I need to use ADMM algorithm to minimize a function of the form
$$f(x,\theta)+g(z,\eta)$$
subject to:
$$x-z=0$$ $$\theta-\eta=0$$
where $x,z \in \mathbb{R}^n$ and $\theta,\eta$ are positive scalar coefficients. How do I formulate augmented Lagrangian for this problem?
A good hint was given by @LinAlg. I give here the expression of the augmented Lagrangian, but you should definitely follow that hint to obtain the result: $$L_\rho(x,\theta,z,\eta,\lambda,\mu) = f(x,\theta)+g(z,\eta) + \lambda^\top(x -z) + \mu(\theta-\eta) + \frac{\rho}{2}\|x-z\|^2 + \frac{\rho}{2}(\theta - \eta)^2.$$