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where
s
e
is the free link capacity of link
e
and
v
ʲ
(
s
e
) is an increasing concave
function. In this paper, we adopt the form as stated in [11].
log(
s
e
)
,β
=1
s
1
−β
1
−ʲ
,
v
ʲ
(
s
e
)=
β
=1
.
The Lagrange dual function of (9) is as follows:
v
ʲ
(
s
e
)
D
(
p
)=
max
{s
e
},t
k
∈T
k
,∀k
e
p
e
s
e
−
t
e
+
b
e
c
e
+
k
−
e
(10)
v
ʲ
(
s
e
)
p
e
s
e
−
=max
s
e
e
+
k
p
e
t
e
+
e
p
e
c
e
−
b
e
.
min
t
k
∈T
k
e
The dual problem of (9) is as follows:
min
p≥
0
D
(
p
)
.
(11)
Because
D
(
p
) is not differentiable and (11) cannot be solved with the gradient
method directly, we solve the problem by using the subgradient method. We can
obtain the subgradient of
D
(
p
)from[10],
t
e
,
ζ
e
=
c
e
−
b
e
−
s
e
−
∀
e
∈
E,
k
t
e
}
where
s
e
,
{
is the solution of
maximize
c
e
−b
e
≥s
e
>
0
v
ʲ
(
s
e
)
p
e
s
e
,
−
∀
e
∈
E
(12)
and
p
e
t
e
,
minimize
t
k
∈T
k
∀
k.
(13)
e
On the basis of the subgradient projection method,
p
e
can be updated as
follows,
p
e
(
τ
+1)=
p
e
(
τ
)
−
μ
(
τ
)
ζ
e
(
τ
)
,p
e
(
τ
)
>μ
(
τ
)
ζ
e
(
τ
)
μ
(
τ
)
ζ
e
(
τ
)
where
ζ
e
is the subgradient and
μ
(
τ
) is the step parameter. Theoretically, the
step parameter
μ
(
τ
) is of vital importance to the convergence of this algorithm.
However, practically, owing to the continuous evolving of the network and the
P2P applications, we can set the step parameter as a constant value.
After solving (11) by using the subgradient method, we obtain the distributed
algorithm for solving (9), which is the interactive optimization algorithm of the
0
,
p
e
(
τ
)
≤
