Use the augmented Lagrangian method to solve the problem
$ \min \ f(x); f(x)= \|x\|^4, x \in\mathbb R^2 \ $ subject to a constraint $ \ x_1 + x_2- 3 x_3= 2 \ $.
Answer:
we have to minimize $ f(x)=\|x\|^4 \ $ subject to the constraint $ \ x_1+x_2-3x_3 =2 \ $ .
Here $ f(x)=f(x_1,x_2,x_3)=\|x\|^4=(x_1^2+x_2^2+x_3^2)^2 $.
Now , how to set the Lagrangian function $ L(x,\lambda , \rho ) \ $ ?
I need help in setting up the augmented function .
We want to minimize $$ f \colon \mathbb{R}^3\rightarrow\mathbb{R} \colon (x_1,x_2,x_3) \mapsto \left(x_1^2 + x_2^2 + x_3^2\right)^2 $$ subject to $$ g(x) = x_1 + x_2 - 3x_3 -2 = 0. $$ Because there is no inequality constraint and only one equality constraint, then the Lagrangian $\mathcal{L}$ has the form $$ \mathcal{L}\colon \mathbb{R}^4\rightarrow\mathbb{R} \colon (x_1,x_2,x_3,\lambda) \mapsto f(x_1,x_2,x_3) + \lambda g(x_1, x_2, x_3), $$ thus $$ \mathcal{L}(x_1,x_2,x_3,\lambda)=\left(x_1^2 + x_2^2 + x_3^2\right) + \lambda(x_1+x_2-3x_3-2). $$