the original problem in (3) can be expressed as A straightforward approach to determine the optimal Macro resource share among the TPs is to optimize { Z$_{b}$ } using the generic subgradient method. However, we will show that exploiting the structure of the problem at hand leads directly to a very simple algorithm. First, let us define the function, ¯ h : I R$_{+}$ × B ′ 1 → I R$_{+}$ , such that ¯ h (Γ$_{b}$ , b ) for any given TP b and corresponding budget, Γ$_{b}$ , yields the minimum Macro resource needed (in addition to the available budget Γ$_{b}$ for the pico TP b ) accommodate the minimum rates of all users in U ( b $^{)}$. In particular, we can invoke Proposition 1 to explicitly detail ¯ h (Γ$_{b}$ , b ) after recalling the labelling we have adopted, as In (11) we use the convention that ˜ k + 1 reurns the user with the lowest index in U ( b ) whenever ˜ k is null on account of ∑ j ∈U ( b $^{)}$: j ≤ k R min j R$_{j,b}$ > Γ$_{b}$ for all k ∈ U ( b $^{)}$. In a similar manner we define h : I R$_{+}$ × B ′ 1 → I R$_{+}$ , such that h ( Z$_{b}$, b ) for any given TP b and a given Macro budget, Z$_{b}$ , yields the additional minimum resource needed by TP b to accommodate the minimum rates of max γ$_{u,b}$,θ$_{u,}$$_{1}$ ∈ [0 , 1] ∀ u ∈U ( b )    ∑ u ∈U ( b ) ( w$_{u}$R$_{u,}$$_{1}$θ$_{u,}$$_{1}$ + w$_{u}$R$_{u,b}$γ$_{u,b}$ )$^{}$   s . t . ∑ u ∈U ( b ) θ$_{u,}$$_{1}$ ≤ Z$_{b}$ & ∑ u ∈U ( b ) γ$_{u,b}$ ≤ Γ$_{b}$; R$_{u}$ ∈ [ R min u , R max u ] , ∀ u ∈ U ( b $^{)}$; (9) max { Z$_{b}$ ∈ I R$_{+}$ } : ∑ b ∈B ′ 1 Z$_{b}$ ≤ Γ    ∑ b ∈B ′ 1 ˆ O ( Z$_{b}$, Γ$_{b}$ , b )$^{}$   (10) ¯ h (Γ$_{b}$ , b ) =    R min ˜ k $_{+1}$− Ξ$_{b}$ R$_{˜}$ k +1 ,b R$_{˜}$ k +1 , 1 + ∑ j ∈U ( b $^{)}$: j> ˜ k +1 R min j R$_{j,}$$_{1}$ , ˜ k + 1 ∈ U ( b ) 0 , Else ˜ k = max$^{}$   k : k ∈ U ( b ) & ∑ j ∈U ( b $^{)}$: j ≤ k R min j R$_{j,b}$ ≤ Γ$_{b}$$^{}$   , Ξ$_{b}$ = Γ$_{b}$ − ∑ j ∈U ( b $^{)}$: j ≤ ˜ k R min j R$_{j,b}$ (11)