a non-smooth optimization problem




Hi,

(in the following k and N and q are fixed integers)

Assume I have fixed reel numbers
A_1,...,A_N
and fixed SMOOTH functions
g_1(theta),...,g_N(theta)
with a finite dimensional parameter theta in a subset of R^q

I'd like to minimize the following expression

sum_{n=1}^N sum_{j=0}^k 1_{[a_j,a_{j+1}[}(A_n) g_n(theta_j)

where 1_M is the indicator function of a set M; the
sets M are intervals.

The expression is simply smooth if only
theta_1,...,theta_k
are free variables and the a_j fixed; classical calculus
works in principle.

But I'd like to minimize the expression in the free
variables
theta_1,...,theta_k
as well as the k free variables a_j, j=1,...,k, with
0=a_0<a_1<...<a_k<a_{k+1}=infinity

In the a_j, j=1,...,k, the function is not smooth.

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.

.

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