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NE
W
A
L
G
O
RI
T
H
M
O
F
F
RA
W
e
p
r
o
p
o
s
e
a
n
ew
f
o
r
th
e
C
G
m
et
h
o
d
.
T
h
e
s
eq
u
en
ce
o
f
iter
atio
n
in
th
e
n
e
w
m
et
h
o
d
is
o
b
tain
ed
f
r
o
m
(
2
)
f
o
r
w
h
ic
h
t
h
e
d
ir
ec
tio
n
d
_
k
is
co
m
p
u
ted
b
y
(
3
)
.
W
h
ile
th
e
p
ar
am
eter
p
ar
am
eter
B
k
i
n
th
e
n
e
w
m
et
h
o
d
is
;
=
‖
‖
2
‖
−
1
‖
2
∈
(
0
;
1
)
(
1
0
)
w
h
er
e
FR
A
d
es
ig
n
ed
th
e
n
e
w
m
o
d
i
f
ied
m
eth
o
d
b
y
Ah
m
ed
C
h
er
g
u
i
.
No
te
th
at,
f
o
r
th
e
d
ir
ec
tio
n
d
ef
i
n
ed
b
y
(
3
)
,
w
i
th
t
h
e
C
G
p
ar
am
eter
co
m
p
u
ted
b
y
(
1
0
)
,
w
e
h
av
e
,
=
−
‖
‖
2
+
‖
‖
2
‖
−
1
‖
2
−
1
(
1
1
)
Evaluation Warning : The document was created with Spire.PDF for Python.
I
n
t J
E
lec
&
C
o
m
p
E
n
g
I
SS
N:
2
0
8
8
-
8708
Glo
b
a
l c
o
n
ve
r
g
e
n
ce
o
f n
ew co
n
ju
g
a
te
g
r
a
d
ien
t m
eth
o
d
w
ith
in
ex
a
ct
lin
e
s
ea
r
ch
(
C
h
erg
u
i A
h
med
)
1471
B
y
t
h
e
C
a
u
c
h
y
-
Sc
h
w
ar
z
i
n
eq
u
alit
y
,
it c
a
n
b
e
co
n
clu
d
ed
th
at,
≤
−
‖
‖
2
+
‖
‖
2
=
(
−
1
+
)
‖
‖
2
<
0
(
1
2
)
So
,
th
e
n
e
w
d
ir
ec
tio
n
is
s
atis
f
ied
.
I
n
th
e
n
e
w
C
G
m
et
h
o
d
,
th
e
s
te
p
is
d
eter
m
i
n
ed
b
y
t
h
e
(
SW
P
)
.
T
o
th
is
ai
m
,
w
e
u
s
e
a
b
ac
k
tr
a
ck
in
g
ap
p
r
o
ac
h
to
co
m
p
u
te
t
h
e
s
tep
l
en
g
t
h
.
No
w
w
e
ar
e
r
ea
d
y
to
p
r
o
p
o
s
e
th
e
alg
o
r
ith
m
o
f
t
h
e
n
e
w
C
G
m
e
th
o
d
(
1
0
)
A
l
g
o
r
ith
m
1
Step 1
:
Given
0
∈
s
e
t
k
=
1
.
∈
(
0
,
1
)
set
0
=
−
0
=
−
∇
(
0
)
Step 2: Compute
by (10), (4), (5), (6); (7)
Step 3: Compute
by (3); if
‖
‖
=
0
, then stop.
Step 4: Calculate step length
by (8) and (9) line search,
=
0
.
1
,
=
0
.
01
Step 5:
Let
+
1
=
+
.
Step 6:
if
(
)
<
(
−
1
)
and
‖
∇
(
)
‖
<
, then stop,
Otherwise
Set
=
+
1
go to step 2
3.
T
H
E
G
L
O
B
AL
CO
N
VE
R
G
E
NC
E
P
RO
P
RIE
T
E
S
I
n
th
i
s
s
ec
t
io
n
,
w
e
a
n
al
y
ze
th
e
co
n
v
er
g
e
n
ce
o
f
F
R
A
m
eth
o
d
.
