by Robert Gowers, Roger Hill, Sami Al-Izzi, Timothy Pollington and Keith Briggs
from Boyd and Vandenberghe, Convex Optimization, exercise 4.62 page 210
Consider a system in which a central node transmits messages to $n$ receivers. Each receiver channel $i \in \{1,...,n\}$ has a transmit power $P_i$ and bandwidth $W_i$. A fraction of the total power and bandwidth is allocated to each channel, such that $\sum_{i=1}^{n}P_i = P_{tot}$ and $\sum_{i=1}^{n}W_i = W_{tot}$. Given some utility function of the bit rate of each channel, $u_i(R_i)$, the objective is to maximise the total utility $U = \sum_{i=1}^{n}u_i(R_i)$.
Assuming that each channel is corrupted by Gaussian white noise, the signal to noise ratio is given by $\beta_i P_i/W_i$. This means that the bit rate is given by:
$R_i = \alpha_i W_i \log_2(1+\beta_iP_i/W_i)$
where $\alpha_i$ and $\beta_i$ are known positive constants.
One of the simplest utility functions is the data rate itself, which also gives a convex objective function.
The optimisation problem can be thus be formulated as:
minimise $\sum_{i=1}^{n}-\alpha_i W_i \log_2(1+\beta_iP_i/W_i)$
subject to $\sum_{i=1}^{n}P_i = P_{tot} \quad \sum_{i=1}^{n}W_i = W_{tot} \quad P \succeq 0 \quad W \succeq 0$
Although this is a convex optimisation problem, it must be rewritten in DCP form since $P_i$ and $W_i$ are variables and DCP prohibits dividing one variable by another directly. In order to rewrite the problem in DCP format, we utilise the $\texttt{kl_div}$ function in CVXPY, which calculates the Kullback-Leibler divergence.
$\text{kl_div}(x,y) = x\log(x/y)-x+y$
$-R_i = \text{kl_div}(\alpha_i W_i, \alpha_i(W_i+\beta_iP_i)) - \alpha_i\beta_iP_i$
Now that the objective function is in DCP form, the problem can be solved using CVXPY.
In [1]:
#!/usr/bin/env python3
# @author: R. Gowers, S. Al-Izzi, T. Pollington, R. Hill & K. Briggs
import numpy as np
import cvxpy as cp
In [2]:
def optimal_power(n, a_val, b_val, P_tot=1.0, W_tot=1.0):
# Input parameters: α and β are constants from R_i equation
n = len(a_val)
if n != len(b_val):
print('alpha and beta vectors must have same length!')
return 'failed', np.nan, np.nan, np.nan
P = cp.Variable(shape=n)
W = cp.Variable(shape=n)
alpha = cp.Parameter(shape=n)
beta = cp.Parameter(shape=n)
alpha.value = np.array(a_val)
beta.value = np.array(b_val)
# This function will be used as the objective so must be DCP;
# i.e. elementwise multiplication must occur inside kl_div,
# not outside otherwise the solver does not know if it is DCP...
R = cp.kl_div(cp.multiply(alpha, W),
cp.multiply(alpha, W + cp.multiply(beta, P))) - \
cp.multiply(alpha, cp.multiply(beta, P))
objective = cp.Minimize(cp.sum(R))
constraints = [P>=0.0,
W>=0.0,
cp.sum(P)-P_tot==0.0,
cp.sum(W)-W_tot==0.0]
prob = cp.Problem(objective, constraints)
prob.solve()
return prob.status, -prob.value, P.value, W.value
In [3]:
np.set_printoptions(precision=3)
n = 5 # number of receivers in the system
a_val = np.arange(10,n+10)/(1.0*n) # α
b_val = np.arange(10,n+10)/(1.0*n) # β
P_tot = 0.5
W_tot = 1.0
status, utility, power, bandwidth = optimal_power(n, a_val, b_val, P_tot, W_tot)
print('Status: {}'.format(status))
print('Optimal utility value = {:.4g}'.format(utility))
print('Optimal power level:\n{}'.format(power))
print('Optimal bandwidth:\n{}'.format(bandwidth))