How can I express this objective function in CVX?

MIDCP
1

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_{b=1}^{B} D_{u,b}H_{u,b}}{\sum_{b=1}^{B}H_{u,b}-\sum_{b=1}^{B} D_{u,b}H_{u,b}}, \forall u, u=1,\cdots, U

And I want to maximize \sum_{u=1}^US_u

I am doing the following,

  Num=sum(D.*H,2);
  Denom=sum(H,2)-Num;
  Ratio=Num./Denom;
  
  maximize sum(Ratio)

it is throwing me the following error

Disciplined convex programming error:
Cannot perform the operation: {real affine} ./ {real affine}

Error in ./ (line 19)
z = times( x, y, ‘./’ );

How can I get rid of this? Any alternative formulation of the objective function!

2

Please show us your proof that the objective is concave.

3

Can I perform some convex approximation of the objective function so that formulated as a DCP?

4

You can perform any Approximation you want. Whether it’s any good is another matter.

Using DCP capability, you can do piecewise linear model per section 9.1.13 “Continuous piecewise-linear functions” of the Mosek Cookbook https://docs.mosek.com/modeling-cookbook/mio.html#integer-modeling