Convert fractional to quadratic integer programming > Maximize $\frac{\sum_i\sum_ja_{ij}x_iy_j}{(\sum_ix_i)(\sum_jy_j)}+ \frac{\sum_i\sum_jb_{ij}x_iz_j}{(\sum_ix_i)(\sum_jz_j)}$, subject to $Ax+By+Cz \leq d, \quad x,y,z\in \\{0,1\\}$. Can we convert the above problem to a quadratic integer programming? I tried but I could find noway. Does anyone have any idea?

Yes, one way is to introduce two new variables $t_1$and $t_2$ and maximize $t_1+t_2$with the constraints $f_1(x,y) \geq t_1$ and $f_2(x,z) \geq t_2$. Now multiply with denominators and you will have polynomials of $x,y$ and $t$ in your constraints.

These nonlinear constraints can all be linearized using standard strategies. For instance, $ab$ where $a$ and $b$ are binary can be written as $w$ where the new variable $w$ satisfies $w \leq a$, $w \leq b$ and $w \geq a +b -1$. A product between binary $a$ and continuous variable $b$ is modelled using a big-M reformulation where the product is replaced with $w$ where $-Ma \leq w \leq Ma$ and $-M(1-a) \leq w-b \leq M(1-a)$

Of course, the special case here with with a cubic $abc$ involving two binaries $a$ and $b$ and a continuous variable $c$, can be done in one step as $-Ma \leq w \leq Ma$, $-Mb \leq w \leq Mb$ and $-M(2-a-b) \leq w-c \leq M(2-a-b)$

At this point, you have a mixed-integer linear program.