I have a nonlinear problem as follows:

$$ min sum_{k=1}^{K}|y_k – sum_{i=1}^{N} frac{e^{x_{k}^{i}}}{sum_{j=1}^{K} e^{x^{i}_{j}}}| \ st quad x^i_{j} ge 0$$

Essentially, there are $k$ buckets with a desired value of $y_k$ for each. There are $N$ agents, each of which make a choice based on a multinomial logit function.

I think I can get rid of the absolute value using the common trick:

$$ min quad sum_{k=1}^{K}t_k \

t_k ge y_k – sum_{i=1}^{N} frac{e^{x_{k}^{i}}}{sum_{j=1}^{K} e^{x^{i}_{j}}} \

t_k ge -(y_k – sum_{i=1}^{N} frac{e^{x_{k}^{i}}}{sum_{j=1}^{K} e^{x^{i}_{j}}})

$$

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 than encompass this?