Convex Optimization · Problem 4.62

Duality — Boyd & Vandenberghe

Mathematics — Optimization · optimal-power-bandwidth-allocation-gaussian-broadcast

Problem

Optimal power and bandwidth allocation in a Gaussian broadcast channel. We consider a communication system in which a central node transmits messages to

n

receivers. ('Gaussian' refers to the type of noise that corrupts the transmissions.) Each receiver channel is characterized by its (transmit) power level

P_i \geq 0

and its bandwidth

W_i \geq 0

. The power and bandwidth of a receiver channel determine its bit rate

R_i

(the rate at which information can be sent) via

R_i = \alpha_i W_i \log(1 + \beta_i P_i / W_i),

where

\alpha_i

and

\beta_i

are known positive constants. For

W_i = 0

, we take

R_i = 0

(which is what you get if you take the limit as

W_i \to 0

). The powers must satisfy a total power constraint, which has the form

P_1 + \cdots + P_n = P_{\mathrm{tot}},

where

P_{\mathrm{tot}} > 0

is a given total power available to allocate among the channels. Similarly, the bandwidths must satisfy

W_1 + \cdots + W_n = W_{\mathrm{tot}},

where

W_{\mathrm{tot}} > 0

is the (given) total available bandwidth. The optimization variables in this problem are the powers and bandwidths, i.e.,

P_1, \ldots, P_n, W_1, \ldots, W_n

. The objective is to maximize the total utility,

\sum_{i=1}^n u_i(R_i),

where

u_i : \mathbf{R} \to \mathbf{R}

is the utility function associated with the

i

th receiver. (You can think of

u_i(R_i)

as the revenue obtained for providing a bit rate

R_i

to receiver

i

, so the objective is to maximize the total revenue.) You can assume that the utility functions

u_i

are nondecreasing and concave. Pose this problem as a convex optimization problem.

Related theory

Power In Ac Circuits And Power Factor

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