T
o
th
is
ai
m
,
w
e
m
ad
e
t
h
e
f
o
llo
w
in
g
ass
u
m
p
tio
n
:
Ass
u
m
p
t
io
n
1
(
H1
)
T
h
e
o
b
j
ec
tiv
e
f
u
n
ctio
n
is
b
o
u
n
d
ed
b
elo
w
o
n
th
e
le
v
el
s
e
t
an
d
is
co
n
ti
n
u
o
u
s
a
n
d
d
if
f
er
en
ti
ab
le
in
n
eig
h
b
o
r
h
o
o
d
o
f
th
e
lev
el
s
et
=
{
∈
;
(
)
<
(
0
)
}
(
H2
)
T
h
e
g
r
ad
ien
t
is
L
ip
s
c
h
itz
co
n
ti
n
u
o
u
s
i
n
, s
o
a
co
n
s
tan
t M
≥
0
ex
is
ts
,
s
u
ch
t
h
at
‖
(
)
−
(
)
‖
≤
‖
−
‖
,
∈
(
1
3
)
T
h
e
f
o
llo
w
i
n
g
le
m
m
a
p
r
o
v
id
e
s
a
lo
w
er
b
o
u
n
d
f
o
r
th
e
s
tep
le
n
g
t
h
(
g
en
er
ated
b
y
A
l
g
o
r
ith
m
1
)
.
T
h
e
r
esu
lt o
f
t
h
i
s
le
m
m
a
w
i
ll b
e
n
e
ed
ed
in
th
e
r
est o
f
t
h
is
s
ec
tio
n
.
3
.
1
.
Su
f
f
icient
des
ce
nt
co
nd
it
io
n
T
h
eo
r
em
1
:
s
u
p
p
o
s
e
th
at
th
e
s
eq
u
e
n
ce
{
}
an
d
{
}
ar
e
g
en
er
ated
b
y
(
2
)
(
3
)
an
d
FR
A
.
th
e
s
tep
len
g
t
h
,
is
d
eter
m
in
ed
b
y
i
n
e
x
ac
t
li
n
e
s
e
ar
ch
(
9
)
an
d
(
1
0
)
if
‡
0
,
th
en
p
o
s
s
es
s
es
t
h
e
s
u
f
f
icie
n
t
d
esce
n
t
co
n
d
itio
n
:
≤
−
‖
‖
2
P
r
o
o
f
:
B
y
t
h
e
f
o
r
m
u
la
(
1
0
)
,
w
e
h
av
e
t
h
e
f
o
llo
w
i
n
g
:
=
‖
‖
2
‖
−
1
‖
2
≥
0
Hen
ce
w
e
o
b
tain
0
≤
≤
‖
‖
2
‖
−
1
‖
2
(
1
4
)
Usi
n
g
(
9
)
an
d
(
1
4
)
,
w
e
g
et,
|
.
−
1
|
≤
‖
‖
2
‖
−
1
‖
2
|
.
−
1
|
(
1
5
)
B
y
(
3
)
,
w
e
h
av
e
=
−
+
−
1
.
‖
‖
2
=
−
1
+
−
1
‖
‖
2
(
1
6
)
w
e
h
av
e
0
0
≤
−
‖
0
‖
2
<
0
If
0
‡
0
; su
p
p
o
s
e
th
at
d
i
; i
=
1
,
2
,
…,
k
; a
r
e
all
d
escen
te
d
ir
ec
tio
n
s
,
t
h
at
is
<
0
B
y
(
1
6
)
;
w
e
g
et;
|
.
−
1
|
≤
−
‖
‖
2
‖
−
1
‖
2
.
−
1
(
1
7
)
T
h
at
is
;
‖
‖
2
‖
−
1
‖
2
.
−
1
≤
.
−
1
≤
−
‖
‖
2
‖
−
1
‖
2
.
−
1
(
1
8
)
Evaluation Warning : The document was created with Spire.PDF for Python.
I
SS
N
:
2
0
8
8
-
8708
I
n
t J
E
lec
&
C
o
m
p
E
n
g
,
Vo
l.
11
,
No
.
2
,
A
p
r
il 2
0
2
1
:
1
4
6
9
-
1475
1472
As s
h
o
w
n
i
n
(
1
7
)
an
d
(
1
8
)
d
e
d
u
ce
−
1
+
−
1
.
−
1
‖
−
1
‖
2
≤
.
‖
‖
2
≤
−
1
−
−
1
.
−
1
‖
−
1
‖
2
B
y
r
ep
ea
tin
g
t
h
is
p
r
o
ce
s
s
a
n
d
th
e
f
ac
t
0
0
=
−
‖
0
‖
2
,
w
e
h
a
v
e
−
∑
(
)
≤
−
1
=
0
‖
‖
2
≤
−
2
+
∑
(
)
−
1
=
0
(
1
9
)
As s
h
o
w
n
i
n
(
1
9
)
.
