How do you turn an abstract set constraint into equality constraints?

Question

I am reading from Nonlinear Programming by Bertsekas, and in the section on ADMM, he says:

Consider the problem $$\text{min} \sum _{i=1} ^ m f_i(x)$$ $$\text{s.t. }x \in \cap _{i = 1}^m X_i,$$ where $f_i : \mathbb{R}^n \to \mathbb{R}$ are convex functions and $X_i$ are closed, convex sets with nonempty intersection. We can reformulate this as an equality constrained problem, by introducing additional artificial variables $z_i, i = 1,..., m$ and the equality constraints $x = z_i$: $$\text{min} \sum _{i=1} ^ m f_i(z_i)$$ $$\text{s.t.} \hspace{0.3 cm} x = z_i, \hspace{0.6cm} z_i \in X_i,\hspace{0.6cm} i = 1, ..., m$$

I don't quite understand this transformation. I see that we have $m$ different functions, and for each function $f_i$ its input must come from a (possibly) different set $X_i$. That makes sense. But what does $x$ mean in this second problem? My only idea is that $x$ doesn't really stand for anything; it is just there as a short way of saying $z_1 = z_2 = ... = z_m$. Is this correct?

Thank you very much!

Answer 1

Yes, $x$ is the common value of the $z_i$s. This idea of introducing multiple copies of a variable is known as Lagrangian Decomposition. The $x=z_i$ equalities are linking constraints for what otherwise would decompose into $m$ disjoint subproblems.