Resource Allocation in OFDM Based Wireless Relay Networks ...

2.3 Resource Allocation Schemes
Let D(ν, λ) be the dual function for some λ, i.e
D ( ν, λ ) K∑ K∑
(
(√
pm,k |h k |
= max π (k,j) log 2
(1 √ q j |g j | ) 2 ) )
+
p,q
(√
pm,k |h k | ) 2
+ σ
2
(√
k
qj |g j | ) 2
− ν m p m,k
k=1 j=1
(
+ λ Q −
K∑
)
q j +
j=1
σ 2 j
M∑
ν m P m . (2.38)
m=1
Let p ∗ m,k , q∗ j be the solution to problem (2.37) and p m,k , q j be the solution to problem
(2.38), then
D ( ν, λ ) =
≥
=
K∑
K∑
k=1 j=1
K∑
k=1 j=1
K∑
π (k,j)
(
log 2
(1 +
(
+ λ Q −
K∑
K∑
)
q j +
j=1
π (k,j)
(
log 2
(1 +
(
+ λ Q −
K∑
k=1 j=1
K∑
j=1
q ∗ j
)
+
π (k,j)
(
log 2
(1 +
(
+ λ Q −
K∑
j=1
q ∗ j
(
=D (ν, λ) + (λ − λ) Q −
σ 2 j
( √pm,k
|h k | √ 2
q j |g j |)
) )
( √pm,k
) 2 ( √qj ) 2
− ν m p m,k
|h k | + σ
2
k
|g j |
M∑
ν m P m
m=1
σ 2 j
( √p
∗
m,k
|h k | √ q ∗ j |g j|) 2
)
( √p ) 2 (
∗
m,k
|h k | + σ
2 √q ) 2
− ν m p ∗ m,k
∗
k j |g j |
M∑
ν m P m
m=1
σ 2 j
( √p
∗
m,k
|h k | √ q ∗ j |g j|) 2
)
( √p ) 2 (
∗
m,k
|h k | + σ
2 √q ) 2
− ν m p ∗ m,k
∗
k j |g j |
) (
+ (λ − λ) Q −
K∑
j=1
q ∗ j
)
.
K∑
j=1
q ∗ j
)
+
M∑
ν m P m
Since the subgradient of a function
(
f at x is c that satisfies the inequality f(y) ≥
f(x) + c(y − x) for any y [40], Q − ∑ )
K
j=1 q∗ j is exactly the subgradient ∆(λ). The
subgradient ∆(ν m ) can be proved in a similar way.
The optimal power allocation has complexity of O(MK 2 ) and each
maximization operation in (2.29) requires a complexity of O(M). If the sub-gradient
method requires I iterations to converge, the total complexity of the proposed
scheme becomes O(IK 2 (2M + K)).
m=1
)
)
29