C
an
b
e
w
r
it
ten
as;
−
1
1
−
≤
‖
‖
2
≤
−
2
+
1
1
−
(
2
0
)
B
y
m
a
k
in
g
t
h
e
r
estrictio
n
∈
(
0
,
0
.
1
)
w
e
h
av
e
<
0
.
No
w
,
w
e
p
r
o
v
e
th
e
s
u
f
f
icie
n
t
d
escen
t c
o
n
d
itio
n
o
f
if
∈
(
0
,
1
)
Set
=
−
2
+
1
1
−
th
e
n
0
<
<
1
,
an
d
(
1
7
)
tu
r
n
s
o
u
t to
b
e
;
−
2
≤
‖
‖
2
≤
−
(
2
1
)
T
h
u
s
w
e
o
b
tain
≤
−
‖
‖
2
Or
C
=
−
2
+
1
1
−
.
3
.
2
.
Co
nv
er
g
ent
a
na
ly
s
is
L
e
m
m
a
1
L
et
th
e
s
tep
len
g
t
h
is
g
en
er
ated
b
y
Alg
o
r
it
h
m
1
.
T
h
en
,
u
n
d
er
th
e
ass
u
m
p
tio
n
s
H1
an
d
H2
,
th
er
e
is
a
p
o
s
itiv
e
co
n
s
tan
t C s
u
ch
t
h
at,
≥
‖
‖
2
‖
‖
2
(
2
2
)
P
r
o
o
f
:
Su
b
tr
ac
ti
n
g
f
r
o
m
b
o
th
s
id
es o
f
(
1
0
)
an
d
u
s
i
n
g
(
1
9
)
w
e
h
av
e
−
(
1
−
)
≤
(
+
1
−
)
≤
‖
‖
2
(
2
3
)
th
er
ef
o
r
e;
≥
−
(
1
−
)
‖
‖
2
(
2
4
)
w
it
h
(
1
0
)
w
e
o
b
tain
:
≥
−
(
1
−
)
‖
‖
2
‖
‖
2
(
2
5
)
T
h
is
in
eq
u
ali
t
y
m
ea
n
s
t
h
at
(
2
5
)
s
atis
f
ies
w
it
h
C
=
−
(
1
−
)
,
th
e
p
r
o
o
f
is
co
m
p
leted
.
T
h
e
n
ex
t le
m
m
a
is
k
n
o
w
n
a
s
Z
o
u
ten
d
ij
k
co
n
d
it
io
n
[
2
4
]
.
L
e
m
m
a
2
: Su
p
p
o
s
e
ass
u
m
p
t
io
n
1
h
o
ld
an
d
is
g
e
n
er
ated
b
y
A
l
g
o
r
ith
m
1
,
th
en
;
∑
‖
‖
4
‖
‖
2
∞
=
0
<
∞
(
2
6
)
P
r
o
o
f
:
Fro
m
(
1
0
)
f
o
r
an
y
w
e
h
av
e
;
(
)
−
(
+
)
≥
−
≥
−
(
1
−
)
(
)
2
‖
‖
2
(
2
7
)
Mo
r
eo
v
er
,
f
r
o
m
t
h
e
h
y
p
o
th
e
s
i
s
(
1
)
,
w
e
h
av
e
t
h
at
{
(
)
}
is
a
d
ec
r
ea
s
in
g
s
eq
u
e
n
ce
an
d
h
a
s
a
li
m
it i
n
,
w
h
ic
h
s
h
o
w
s
t
h
at
l
im
→
∞
(
+
1
)
<
+
∞
an
d
af
ter
(
2
8
)
w
e
h
av
e
;
Evaluation Warning : The document was created with Spire.PDF for Python.
I
n
t J
E
lec
&
C
o
m
p
E
n
g
I
SS
N:
2
0
8
8
-
8708
Glo
b
a
l c
o
n
ve
r
g
e
n
ce
o
f n
ew co
n
ju
g
a
te
g
r
a
d
ien
t m
eth
o
d
w
ith
in
ex
a
ct
lin
e
s
ea
r
ch
(
C
h
erg
u
i A
h
med
)
1473
+
∞
>
(
1
)
−
li
m
→
∞
(
+
1
)
≥
(
1
−
)
∑
(
)
2
‖
‖
2
(
2
8
)
T
h
en
∑
(
)
2
‖
‖
2
≤
+
∞
,
s
o
,
th
e
p
r
o
o
f
is
co
m
p
let
ed
.
