Let N and k be fixed integers.
Let be given N positive real numbers
A_l,,A_N
and k real functions
g_1(theta),,g_k(theta)
of a parameter theta in R^q, q a fixed integer
I'm searching for a partition of the interval [0,infinity[
[0,a_1[,[a_1,a_2[,,[a_{k-1},a_k[
with
0=a_0<=a_1<= <=a_k
and for a
theta
such that the sum
sum_{l=1,,N} g_{j(l)}(theta) where j(l) is the number of the
interval containing A_l, i.e., with
A_l in [a_{j(l)-1},a_{j(l)}[,
is minimal.
I have no idea, after what I could search.
Have such problems a "name" or classification?
I would appreciate hints (like references)
for a starting point.
Best regards.

A question to narrow down the scope of this problem:
- Assuming parameter R^q is FIXED, or is it also an optimization
variable?

With some adjustment, this problem looks like the optimization of a
free-knot spline (in 1 dimension). The A_I being the points to be fit,
the interval partition points a_i defining the knot locations, and the
g_k defining the (fitting error between an) interval's spline function
(and point A_i). What makes this differ from your problem is that the
R^q would also be variables of the optimization; and moreover, there
being an R^q for each g_k().

But then again, your statement of how the A_I relate to the
minimization sum is ambiguous. You need to clear that up as well, in
order for anyone to be of further help.

Regards,