I’m trying to solve the following mixed-integer non-linear optimization problem:

\underset{\mathbf{p},\mathbf{h},\mathbf{a},\mathbf{b}}{\text{min.}} \quad P= \sum_{\mathcal{K}}p+\sum_{\mathcal{J}}h

\text{subject to}

\quad \quad \quad \ \ \prod_{\mathcal{K}}\big(1+\frac{a* p_{k}* l_{k}}{1+e*P}\big) \geq 2^{c}

\quad \quad \quad \ \ \sum_{\mathcal{J}}b*h_{j}*l_{j}+b*e*P \geq s

\quad \quad \quad \ \ 0 \leq p_{k}+h_{j} \leq f_{max}

\quad \quad \quad \ \ 0 \leq p_{k} \leq f_{max}, \quad \quad \quad \ \ 0 \leq h_{j} \leq f_{max}

\quad \quad \quad \ \ a \in \{0,1\}, b \in \{0,1\}

\quad \quad \quad \ \ \sum_{\mathcal{K}} a+\sum_{\mathcal{J}}b \leq N

l \in \mathbb{R}^{N \times 1} represents the eigenvalues of the matrix \mathbf{H}^{H}\mathbf{H}, where \mathbf{H}\in \mathbb{C}^{N \times N} is a full rank matrix and e is a constant (always we use 0.1 or any number less than 0.1). The problem is not convex due to binary variables multiplication in the first and the second constraints and also due to the product \prod_{\mathcal{K}}... in the first constraint.

To linearize the binary variables, I have used proposition 1 in the following paper SVD-TWC as follows:

p-(1-a)*f_{max} \leq p \leq a.*f_{max}

h-(1-b)*f_{max} \leq h \leq b.*f_{max}

While for the product term in the first constraint, I have used (geo_mean) function as suggested in this post.

The only term that I was not able to deal with is the second term at the second constraint (b*e*P). I used the following code to test the solution without considering b*e*P

```
cvx_begin
cvx_solver gurobi
variables p(N) h(N) y(N)
variable a(N) binary
variable b(N) binary
expression x0
x0=(e*(sum(p)+sum(h)));
minimize sum(p)+sum(h)
subject to
geo_mean(1+p.*l+x0)>=2^(c/N)*(1+x0);
sum(h.*l+y)==s;
p<=a.*fmax;
p>=p-(1-a).*fmax;
h<=b.*fmax;
h>=h-(1-b).*fmax;
0<=p+h<=fmax;
0<=p<=fmax;
0<=h<=fmax;
sum(a)+sum(b)<=N;
0<=y<=b*N*e*fmax;
0<=x0-y<=(1-b)*N*e*fmax;
cvx_end
```

and this works fine. Any idea how to deal with the term b*e*P?

Thanks

A.