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?