_1+x_2^2+4x_3^2\le5$
$x_1+x_2+x_3\leq15$ Find the dual problem
Edit:
Setting $A=\begin{pmatrix}\frac{\sqrt{15}}{2}&0&0\\1/2&0&1\\0&0&0\end{pmatrix},b=\begin{pmatrix}0\\0\\\sqrt{2}\end{pmatrix}$ and $Ax+b=y$
and the lagragian has the form of
$L(x,y,\lambda,\eta,\mu)=2x_2^2+\lambda x_2^2+\eta x_2+x_1+x_3^2+\lambda(x_1+4x_3^2)+\eta(x_1+x_3)+\mu^TAx+||y||-\mu^Ty-5\lambda-15\eta+\mu^Tb$
I know how to solve for every variable just a little confused for $x_1,x_3$.
$\underset{x_1,x_3}{\min}x_1+x_3^2+\lambda(x_1+4x_3^2)+\eta(x_1+x_3)+\mu^TAx=x_1+x_3^2+\lambda(x_1+4x_3^2)+\eta(x_1+x_3)+\frac{\sqrt{15}}{2}\mu_1x_1+\mu_2(1/2x_1+x_3)$ now for $x_1$ we can say that the minimum is $0$ when $1+\lambda+\eta+\frac{\sqrt{15}}{2}\mu_1+1/2\mu_2$ else it's $-\infty$ and for $x_3$ just solving a quadratic function.
Am I right?