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N }, output Y ∈ Y = {1, 2, . . , M }, and channel law Pij = Prob{Y = j|X = i}, i = 1, 2, . . , N, j = 1, 2, . . , M . The channel law forms a channel matrix P ∈ RN ×M , where the (i, j) entry of P is Pij ≥ 0 with P1 = 1. A distribution p ∈ RN ×1 on the input, together with a given channel matrix P, induces a distribution q ∈ RM ×1 on the output by q = PT p, and a joint distribution Q ∈ RN ×M on the input output pair by Qij = pi Pij . We also associate with each input alphabet symbol i an input cost si ≥ 0, forming a cost vector s ∈ RN ×1 .

N j=1 zj w ≥ 1, zj ≥ 0, j = 1, 2, . . 5) where P(i) is the ith row of P. 4) means the following: • Weak duality. 4) produces an upper bound on channel capacity with input cost: log M j=1 exp(αj + γS) ≥ C(S). • Strong duality. 4) is C(S). 1. 1. 3) is the following GP (in convex form) minimize log subject to Pα variables α. 6) The constant parameters are P. An equivalent version of the Lagrange dual problem is the following GP (in standard form): minimize subject to variables M j=1 zj Pij M j=1 zj (i) ≥ e−H(P ) , i = 1, 2, .

22) 46 Geometric Programming Appropriate choice of the stepsize α(t) > 0 leads to convergence of the dual algorithm [16]. 20) involving only local variables, upon receiving the updated dual variables {γji , j : i ∈ I(j)} (note that {γij , j ∈ I(i)} are local dual variables). 22). The amount of message passing overhead in the above distributed algorithm can be substantially reduced using the structures of the coupling variables. 1 through distributed GP-based power control. 1 Information Theory Materials in this subsection are in part based on recent publications [30, 33, 82, 84], and some problems have been studied in the 1980s and 1990s [120, 121, 122].