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 :math:P_1, :math:P_2, :math:\ldots, :math:P_n to :math:n receivers, labeled :math:1, \ldots, n, with receiver :math:i receiving signal from transmitter :math:i. The power received from transmitter :math:j at receiver :math:i is :math:G_{ij} P_{j}, where :math:G_{ij} > 0 represents the path gain from transmitter :math:j to receiver :math:i. The signal power at receiver :math:i is :math:G_{ii} P_i, and the interference power at receiver :math:i is :math:\sum_{k \neq i} G_{ik}P_k. The noise power at receiver :math:i is :math:\sigma_i, and the signal to noise ratio (SINR) of the :math:i\ th receiver-transmitter pair is .. math:: S_i = \frac{G_{ii}P_i}{\sigma_i + \sum_{k \neq i} G_{ik}P_k}. The transmitters and receivers are constrained to have a minimum SINR :math:S^{\text{min}}, and the :math:P_i are bounded between :math:P_i^{\text{min}} and :math:P_i^{\text{max}}. This gives the problem .. math:: \begin{array}{ll} \mbox{minimize} & P_1 + \cdots + P_n \\ \mbox{subject to} & P_i^{\text{min}} \leq P_i \leq P_i^{\text{max}}, \\ & 1/S^{\text{min}} \geq \frac{\sigma_i + \sum_{k \neq i} G_{ik}P_k}{G_{ii}P_i}. \end{array} .. code:: python 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) .. code:: python problem.is_dgp() .. parsed-literal:: True .. code:: python problem.solve(gp=True) problem.value .. parsed-literal:: 0.9615384629119621 .. code:: python p.value .. parsed-literal:: array([0.18653846, 0.16730769, 0.23461538, 0.19615385, 0.17692308]) .. code:: python inverse_sinr.value .. parsed-literal:: array([5., 5., 5., 5., 5.]) .. code:: python (1/sinr_min) .. parsed-literal:: 5.0