I have a nonlinear problem as follows: \begin{align}\min&\quad\sum_{k=1}^{K}\left|y_k - \sum_{i=1}^{N} \frac{e^{x_{k}^{i}}}{\sum_{j=1}^{K} e^{x^{i}_{j}}}\right|\\\text{s.t.}&\quad x^i_{j} \ge 0\end{align}
Essentially, there are $K$ buckets with a desired value of $y_k$ for each. There are $N$ agents, each of which makes a choice based on a multinomial logit function.
I think I can get rid of the absolute value using the common trick: \begin{align}\min&\quad \sum_{k=1}^{K}t_k\\\text{s.t.}&\quad t_k \ge y_k - \sum_{i=1}^{N} \frac{e^{x_{k}^{i}}}{\sum_{j=1}^{K} e^{x^{i}_{j}}}\\&\quad t_k\ge-\left(y_k - \sum_{i=1}^{N} \frac{e^{x_{k}^{i}}}{\sum_{j=1}^{K} e^{x^{i}_{j}}}\right)\end{align} but I don't know how to proceed from here. I have 2 questions.
- Is it possible to linearize the fractional exp and reduce the problem to a linear program?
- If not, how should I try to solve this problem? Is there a class of models that encompass this?