Non-convex optimization over convex set with separable objective function and linear equality constrains

Question

I have a problem as follows.

$\max\limits_{\mathbf{x}_1,\cdots,\mathbf{x}_N}\sum_{n=1}^Na_nf(\mathbf{x}_n)\\ s.t.\sum_{n=1}^N\mathbf{x}_n=\mathbf{c},\\ 0\le\mathbf{x}_n\le1,$

where $0\le a_n\le1$, $\mathbf{c}$, $\mathbf{x}_n\in\mathbb{R}^K$, and $f(\mathbf{x}_n)=\sum_{m=1}^M(-1)^m\frac{\mathbf{q}_1(m)^T\mathbf{x}_n+b_1(m)}{\mathbf{q}_2(m)^T\mathbf{x}_n+b_2(m)}$ with $\mathbf{q}_1(m),\mathbf{q}_2(m),b_1(m),b_2(m)>0$, the gradient of $f$,i.e., $\Delta f$, is Lipschitz continuous.

I am stuck in this problem over days. I appreciate any help you give.

Answer 1

Your problem falls in the class of Fractional Programming Problems. This paper, this one and this one should give you a starting point. Best of lucks!!