I have a minimization problem where the objective is a rational function with a quadratic numerator and linear denominator, and where the constraints are linear inequalities.
$$ \begin{array}{ll} \underset {x_1, x_2} {\text{minimize}} & \dfrac{{\beta }_1{x_1}^2+{\beta }_2x_1x_2+{\beta }_3{x_2}^2}{h-{\alpha }_1x_1-{\alpha }_2x_2} =:z (x_1,x_2) \\ \text{subject to} & 0\le x_1\le 1 \\ & 0\le x_2\le 1 \end{array} $$
Note that for the denominator, $h-{\alpha }_1x_1-{\alpha }_2x_2>0$ is always valid, and all parameters ${\alpha }_1$, ${\alpha }_2$, ${\beta }_1$, ${\beta }_2$, ${\beta }_3$ and $h$ are positive real numbers. Is there any rule to show with what values of parameters the objective function is convex, concave, or non-convex?