4 The average network throughput C is a function of both subcarrier and power allocation variables. The sum rate maximization problem is formulated as follows using the standard Shannon capacity formula, C$_{n,k,l}$ = log$_{2}$(1 + γ$_{n,k,l}$ ) , where C$_{n,k,l}$ and γ$_{n,k,l}$ represent the throughput and SINR of the k th user at n th subcarrier in cell l , respectively: maximize p$_{n,k,l}$,α$_{n,k,l}$ L ∑ l =1 K ∑ k =1 N ∑ n =1 α$_{n,k,l}$ log$_{2}$ ( 1 + p$_{n,k,l}$h$_{n,k,l}$ σ $^{2}$+ I$_{n,l}$ ) (1) subject to N ∑ n =1 p$_{n,k,l}$ ≤ P$_{k,}$$_{max}$ , ∀ k, ∀ l (2) K ∑ k =1 α$_{n,k,l}$ = 1 , ∀ n, ∀ l (3) α$_{n,k,l}$ ∈ { 0 , 1 } , ∀ n, ∀ l, ∀ k (4) In (1), I$_{n,l}$ = ∑ L j =1 ,j = l ∑ K k =1 α$_{n,k,j}$p$_{n,k,j}$g$_{n,k,jl}$ represents the cumulative interference at n th subcarrier in cell l from the users in all other cells, p$_{n,k,l}$ denotes the power transmitted by k th user at the n th subcarrier in cell l , α$_{n,k,l}$ represents the allocation of k th user at the n th subcarrier in cell l and h$_{n,k,l}$ is the channel gain of k th user at the n th subcarrier in cell l . Constraint (2) implies that the power spent by k th user on its allocated subcarriers cannot exceed the maximum available power, P$_{k}$$_{max}$ . For each cell, we collect the power allocation variables p$_{n,k,l}$ in a vector p$_{n,l}$ = [ p$_{n,}$$_{1}$$_{,l}$, p$_{n,}$$_{2}$$_{,l}$, ...., p$_{n,K,l}$ ] and then stack all the vectors in a power matrix P$_{l}$ of cell l where P$_{l}$ ∈ R N × $^{K}$. Constraint (3) restricts the allocation of each subcarrier to only one user. The channel gains h$_{n,k,l}$ and binary allocation variables α$_{n,k,l}$ are stacked up similarly in the matrices H$_{l}$ and A$_{l}$ , respectively, where A$_{l}$ , H$_{l}$ ∈ R N × $^{K}$. Moreover, we define g$_{n,k,lj}$ as the interfering gain from the k th user in cell l to cell j , ∀ j = l at n th subcarrier. We collect these interfering gains into a vector g$_{n,lj}$ = [ g$_{n,}$$_{1}$$_{,lj}$, g$_{n,}$$_{2}$$_{,lj}$...., g$_{n,K,lj}$ ] and then stack all the vectors in a matrix G$_{lj}$ ∈ R N × $^{K}$. DRAFT March 19, 2018