Resource Allocation in OFDM Based Wireless Relay Networks ...

2.3 Resource Allocation Schemes
that is uncovered. Subtract it from all uncovered values and add it to values
at the intersections of lines.
5) Repeat step 3 to step 4 until there is a solution.
Denote the optimal pairing found from Hungarian algorithm as πk,j ∗ .
Substituting πk,j ∗ , τ m,(k,j) ∗ , p∗ m,k , and q∗ j into (2.9) gives the dual function D(λ, ν).
Next we need to find the optimal dual variables from the dual problem (2.11).
From the sub-gradient method [40], we could pick up initial dual variables λ (0) and
ν (0) . Then the dual variables at (i + 1)-th iteration could be updated as
ν (i+1)
m
= ( ν (i)
m + δ (i) ∆ (i) (ν m ) ) +
, ∀m, (2.33)
λ (i+1) = ( λ (i) + δ (i) ∆ (i) (λ) ) +
, (2.34)
where δ (i) is appropriate step size of the ith iteration. Moreover ∆(λ), and ∆(ν m ) are
the subgradients of D(ν, λ) whose expressions can be found through the following
proposition: 1
Proposition 1 : The subgradients of D(ν, λ) at ν and λ are given by, respectively,
∆(ν m ) = P m −
∆(λ) = Q −
K∑
k=1 j=1
K∑
π k,j τ m,(k,j) p ∗ m,k, ∀m, (2.35)
K∑
qj ∗ , (2.36)
j=1
where p ∗ m,k and q∗ j are the optimal power values obtained from (2.24) and (2.25) at
ν and λ.
as
Proof : For the given (π, τ , ν, λ), the dual function in (2.9) can be re-stated
D(ν, λ) = max
p,q
K∑
K∑
k=1 j=1
(
+ λ Q −
π (k,j)
(
log 2
(1 +
K∑
)
q j +
j=1
σ 2 j
(√
pm,k |h k | √ q j |g j | ) 2
(√
pm,k |h k | ) 2
+ σ
2
k
(√
qj |g j | ) 2
)
− ν m p m,k
)
M∑
ν m P m . (2.37)
m=1
1 Note that, for each iteration, all the variables πk,j ∗ , τ m,(k,j) ∗ , p∗ m,k , q∗ j , ∀m, k, j should be
re-computed under λ (i) and ν (i) . The iteration will be stopped once certain criterion is fulfilled.
28