# Power control¶

*This example is adapted from Boyd, Kim, Vandenberghe, and Hassibi,* “A
Tutorial on Geometric
Programming.”

*The problem data is adapted from the corresponding example in CVX’s
example library (Almir Mutapcic).*

This example formulates and solves a power control problem for communication systems, in which the goal is to minimize the total transmitter power across n trasmitters, each trasmitting positive power levels \(P_1\), \(P_2\), \(\ldots\), \(P_n\) to \(n\) receivers, labeled \(1, \ldots, n\), with receiver \(i\) receiving signal from transmitter \(i\).

The power received from transmitter \(j\) at receiver \(i\) is \(G_{ij} P_{j}\), where \(G_{ij} > 0\) represents the path gain from transmitter \(j\) to receiver \(i\). The signal power at receiver \(i\) is \(G_{ii} P_i\), and the interference power at receiver \(i\) is \(\sum_{k \neq i} G_{ik}P_k\). The noise power at receiver \(i\) is \(\sigma_i\), and the signal to noise ratio (SINR) of the \(i\)th receiver-transmitter pair is

The transmitters and receivers are constrained to have a minimum SINR \(S^{\text min}\), and the \(P_i\) are bounded between \(P_i^{\text min}\) and \(P_i^{\text max}\). This gives the problem

```
import cvxpy as cp
import numpy as np
# Problem data
n = 5 # number of transmitters and receivers
sigma = 0.5 * np.ones(n) # noise power at the receiver i
p_min = 0.1 * np.ones(n) # minimum power at the transmitter i
p_max = 5 * np.ones(n) # maximum power at the transmitter i
sinr_min = 0.2 # threshold SINR for each receiver
# Path gain matrix
G = np.array(
[[1.0, 0.1, 0.2, 0.1, 0.05],
[0.1, 1.0, 0.1, 0.1, 0.05],
[0.2, 0.1, 1.0, 0.2, 0.2],
[0.1, 0.1, 0.2, 1.0, 0.1],
[0.05, 0.05, 0.2, 0.1, 1.0]])
p = cp.Variable(shape=(n,), pos=True)
objective = cp.Minimize(cp.sum(p))
S_p = []
for i in range(n):
S_p.append(cp.sum(cp.hstack(G[i, k]*p[k] for k in range(n) if i != k)))
S = sigma + cp.hstack(S_p)
signal_power = cp.multiply(cp.diag(G), p)
inverse_sinr = S/signal_power
constraints = [
p >= p_min,
p <= p_max,
inverse_sinr <= (1/sinr_min),
]
problem = cp.Problem(objective, constraints)
```

```
problem.is_dgp()
```

```
True
```

```
problem.solve(gp=True)
problem.value
```

```
0.9615384629119621
```

```
p.value
```

```
array([0.18653846, 0.16730769, 0.23461538, 0.19615385, 0.17692308])
```

```
inverse_sinr.value
```

```
array([5., 5., 5., 5., 5.])
```

```
(1/sinr_min)
```

```
5.0
```