T
h
eo
r
em
2
:
w
e
a
s
s
u
m
e
t
h
at
H1
,
H2
h
o
ld
,
an
d
t
h
e
s
eq
u
en
ce
{
}
is
g
e
n
er
ated
b
y
t
h
e
Alg
o
r
it
h
m
1
,
th
en
,
l
im
→
∞
‖
∇
(
)
‖
=
0
4.
NUM
E
RICAL
E
XP
E
R
I
M
E
NT
I
n
th
i
s
p
ar
t,
w
e
r
ep
o
r
t
n
u
m
er
i
ca
l
ex
p
er
i
m
e
n
ts
t
h
at
i
n
d
icate
th
e
ef
f
ic
ien
c
y
o
f
t
h
e
n
e
w
a
lg
o
r
ith
m
.
T
o
th
is
ai
m
,
w
e
i
m
p
le
m
e
n
t
t
h
e
n
e
w
al
g
o
r
ith
m
(
Alg
o
r
it
h
m
1
)
,
Fletch
er
a
n
d
R
ee
v
e
s
(
F
R
)
alg
o
r
ith
m
a
n
d
t
h
e
m
o
d
i
f
ied
Fle
tch
er
a
n
d
R
ee
v
e
s
(
FR
)
,
W
YL
[
1
0
]
,
DY
[
9
]
,
P
R
P
[
6
].
T
h
e
n
u
m
er
ical
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m
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u
r
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ased
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r
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r
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3
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iles
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o
r
r
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n
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in
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m
es
Evaluation Warning : The document was created with Spire.PDF for Python.
I
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N
:
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0
8
8
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8708
I
n
t J
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C
o
m
p
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g
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11
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2
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r
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h
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it c
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e
o
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s
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est
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ith
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est th
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s
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1
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x
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ed
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ck
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tio
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n
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1
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e
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A
m
et
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t
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g
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F
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m
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1
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o
m
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ar
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g
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h
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1
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é
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h
o
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R
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n
k
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t
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1
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2
5
5
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R
P
3
4
4
%
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3
3
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u
r
e
4
.
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er
f
o
r
m
a
n
ce
o
f
th
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n
u
m
b
er
o
f
f
u
n
ct
io
n
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lu
at
io
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s
Fig
u
r
e
5
.
P
er
f
o
r
m
a
n
ce
o
f
th
e
n
u
m
b
er
o
f
g
r
ad
ien
t
ev
al
u
atio
n
s
-
R
e
m
ar
k
2:
Fro
m
t
h
e
Fi
g
u
r
e
s
4
an
d
5
,
T
h
e
FR
A
m
et
h
o
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er
f
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r
m
s
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n
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er
m
eth
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y
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elec
ti
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r
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[
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.
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d
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s
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est
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ter
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n
s
an
d
t
h
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n
u
m
b
er
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f
iter
atio
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s
Evaluation Warning : The document was created with Spire.PDF for Python.
I
n
t J
E
lec
&
C
o
m
p
E
n
g
I
SS
N:
2
0
8
8
-
8708
Glo
b
a
l c
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n
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ce
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f n
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d
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in
ex
a
ct
lin
e
s
ea
r
ch
(
C
h
erg
u
i A
h
med
)
1475
5.
CO
NCLU
SI
O
N
in
t
h
is
p
ap
er
,
w
e
h
av
e
p
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ed
a
n
e
w
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o
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l
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s
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n
co
n
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tr
ai
n
ed
o
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ti
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p
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lem
.
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e
p
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e
g
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n
ce
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icie
n
t d
esce
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co
n
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itio
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m
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t th
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n
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h
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ate
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d
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ab
le
r
ea
d
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g
.
Su
g
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s
tio
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to
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co
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r
s
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n
t i
m
p
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m
en
t i
n
t
h
e
p
ap
er
.
RE
F
E
R
E
NC
E
S
[1
]
A
.
P
ietra
b
issa
a
n
d
L
.
Ricc
iard
i
C
e
lsi,
“
Disc
re
te
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T
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m
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l
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ish
Ro
u
t
in
g
Co
n
v
e
rg
in
g
to
t
h
e
W
a
rd
ro
p
E
q
u
il
i
b
ri
u
m
,
”
in
IEE
E
T
ra
n
sa
c
ti
o
n
s
o
n
Au
t
o
ma
t
ic Co
n
tro
l
,
v
o
l.
