Consider the following optimization problem:

$mi{n}_{z_1,z_2},3z_1^2+z_2^2$

$s.t.2z_1+z_2=1$

a$.$ Write the augmented Lagrangian function for this problem and obtain its gradient and Hessian.

b$.$ Solve the first three augmented Lagrangian subproblems using Newton's method to update the

design. Consider ${}_{k+1}=10{}_k$ and ${}_{k+1}=\frac{{}_k}{10},{}_0=1,{}_0=1,{}^{(0)}=0,$ and $z_0={\left[\begin{array}{cc}0& 0\end{array}\right]}^T.$