Resource Allocation in OFDM Based Wireless Relay Networks ...

2.3 Resource Allocation Schemes
where
M∑ K∑ K∑
)
a m,k p m,k b j q j
L (p, q, π, τ , ν, λ) =
π k,j τ m,(k,j) log 2
(1 +
1 + a
m=1 k=1 j=1
m,k p m,k + b j q j
) ( )
M∑
K∑ K∑
K∑
+ ν m
(P m − π k,j τ m,(k,j) p m,k + λ Q − q j ,
m=1
k=1 j=1
and λ, ν = [ν 1 , . . . , ν M ] T are the associated dual variables.
The dual problem is then defined as
j=1
(2.10)
min
ν≥0,λ≥0
D (ν, λ) . (2.11)
2.3.2 Joint Resource Allocation Scheme
Before tackling the dual problem (2.11), we need to first find the dual function (2.9)
for given initial λ and ν. The dual function can be rewritten as
D (ν, λ) = max
p,q,π,τ
s.t.
M∑
K∑
K∑
m=1 k=1 j=1
K∑
π k,j = 1, ∀j,
k=1
π k,j τ m,(k,j) (r m,(k,j) − ν m p m,k − λq j ) +
} {{ }
e m,(k,j)
K∑
π k,j = 1, ∀k,
j=1
M∑
ν m P m + λQ
m=1
M∑
τ m,(k,j) = 1, ∀(k, j),
m=1
(2.12)
where e m,(k,j) is defined as the corresponding item. For given π, τ , the optimal p
and q could be found from
D (ν, λ) = max
p,q
M∑
K∑
m=1 k=1 j=1
K∑
e m,(k,j) +
M∑
ν m P m + λQ, (2.13)
which can be further decomposed into the following MK 2 sub-problems:
m=1
max
p m,k ≥0,q j ≥0
e m,(k,j) , ∀m, (k, j). (2.14)
Although e m,(k,j) in (3.10) is not jointly concave in p m,k and q j , a commonly
adopted technique is to apply the high SNR approximation [37] and replace
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