6
4
,
n
o
.
3
,
p
p
.
1
2
8
8
-
1
2
9
4
,
2
0
1
9
.
[2
]
A
.
Di
G
io
rg
io
,
e
t
a
l.
,
“
On
th
e
o
p
ti
m
iza
ti
o
n
o
f
e
n
e
rg
y
sto
ra
g
e
s
y
st
e
m
p
lac
e
m
e
n
t
f
o
r
p
ro
tec
ti
n
g
p
o
w
e
r
tran
sm
i
ss
io
n
g
rid
s
a
g
a
in
st
d
y
n
a
m
i
c
lo
a
d
a
lt
e
r
in
g
a
tt
a
c
k
s,
”
2
0
1
7
2
5
th
M
e
d
it
e
rr
a
n
e
a
n
Co
n
fer
e
n
c
e
o
n
Co
n
tro
l
a
n
d
A
u
to
m
a
ti
o
n
(
M
ED)
,
V
a
ll
e
tt
a
,
2
0
1
7
,
p
p
.
9
8
6
-
9
9
2
.
[3
]
L
.
R.
Ce
lsi,
e
t
a
l.
,
“
On
th
e
m
a
n
y
-
to
-
m
a
n
y
c
a
rp
o
o
li
n
g
p
r
o
b
lem
in
th
e
c
o
n
tex
t
o
f
m
u
lt
i
-
m
o
d
a
l
tri
p
p
lan
n
i
n
g
,
”
2
0
1
7
2
5
t
h
M
e
d
it
e
rr
a
n
e
a
n
C
o
n
fer
e
n
c
e
o
n
Co
n
tro
l
a
n
d
Au
t
o
ma
ti
o
n
(
M
ED)
,
V
a
ll
e
tt
a
,
2
0
1
7
,
p
p
.
3
0
3
-
3
0
9
.
[4
]
V
.
S
u
ra
c
i,
L
.
R.
Ce
lsi,
A
.
G
iu
se
p
p
i
a
n
d
A
.
Di
G
io
rg
io
,
“
A
d
istri
b
u
ted
w
a
rd
ro
p
c
o
n
tro
l
a
lg
o
rit
h
m
f
o
r
lo
a
d
b
a
lan
c
in
g
i
n
s
m
a
r
t
g
r
i
d
s
,
”
2
0
1
7
2
5
t
h
M
e
d
i
t
e
r
r
a
n
e
a
n
C
o
n
f
e
r
e
n
c
e
o
n
C
o
n
t
r
o
l
a
n
d
A
u
t
o
m
a
t
i
o
n
(
M
E
D)
,
V
a
l
l
e
t
t
a
,
2
0
1
7
,
p
p
.
7
6
1
-
767.
[5
]
R.
F
letc
h
e
r
a
n
d
C.
M
.
Re
e
v
e
s,
“
F
u
n
c
ti
o
n
m
in
im
iz
a
ti
o
n
b
y
c
o
n
ju
g
a
te
g
ra
d
ien
ts,
”
T
h
e
Co
m
p
u
ter
J
o
u
rn
a
l
,
v
o
l.
7
,
p
p
.
1
4
9
-
154
,
1
9
6
4
.
[6
]
B.
T
.
P
o
ly
a
k
,
“
T
h
e
c
o
n
ju
g
a
te
g
ra
d
ien
t
m
e
th
o
d
in
e
x
tre
m
e
m
p
ro
b
lem
s,
”
US
S
R
Co
mp
u
t
a
ti
o
n
a
l
M
a
th
e
ma
t
ics
a
n
d
M
a
th
e
ma
ti
c
a
l
Ph
y
sic
s
,
v
o
l.
9
,
n
o
.
4
,
p
p
.
9
4
-
1
1
2
,
1
9
6
9
.
[7
]
Da
i,
Y
., “
Co
n
v
e
rg
e
n
c
e
o
f
n
o
n
l
in
e
a
r
m
e
th
o
d
s,”
Go
u
rn
a
l
o
f
Co
m
p
u
t
a
ti
o
n
a
l
M
a
th
e
ma
ti
c
s
,
v
o
l.
1
9
,
p
p
.
539
-
5
4
9
,
2
0
0
1
.
