Additive multiattribute nonlinear utility functions I am interested in the cases where a decision maker possesses a multiattribute utility function ($u$) of the form:

$u(x) = \sum\limits_{i=1}^{n} u_i(x)$, with $x=(x_1, \ldots, x_m)$, and where $u_{i \in [1, n]}$ is a multiattribute nonlinear utility function.

In particular, what is their relevance in microeconomics, utility theory, and decision-making in general? Are there any examples or references?

Thanks.
These are sometimes known as "quasi-separable utility functions".  A recent paper that describes the theory is Quasi-Separable Utility by Qin and Rommeswinkel.

Utility of this form arises fairly naturally in macro and finance.  If your current utility depends on lagged consumption as well as current, then you get utility of this form,
$$
\sum_{t=1}^\infty u(C_{t-1}, C_t),
$$
where $C_t$ is time $t$ consumption.  One particularly common form is 
$$
\sum_{t=1}^\infty u(C_{t} - h C_{t-1}),
$$
which is known as habit utility, since it captures (for $h &gt; 0$) the idea that get used to a certain level of consumption, so if it drops you are very unhappy.