I know that on fist glance this seems as already explained problem on many internet places, but I haven't found solution anywhere. Anyway, here is my function:
$$F=\sum_{i=0}^N |S_i|$$ where $$S_i=\sum_{j=0}^M A_kX_j$$ for k=1, 2, ...N*M, and $A_k$ are constants.
The goal is to minimize function F, where $X_j$ is variable, including conditions:
$$|S_i|\le B_i$$ for every i and $$\sum_{j=0}^M X_j=C$$ where $B_i$ and C are constants.
It would be helpful to know what part of mathematics is dealing with these problems, or even better to understand algorithm how to write program code.
Thanks in advance.
Introduce a new variable $q_i$ to represent and replace every $|S_i|$ term, and add the constraint $-q_i \leq S_i \leq q_i$. With that, you have a linear program and when you optimize, you will have $q_i = |S_i|$ at the optimum (otherwise it would be possible to improve the solution by decreasing $q_i$)