Resource Allocation in OFDM Based Wireless Relay Networks ...

5.4 Lifetime Maximization Scheme
condition is satisfied [13]. Therefore, the strong duality holds and we can obtain the
optimal primal solution by solving the corresponding dual problem [13] under the
stated assumptions.
For optimization problem (5.7), the dual function can be defined as
D ( λ, η, ν, ¯λ, ¯η ) =
min L ( p n,k , q n,k , z, λ n , η n , ν n , ¯λ
)
n , ¯η n
p n,k ≥0,q n,k ≥0,z≥0
(5.8)
where
L ( p, q, z, λ, η, ν, ¯λ, ¯η ) (
N∑ K
)
∑
=z + λ n p n,k − zEn
DF +
The dual problem is
+
+
n=1
k=1
N∑
ν n
(R −
n=1
)
K∑
r n,k +
k=1
(
N∑ K
)
∑
¯η n q n,k − Q n
n=1
k=1
(
N∑ ∑ K
η n q n,k − zEn
AF
n=1
k=1
(
N∑ K
)
∑
¯λ n p n,k − P n
n=1
k=1
)
(5.9)
max
λ,η,ν,¯λ,¯η
D ( λ, η, ν, ¯λ, ¯η ) (5.10)
s.t. λ n ≥ 0, η n ≥ 0, ν n ≥ 0, ¯λn ≥ 0, ¯η n ≥ 0 ∀ n.
To facilitate the immediate recovery of auxiliary variables z, we change the
primal objective function in (5.7) to z 2 , since for z ≥ 0 minimizing z is equivalent
to minimizing z 2 . Then the Lagrangian L in (5.9) can be re-written as
(
N∑ ( ) ) N∑ K∑
L = z 2 − z λn En
DF + η n En
AF + (( λ n + ¯λ n )p n,k + (η n + ¯η n ) q n,k − ν n r n,k )
+
n=1
n=1 k=1
N∑ (
νn R − ¯λ
)
n P n − ¯η n Q n , (5.11)
n=1
and the dual function becomes
D ( λ, η, ν, ¯λ, ¯η ) (
N∑ (
= min z 2 − z λn En
DF
z≥0
+
N∑
K∑
n=1 k=1
n=1
) )
+ η n En
AF +
N∑ (
νn R − ¯λ
)
n P n − ¯η n Q n
n=1
min (Λ np n,k + Γ n q n,k − ν n r n,k ) (5.12)
p n,k ≥0, q n,k ≥0
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