Explanation needed for coupling constraints

Question

What is the definition of a coupling constraint in optimization problems? Please give an easy to follow example also. Thanks.

Answer 1

Suppose an optimization problem has the structure $\min_{x,y} \{ f(x) + g(y) : c(x) \leq 0, d(y) \leq 0 \}$, then you can split up the problem and solve the subproblems $\min_x \{ f(x) : c(x) \leq 0 \}$ and $\min_y \{ g(y) : d(y) \leq 0 \}$. This results in an exact solution of the original optimization problem in less time (since you can parallelize and a problem that is twice as small can typically be solved more than twice as fast).

A coupling constraint is a constraint that couples $x$ and $y$ so you can no longer split up the problem. For example, in ADMM the coupling constraint is $Ax=y$.