Imagine you have e.g. the following optimization objective over a joint $D$-dimensional real-valued space, i.e.
$x^* = \min_{x} f(x), \quad x \in \mathbb{R}^D$
And then you have the following relaxation over the $D$ individual, separate components:
$\hat{x}_i^* = \min_{\hat{x}_i} g(\hat{x}_i), \quad \hat{x}_i \in \mathbb{R}, \; i \in \{1, \dots, D \}$
Obviously, these are not necessarily equivalent since the relaxation doesn't take interactions among components into account. As with all relaxations, it is expected to be easier to solve, since all $\hat{x}_i$ components are treated independently and the respective search spaces are exponentially smaller. And the relaxed optimum $\hat{x}$ might be off from the original one $x$, leading to worse results.
An example would be assuming that a multivariate Random Variable has a diagonal covariance matrix, when fitting said RV to a given dataset. This is usual e.g. in variational approaches.
But independently of the application, and in its most general sense, I'd assume this to be a fairly popular relaxation, does it have a name? Could anyone point me to materials in the general context of optimization theory? (i.e. specific instances of relaxations are fine as long as this joint-to-parts element is treated on the foreground and in a general manner)
