Methodology in the Dual Domain Sample Clauses

Methodology in the Dual Domain. We apply the Lagrangian dual decomposition method to solve the above RA problem. In order to solve the problem in continuous space, all inequality con- straints are temporarily treated as equalities. The Lagrangian is formed by first relaxing the coupling constraints in the above primal problem [J1] as Γ|hm |2pm xm B log (1 + n,c n,c ) L(x, P, λ, α, µ, β, θ) = Σ Σ Σ n,c RB 2 τ¯ I0 + δ2 m n c m m c1 c2 c1 c2 + Σ αm (Υα) − µΥµ + Σ Σ βc ,c Υβ − Σ Σ θc ,c Υθ n,c − Σ Σ λn,c Σ xm − 1!, (4.8) n c m where (c1, c2) ∈ C and c1 c2. Additionally, for ease in notation, I define the following terms of the Lagrangian used above which will also be used later in the subgradient method [J1] in Section 4.4.2 as Υα = xm B log (1 + n,c n,c ) − τmin, (4.9) I + δ2 m Σ Σ Γ|hm |2pm n n,c RB 2 c n,c n,c Υµ = Σ Σ Σ xm pm − Pmax, (4.10) m n c n,c1 n,c2 Υβ = Σ Σ xm − (1 − ϑ) Σ Σ xm , (4.11) m n m n n,c1 n,c2 Υθ = Σ Σ xm − (1 + ϑ) Σ Σ xm . (4.12) m n m n Lagrange dual variables are represented in matrices λ , [λn,c]N×C , β , [βc1,c2 ]C×C, θ , [θc1,c2 ]C×C, vector α , [αm]M×1 and the constant µ. Now the primal problem decouples into a Lagrange dual function and a dual problem. The Lagrange dual problem as in [87] is defined as max L(x, P, λ, α, µ, β, θ) , x,P g(λ, α, µ, β, θ) = s.t. ∀(λ, µ, β, θ) ≥ 0 (4.13) ,, m m n,c n,c ∈ {0, 1}, P ∈ R+. , ΣM xm ≤ 1 x The resource utilization costs are determined by the Lagrange dual function g(λ, α, µ, β, θ) on a higher level [88] in (4.13) for given values of the Lagrangian variables. Next, the dual problem at the lower level [87] [J1] is given by G = min λ≥0,µ≥0,β≥0,θ≥0