Resource Allocation in OFDM Based Wireless Relay Networks ...

5.4 Lifetime Maximization Scheme
5.4 Lifetime Maximization Scheme
Introducing an auxiliary variable t, we first reformulate the problem as
max
p n,k ,q n,k ,t
t (5.6)
s.t. t ≤ EDF n
∑ K
k=1 p , t ≤ EAF n
∑ K
n,k
k=1 q , ∀n
n,k
K∑
K∑
r n,k ≥ R, p n,k ≤ P n ,
k=1
K∑
q n,k ≤ Q n ,
k=1
k=1
∀n
p n,k ≥ 0, q n,k ≥ 0, ∀n, k.
∀n
Changing the variable t = 1 , we can reformulate the above problem into an
z
equivalent standard convex optimization problem:
min
p n,k ,q n,k ,z
s.t.
z (5.7)
K∑
k=1
K∑
k=1
R −
p n,k − zE DF
n ≤ 0, , ∀ n
q n,k − zE AF
n ≤ 0, ∀ n
K∑
r n,k ≤ 0,
k=1
K∑
q n,k − Q n ≤ 0,
k=1
K∑
p n,k − P n ≤ 0,
k=1
∀n
z ≥ 0, p n,k ≥ 0, q n,k ≥ 0 ∀ n, k,
∀n
where the first two constraints are linear in z.
In problem (5.7), we can always choose a large enough value of z such that
the energy constraints are satisfied with strict inequality. In addition, we assume
that there always exists a feasible solution such that the rate and the maximum
power constraints are satisfied with strict inequality (However, at optimality the
rate constraint R − ∑ K
k=1 r n,k ≤ 0, ∀n holds with equality). Thus the Slater’s
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