Information Technology Reference
In-Depth Information
2
4
3
5
e
i
X
M
V
i
Da
.x
i
x
j
/
j
D
1;j
¤
i
2
4
r
x
i
V
i
.x
i
/
3
5
T
X
M
X
M
1
M
g
r
.jjx
i
x
j
jj/.x
i
x
j
/
T
e
i
1
r
x
j
V
j
.x
j
/
e
i
:
C
j
D
j
D
1;j
¤
i
It holds that
P
jD1
.x
i
M
M
P
jD1
x
j
x
j
/ D Mx
i
D Mx
i
M x D M.x
i
x/ D Me
i
therefore
M
X
V
i
D
aM
jje
i
2
g
r
.jjx
i
x
j
jj/.x
i
x
j
/
T
e
i
jj
C
jD1;j¤i
2
3
T
X
jD1
r
x
j
V
j
.x
j
/
M
1
M
4
r
x
i
V
i
.x
i
/
5
e
i
(8.24)
It assumed that for all x
i
there is a constant
N such that
jjr
x
i
V
i
.x
i
/jj N
(8.25)
Equation (
8.25
) is reasonable since for a particle moving on a 2-D plane, the gradient
of the cost function r
x
i
V
i
.x
i
/ is expected to be bounded. Moreover it is known that
the following inequality holds:
P
jD1;j¤i
g
r
.x
i
P
jD1;j¤i
be
i
P
jD1;j¤i
bjje
i
x
j
/
T
e
i
jj:
Thus the application of Eq. (
8.24
)gives:
C
P
jD1;j¤i
g
r
.jjx
i
V
i
aM
jje
i
2
x
j
jj/jjx
i
x
j
jj jje
i
jj
jj
M
P
jD1
r
x
j
V
j
.x
j
/jjjje
i
1
Cjjr
x
i
V
i
.x
i
/
jj
) V
i
aM
jje
i
jj
2
C b.M 1/jje
i
jj C 2N jje
i
jj
where it has been taken into account that
P
jD1;j¤i
g
r
.jjx
i
jj
P
jD1;j¤i
bjje
i
x
j
jj/
T
jje
i
jj D b.M 1/jje
i
jj;
and from Eq. (
8.25
)
M
P
jD1
r
x
i
V
j
.x
j
/jjjjr
x
i
V
i
.x
i
/jj
jjr
x
i
V
i
.x
i
/
1
M
jj
P
jD1
r
x
i
V
j
.x
j
/jj N
1
1
C
C
M
M N
2N :
