locally made at each BS and do not need collaboration. Once the allocations are decided, each cell shares them with all other interfering cells. The GP based optimal powers in (7) can then be evaluated in a distributed way using dual decomposition methods by first performing the log transformation of the variables, i.e., ln p$_{n,k,l}$ = ˜ p$_{n,k,l}$ and ln p$_{n,k,j}$ = ˜ p$_{n,k,j}$ , then adding auxiliary variable ln z$_{n,lj}$ = ˜ z$_{n,lj}$ where z$_{n,lj}$ = p$_{n,k,j}$ in order to transfer the coupling in the objective to coupling in the constraints [14]. For given allocations, the problem in (7) can thus be written in a distributed way as follows: 14 minimize ˜ z$_{n,lj}$, ˜ p$_{n,k,l}$ L ∑ l =1 N ∑ n =1 log$_{2}$ ( σ 2 + ∑ L j =1 ,j = $_{l}$g$_{n,k,jl}$ e ˜ z$_{n,lj}$ e ˜ $^{p$_{n,k,l}$}$h$_{n,k,l}$ ) subject to N ∑ n =1 e ˜ p$_{n,k,l}$ ≤ P$_{k,}$$_{max}$ , ∀ k, ∀ l z$_{n,lj}$ = p$_{n,k,j}$, ∀ n, ∀ l (15) Since the computational complexity of (15) is high as it has LK power constraints and LKN variables, we present the dual decomposition of (14) which is more suitable for practical scenarios and has a lower computational complexity. Moreover, the objective function in (15) not only depends on the powers of local users p$_{n,k,l}$ but also on the power of users sharing the same subcarrier in neighboring cells p$_{n,k,j}$ . Thus, in order to minimize the objective in (15), each BS requires the knowledge of interfering gains and interfering transmit powers, that may lead to significant overhead to exchange control information. Th us, in order to obtain a practical distributed solution, we keep a local copy of each of the effective received powers i.e., z$_{n,lj}$ = g$_{n,k,jl}$p$_{n,k,j}$ [14]. (14) can then be formulated in a distributed way as follows: minimize ˜ z$_{n,lj}$, ˜ p$_{n,k,l}$ L ∑ l =1 log$_{2}$ ( σ 2 + ∑ L j =1 ,j = $_{l}$e ˜ z$_{n,lj}$ e ˜ $^{p$_{n,k,l}$}$h$_{n,k,l}$ ) subject to e ˜ p$_{n,k,l}$ ≤ P$_{n,k,l,}$$_{eq}$ , ∀ l ˜ z$_{n,lj}$ = ˜ g$_{n,k,jl}$ + ˜ p$_{n,k,j}$ (16) DRAFT March 19, 2018