Consider the function \begin{eqnarray} f(x_1,x_2,\cdots, x_n) = \frac{\sum_{i}^{n}a_ix_i}{\sum_{i}^{n}b_ix_i}, \end{eqnarray} over the set $S = \{x := (x_1,x_2,\cdots, x_n):-1 \leq x_i \leq 1,\; \forall 1 \leq i \leq n,\; \sum_{i}^{n}b_ix_i \neq 0, \sum_{i=1}^{n}c_i^{k}x_i = 0, 1 \leq x_i \leq m, \}$. It is given that the vector $b:= (b_1,b_2,\cdots,b_n)$ does not lie in the span of the vectors $c^{k}, \forall 1 \leq k \leq m$. Assume that the constants $(a_i, b_i, c_i^{k} ), 1 \leq i \leq n$,, $1 \leq k \leq m$ are all strictly positive. Is the function $f(x)$ bounded over the set $S$. I am trying to guess sufficient conditions under on the constraint set under which this ratio of affine functions is bounded.

Without the additional constraints $\sum_{i=1}^{n}c_i^{k}x_i = 0$ which are linearly independent of the vector $b$ , the function is does not seem to be bounded.