# nonlinear optimization – Linearizing a program with multinomial logit in the objective

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.

1. is it possible to linearize the fractional exp and reduce the problem to a linear program?
2. If not, how should I try to solve this problem? Is there a class of models than encompass this?