How can I convexify (allowed some approximation) the objective function?

Question

I have a known matrix, $H$ of size $U\times B$. The optimization variable is $D$ of same size, which is binary

Now I have $$S_u=\frac{\sum\limits_{b=1}^{B} D_{u,b}H_{u,b}}{\sum\limits_{b=1}^{B}H_{u,b}-\sum\limits_{b=1}^{B} D_{u,b}H_{u,b}},\quad \forall u\in\{1,\cdots, U\}$$ and I want to maximize $\sum\limits_{u=1}^US_u$.

UPDATE:

with $b_\min\le \sum_{b=1}^B D_{u,b}\le b_\max, \forall u$ and $\sum_{u=1}^{U}D_{u,b}\le u_\max, \forall b$

Can I perform some alternative formulation so that the function becomes convex, or any convex approximation?

EDIT:

The denominator is strictly non-negative. The first or positive part of the denominator denotes the case where $D_{u,b}=1, \forall b, b=1,\cdots, B$

Also, the elements in $H$ are non-negative.

Answer 1

You can reformulate exactly as a MILP problem by performing a Charnes-Cooper transformation and then a linearization of the resulting products of binary and continuous variables, as described in my answer here.

Because you have a sum of ratios here, introduce a new variable $T_u$ for each summand $S_u$. The idea is to multiply numerator and denominator by $T_u$ so that the denominator becomes 1. You want to maximize $$\sum_{u,b} D_{u,b} H_{u,b} T_u$$ subject to $$\sum_b H_{u,b} T_u - \sum_b D_{u,b} H_{u,b} T_u = 1$$ for each $u$. Now introduce $Y_{u,b} = T_u\cdot D_{u,b}$ to linearize both objective and constraint: $$\text{maximize $\sum_{u,b} H_{u,b} Y_{u,b}$ subject to $\sum_b H_{u,b} T_u - \sum_b H_{u,b} Y_{u,b}=1$}$$ Finally, linearize the relationship between $Y$ and $D$.

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For your new question in EDIT 2, first linearize $P_{u,b}D_{u,b}$ and then apply Charnes-Cooper.