15 The partial lagrange L (˜ p$_{n,k,l}$, ˜ z$_{n,lj}$, λ$_{l}$, η$_{n,lj}$ ) for (16) can then be written explicitly as follows: Eq. (17) can be decomposed into L sub-problems with local variables ˜ p$_{n,k,l}$, ˜ z$_{n,lj}$, λ$_{l}$ and coupling variable η$_{n,lj}$ . The simple lagrangian L$_{l}$ for each cell l can then be written as follows: where λ$_{l}$ is the lagrange multiplier for the inequality constraints and η$_{n,lj}$ are the consistency prices. Thus, the minimization of (18) with respect to the local variables can be done in a distributed way at all BSs. At every iteration, each cell l receives the term ( L ∑ j =1 ,j = l η$_{n,jl}$ ) by message passing and minimizes the local Lagrangian (18) with respect to the local variables ˜ p$_{n,k,l}$, ˜ z$_{n,lj}$, λ$_{l}$ subject to the local constraints. In order to obtain η$_{n,lj}$ the following master lagrange dual problem has to be solved: L ∑ l =1 log$_{2}$ ( σ 2 + ∑ L j =1 ,j = l e ˜ z$_{n,lj}$ e ˜ $^{p$_{n,k,l}$}$h$_{n,k,l}$ ) + L ∑ l =1 L ∑ j =1 ,j = l η$_{n,lj}$ (˜ z$_{n,lj}$ − ˜ g$_{n,k,jl}$ − ˜ p$_{n,k,j}$ )+ L ∑ l =1 λ$_{l}$ ( e ˜ p$_{n,k,l}$ − P$_{n,k,l,}$$_{eq}$ ) (17) L$_{l}$ = log$_{2}$ ( σ 2 + ∑ L j =1 ,j = l e ˜ z$_{n,lj}$ e ˜ $^{p$_{n,k,l}$}$h$_{n,k,l}$ ) + L ∑ j =1 ,j = l η$_{n,lj}$ ˜ z$_{n,lj}$ − ( L ∑ j =1 ,j = l η$_{n,jl}$ ) ˜ p$_{n,k,l}$ + λ$_{l}$ (˜ p$_{n,k,l}$ − P$_{n,k,l,}$$_{eq}$) (18) maximize η$_{n,lj}$ L ∑ l =1 minimize ˜ p$_{n,k,l}$, ˜ z$_{n,lj}$,λ$_{l}$ L$_{l}$ (19) η$_{n,lj}$ ( t + 1) = η$_{n,lj}$ ( t ) + ( δ/t ) (˜ z$_{n,lj}$ − log$_{2}$ p$_{n,k,j}$g$_{n,k,jl}$ ) (20) A simple way to solve (19) is to use the following subgradient update for the consistency prices: In summary, each BS minimizes (18) in parallel with respect to the local variables after receiving the term ∑ L j =1 ,j = l η$_{n,jl}$ . Each BS then estimates the received interference z$_{n,lj}$ from each cell and update the local consistency prices using (20). Finally, each BS broadcast them by message passing to all BSs. Note that δ in (20) represents the step size and is non-negative. A cellular OFDMA network is considered where the radius of each cell is assumed to be 1 km. The maximum user transmit power is considered to be 1 W. The channel gain is defined VI. PERFORMANCE EVALUATION DRAFT March 19, 2018