Similarly, we define function g : I R$_{++}$ × B ′ 1 → I R$_{+}$ such that, g ( λ, b ) , ∀ λ > 0 , b ∈ B ′ 1 equals The following result is proved in the appendix. Theorem 1. The optimal objective value of (16) is given by where ˆ λ ∈ (0 , ∞ ) is the unique solution to the relation which can be determined via bisection search. Corollary 3. Suppose that the optimal ˆ λ satisfying (20) is given. Then, if ˆ λ ∈ (( m − 1) µ$_{m,b}$, mµ$_{m,b}$ ) for some m = 1 , · · · , N$_{b}$ , the optimal solution comprises of assigning an identical resource share γ$_{k,b}$ = µ$_{m,b}$/ ˆ λ with θ$_{k,}$$_{1}$ = 0 for all users k ∈ U ( b ) : R$_{k,}$$_{1}$ R$_{k,b}$ < µ$_{m,b}$ , whereas all users k ∈ U ( b ) : R$_{k,}$$_{1}$ R$_{k,b}$ > µ$_{m,b}$ are assigned θ$_{k,}$$_{1}$ = 1 / ˆ λ with γ$_{k,b}$ = 0 . The user k ∈ U ( b ) : R$_{k,}$$_{1}$ R$_{k,b}$ = µ$_{m,b}$ is assigned θ$_{k,}$$_{1}$ = m/ ˆ λ − 1 /µ$_{m,b}$ with γ$_{k,b}$ = 1 − ( m − 1) µ$_{m,b}$/ ˆ λ . On max γ$_{u,b}$,θ$_{u,}$$_{1}$ ∈ [0 , 1] ∀ u ∈U $^{′}$,b ∈B ′ 1    ∑ u ∈U ′ ∑ b ∈B ′ 1 x$_{u,b}$ ln ( R$_{u,}$$_{1}$θ$_{u,}$$_{1}$ + R$_{u,b}$γ$_{u,b}$ )$^{}$   s . t . ∑ u ∈U ′ θ$_{u,}$$_{1}$ ≤ 1 & ∑ u ∈U ′ γ$_{u,b}$ ≤ 1 ∀ b ∈ B ′ $_{1}$. (16) h ( λ, b ) =    1 $_{µ$_{m,b}$}$, m = 1 , · · · , N$_{b}$ : λ ∈ (( m − 1) µ$_{m,b}$, mµ$_{m,b}$ ) m − 1 λ , m = 2 , · · · , N$_{b}$ + 1 : λ ∈ [( m − 1) µ$_{m}$$_{−}$$_{1}$$_{,b}$, ( m − 1) µ$_{m,b}$ ] (17) set { R$_{u,}$$_{1}$ $_{R$_{u,b}$}$}$_{u}$$_{∈U}$ ( b ) and recall that these ratios are all strictly positive and distinct. Then, defining µ$_{N}$$_{b}$$_{+1}$$_{,b}$ = ∞ , we have 0 < µ$_{1}$$_{,b}$ < µ$_{2}$$_{,b}$ < · · · < µ$_{N}$$_{b}$$_{,b}$ < µ$_{N}$$_{b}$$_{+1}$$_{,b}$ = ∞ . Next, we define a function h : I R$_{++}$ × B ′ 1 → I R$_{+}$ as    ∑ N$_{b}$ j = $_{m}$ln( µ$_{j,b}$/λ ) + ( m − 1) ln( µ$_{m,b}$/λ ) , m = 1 , · · · , N$_{b}$ : λ ∈ (( m − 1) µ$_{m,b}$, mµ$_{m,b}$ ) − ( m − 1) ln( m − 1) + ∑ N$_{b}$ q = $_{m}$ln( µ$_{q,b}$/λ ) , m = 2 , · · · , N$_{b}$ + 1 : λ ∈ [( m − 1) µ$_{m}$$_{−}$$_{1}$$_{,b}$, ( m − 1) µ$_{m,b}$ ] (18) ∑ b ∈B ′ 1  $_{}$g ( ˆ λ, b ) + ∑ k ∈U ( b ) ln( R$_{k,b}$ )$^{}$ $_{}$, (19) 1 + ∑ b ∈B ′ 1 h ( λ, b ) = ∑ b ∈B ′ 1 N$_{b}$/λ, (20)