Let \(X\subseteq\mathbb{R}^n\). I have the following function \(f:X\rightarrow\mathbb{R}\): $$ f({\bf x})= \frac{c_1 + \sum_{i=1}^n a_ix_i +\sum_{i=1}^n b_ix_i^2}{c_2+\sum_{i=1}^n d_i x_i}\enspace.$$ All the \(a_i\), \(b_i\), \(c_i\), and \(d_i\) are strictly greater than 0, and X is such that $$ c_2+\sum_{i=1}^n d_i x_i>0, \forall {\bf x}\in X\enspace.$$ Is \(f\) convex or at least pseudo or quasiconvex? Note that $$f({\bf x})= \frac{c_1 + \sum_{i=1}^n a_ix_i }{c_2+\sum_{i=1}^n d_i x_i}+\frac{\sum_{i=1}^n b_ix_i^2}{c_2+\sum_{i=1}^n d_i x_i}$$ and the first term is on the right side is pseudolinear (hence pseudoconvex, hence quasiconvex) and third term is convex (hence pseudoconvex, hence quasiconvex). I know that the sum of quasiconvex functions is not in general quasiconvex, but I wonder whether something else was known that can help me show that \(f\) is. asked 28 Oct '13, 18:29 MatteoR Paul Rubin ♦♦ 
In fact your constraint belongs to the class cf conic quadratic (aka. SOCP) representable sets that is known to be convex. Also problems with conic quadratic representable sets can be solved efficiently. It is easy to your constraint can be represented as follows $$ \left[ \begin{array}{c} 0.5 \\ dx+c \\ Ax+b \end{array} \right] \in K $$ where $$ K := \{y \mid 2y_1 y_2 \geq \y_{3:n}\^2, \quad y_1,y_2 \geq 0\}. $$ K is the so called rotated quadratic cone. For more information see the MOSEK modelling manual. answered 31 Oct '13, 03:49 Erling_MOSEK fbahr ♦ I have tested the LaTex code on mathjax.org where shows correct but here are problems. Do not know why.
(31 Oct '13, 03:52)
Erling_MOSEK
@Erling: ORX FAQ: Hey, how do I get that fancy math stuff? \(\rightarrow\) "Within latex, all backslashes must be doubled."
(31 Oct '13, 05:36)
fbahr ♦
Thanks a lot Florian.
(31 Oct '13, 06:19)
Erling_MOSEK

Never mind: my original function was actually
\[ \frac{\sum_{i=1}^\ell (b_i + \sum_{j=1}^\ell a_j^{(i)} x_j)^2}{c_1+\sum_{i=1}^n d_ix_i}\]
so I can see it as a sum of \(g_i^2/h\), where \(g_i\) is affine (and actually nonnegative in my domain) and \(h\) is positive affine. Then each of the \(g_i^2/h\) is convex, and so is their sum.