# Does this relaxation have a name?

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)

• I think a general framework to describe every instance of an optimization problem being decomposed in such a way (which can take such forms as by a simplifying independence assumption, coordinate descent as mentioned below, etc.) is perhaps slightly ambitious Commented May 17, 2021 at 15:57
• Note that I'm not asking for ways to solve this, rather pointers to previous work studying this and ideally giving it a name Commented May 17, 2021 at 16:28
• What I mean is such a decomposition can occur in such varied settings that to classify all such instances under one general class of optimization problems and expecting that classification to be meaningful seems ambitious. Commented May 18, 2021 at 9:20