Minimizing the sum of linear ratios Is there a way to transform the following optimization problem into an equivalent convex problem:
$ \mathrm{min}_x \, \Large \left\{ \sum\limits_{i=1}^n \frac{\langle x,a_i \rangle}{\langle x,b_i\rangle} \right\} $
with $x \in \mathcal{X}$, and $ \mathcal{X} \subset \mathbb{R}^n$ is a convex subset of $\mathbb{R}^n$, and $\langle\cdot,\cdot\rangle$ denotes the standard scalar-product in $\mathbb{R}^n$. Moreover, $a_i, b_i \in \mathbb{R}^n$ and $b_i > 0 \,\,\, \forall \, i$ (i.e. all components of the vectors $b_i$ are positive. If necessary, the problem could be reformulated such that also the vectors $a_i$ have only positive components). 
The convex set $\mathcal{X}$ is defined via the inequalities
$c_1 \leq x \leq c_2, \,\,\,\,\,\, c_1,c_2 >0$
$\sum_{i=1}^n \langle x, A_i\, x \rangle \,\, \leq c_3$
with positive vectors $c_1$ and $c_2$ and a positive constant $c_3$ and symmetric, positive semi-definite matrices $A_i$. Moreover, there are further linear equality constraints of the form
$f(x) =c_4$ with a linear function $f$.
I want to transform this problem to an equivalent convex optimization problem.
I have constructed several possibilities to transform this problem to a convex one for $n=1$. In this case I can also prove several nice properties.
However, for $n>1$ I was not able to transform this problem to a convex one.
Does anybody know whether this is possible, and if yes how can it be done?
Thank you very much!
 A: General problems of the type you give are NP-hard.  However, structured classes can be solved. Here is an argument that shows hardness in general: 
Consider the (hard) integer program of finding a vector $x \in \mathbb{R}^n$ to satisfy the following constraints:
\begin{align}
&Ax \leq b \\
&\sum_{i=1}^n x_i = k\\
&x_i \in \{0, 1\} \quad, \forall i \in \{1, ..., n\}
\end{align}
Assume the integer program is feasible (so there exists an integer solution that satisfies the constraints). Let's define a new (non-integer) problem of the general type you pose, then show it could be used to solve the above (hard) problem. The new problem is: 
\begin{align}
\mbox{Minimize:} \quad & \sum_{i=1}^n \frac{x_i}{1+x_i} \\
\mbox{Subject to:} \quad & A x\leq b\\
&\sum_{i=1}^n x_i = k \\
&x_i \in [0,1] \quad , \forall i \in \{1, ..., n\}
\end{align}
Claim: If the integer program is feasible, then the new problem is also feasible and its solution satisfies the constraints of the integer program.
Proof: Let $y^*$ be a solution to the integer problem.  Then $y_i^* \in \{0,1\}$ and $\sum_{i=1}^n y_i^*=k$.  Notice that the function $x/(1+x)$ satisfies
$$ (Fact 1): \quad \frac{x}{1+x} \geq \frac{x}{2} \quad , \forall x \in [0,1], \mbox{with equality iff $x\in \{0,1\}$} $$
Hence
$$ \sum_{i=1}^n \frac{y_i^*}{1+y_i^*} = \sum_{i=1}^n \frac{y_i^*}{2} = k/2 $$
Notice that $y^*$ satisfies the constraints of the new problem and achieves an objective value of $k/2$. So the new problem is feasible, and its optimal objective value is less than or equal to $k/2$. 
Now let $x^*$ be an optimal solution to the new problem. It suffices to show $x^*$ is a binary vector. Since the optimal objective value is no more than $k/2$, we have:
$$ \sum_{i=1}^n \frac{x_i^*}{1+x_i^*} \leq k/2 $$
On the other hand: 
$$ k/2 \geq \sum_{i=1}^n \frac{x_i^*}{1+x_i^*} \overset{(a)}{\geq} \sum_{i=1}^n \frac{x_i^*}{2}  \overset{(b)}{=} k/2 $$
where (a) holds by Fact 1 and (b) holds  because $x^*$ satisfies the constraint $\sum_{i=1}^n x_i^*=k$.  It follows that 
$$ \sum_{i=1}^n\frac{x_i^*}{1+x_i^*} = \sum_{i=1}^n \frac{x_i^*}{2}$$
But for each $i \in \{1, ..., n\}$, we know by Fact 1 that 
$$\frac{x_i^*}{2} \geq \frac{x_i^*}{1+x_i^*}$$
So equality must hold in each term $i$, that is, $\frac{x_i^*}{2} = \frac{x_i^*}{1+x_i^*}$, which implies $x_i^* \in \{0,1\}$ by Fact 1. $\Box$

The "structured classes" of problems I mentioned above are generalizations of linear fractional problems where denominator terms can be grouped over separate variables, such as 
\begin{align}
\mbox{Minimize:} \quad & \sum_{i=1}^G \frac{g_i(x)}{T_i(x)} \\
\mbox{Subject to:} \quad & \sum_{i=1}^G \frac{h_{i,k}(x)}{T_i(x)} \leq c_k, \quad \forall k \in \{1, ..., K\} \\
\end{align}
where $G$ is the number of groups, and 
where for each group $i \in \{1, ..,G\}$, the functions $g_i(x)$ and $T_i(x)$, $h_{i,k}(x)$ are functions of a disjoint set of components of the $x$ vector.  For example, group 1 uses only components $x_1, x_2$.  Group 2 uses components $x_3, x_4$, and so on. I've done some work in more complicated stochastic scenarios that use this structure, for example Theorem 1 here (which is unfortunately written for a more complicated system, a simplified version could be written for this problem): 
http://ee.usc.edu/stochastic-nets/docs/asynch-markov-v8.pdf 
