n the vector x will be the variable in this problem

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Unformatted text preview: 12.3 Power control for sum rate maximization in interference channel. We consider the optimization problem n pi log 1 + maximize Aij pj + vi j =i i=1 n pi = 1 subject to i=1 pi ≥ 0, i = 1, . . . , n with variables p ∈ Rn . The problem data are the matrix A ∈ Rn×n and the vector v ∈ Rn . We assume A and v are componentwise nonnegative (Aij ≥ 0 and vi ≥ 0), and that the diagonal elements of A are equal to one. If the off-diagonal elements of A are zero (A = I ), the problem has a simple solution, given by the waterfilling method. We are interested in the case where the off-diagonal elements are nonzero. We can give the following interpretation of the problem, which is not needed below. The variables in the problem are the transmission powers in a communications system. We limit the total power to one (for simplicity; we could have used any other number). The ith term in the objective is 94 the Shannon capacity of the ith channel; the fraction in the argument of the log is the signal to interference plus noise ratio. We can express the problem as n j =1 Bij pj n log maximi...
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