I want to formulate the well-known max-min (fairness) beamforming problem for multi user multiple input single output (MU-MISO) antenna system. Where {\bf{Q}}_k \in \mathbb{C}^{Mt \times Mt}, k=1,\cdots,Mr, are given channel correlation matrices for all users and `noise`

is the variance of noise which is taken equal for all users.

I want to solve the max-min optimization problem for beamforming matrices ${\bf{X}}_k$s such that it gives the maximum `SINR`

for all users, subject to total transmit power constraint, `P`

.

This problem has been repeatedly announced as a convex problem in the literature but I do not know how can I formulate it as a CVX problem because the following form gives the DCP programming error for the first equality constraint, `SINR(k) == sp/ip;`

, is there another way to write this constraint?

One may write the `SINR(k) == sp-ip;`

, but this expression is typically negative and leads to close to zero ${\bf{X}}_k$s.

```
cvx_begin
variable X(Mt,Mt,Mr) complex semidefinite;
variable Power(Mr,1);
variable SINR(Mr,1);
maximize (min(SINR))
subject to
for k=1:Mr
sp = trace(Q(:,:,k)*Xs(:,:,k));
ip = noise;
for j=1:k-1
ip = ip + trace(Q(:,:,k)*X(:,:,j));
end
for j=k+1:Mr
ip = ip + trace(Q(:,:,k)*X(:,:,j));
end
SINR(k) == sp/ip; % signal to interference ratio on kth user
X(:,:,k)==hermitian_semidefinite(Mt);
Power(k)==trace(X(:,:,k)); % transmit power to kth user
end
sum(Power)<=P; % total power constraint
cvx-end
```