[8
]
M.
R.
He
ste
n
e
s
a
n
d
E.
S
ti
e
f
e
l,
“
M
e
th
o
d
s
o
f
c
o
n
ju
g
a
te
g
ra
d
ien
ts
f
o
r
so
lv
i
n
g
li
n
e
a
r
sy
ste
m
s,
”
J
o
u
rn
a
l
o
f
Res
e
a
rc
h
o
f
th
e
Na
ti
o
n
a
l
B
u
re
a
u
o
f
S
t
a
n
d
a
r
d
s
,
v
o
l.
4
9
,
n
o
.
6
,
p
p
.
4
0
9
-
4
3
6
,
1
9
5
2
.
[9
]
Y.
H.
Da
i
a
n
d
Y.
Yu
a
n
,
“
A
n
o
n
li
n
e
a
r
c
o
n
ju
g
a
te
g
ra
d
ien
t
m
e
th
o
d
w
it
h
a
stro
n
g
g
lo
b
a
l
c
o
n
v
e
rg
e
n
c
e
p
ro
p
e
rty
,
”
S
IAM
J
o
u
rn
a
l
o
n
Op
ti
miza
ti
o
n
,
v
o
l.
1
0
,
n
o
.
1
,
p
p
.
1
7
7
-
1
8
2
,
1
9
9
9
.
[1
0
]
W
e
iet
,
e
t
a
l.
,
“
T
h
e
c
o
n
v
e
rg
e
n
c
e
p
ro
p
e
rti
e
s
o
f
so
m
e
c
o
n
ju
g
a
te
g
ra
d
ien
t
m
e
th
o
d
s,
”
A
p
p
li
e
d
M
a
th
e
ma
t
ics
a
n
d
Co
mp
u
t
a
ti
o
n
,
v
o
l.
1
8
3
,
p
p
.
1
3
4
1
-
1
3
5
0
,
2
0
0
6
.
[1
1
]
Da
i,
Y.
a
n
d
Y
u
a
n
,
Y.,
“
No
n
li
n
e
a
r
c
o
n
j
u
g
a
te g
ra
d
i
e
n
t
m
e
th
o
d
s,”
sh
a
n
g
h
a
i
sc
ien
t
ic
,
2
0
0
0
.
[1
2
]
G
il
b
e
rt
,
J.
C.
a
n
d
N
o
c
e
d
a
l,
J.,
“
G
lo
b
a
l
c
o
n
v
e
rg
e
n
c
e
p
ro
p
e
rti
e
s
o
f
c
o
n
ju
g
a
te
g
ra
d
ien
t
m
e
th
o
d
s
f
o
r
o
p
ti
m
iza
ti
o
n
,
”
S
IAM
J
o
u
rn
a
l
o
n
o
p
ti
miza
t
io
n
,
v
o
l.
2
,
n
o
.
1
,
p
p
.
21
-
42
,
1
9
9
2
.
[1
3
]
G
u
a
n
g
h
u
i,
L
.
,
Ji
y
e
,
H.,
a
n
d
Ho
n
g
x
ia,
Y.,
“
G
lo
b
a
l
c
o
n
v
e
rg
e
n
c
e
o
f
th
e
.
f
letc
h
e
r
-
re
e
v
e
s
a
l
g
o
rit
h
m
w
it
h
in
e
x
a
c
t
li
n
e
se
a
rc
h
,
”
Ap
p
li
e
d
M
a
t
h
e
ma
ti
c
s
-
A
J
o
u
r
n
a
l
o
f
Ch
i
n
e
se
Un
ive
rs
it
ies
,
v
o
l.
1
0
,
n
o
.
1
,
p
p
.
7
5
-
8
2
,
1
9
9
5
.
[1
4
]
Ha
g
e
r,
W
.
W
.
a
n
d
Zh
a
n
g
,
H.,
“
A
su
rv
e
y
o
f
n
o
n
li
n
e
a
r
c
o
n
j
u
g
a
te
g
r
a
d
ien
t
m
e
th
o
d
s,”
Pa
c
i
c
jo
u
r
n
a
l
o
f
Op
ti
miza
ti
o
n
,
v
o
l.
2
,
n
o
.
1
,
p
p
.
35
-
5
8
,
2
0
0
6
.
[1
5
]
L
iu
,
H.,
Wan
g
,
H.
,
Qia
n
,
X
.
,
a
n
d
Ra
o
,
F
.
,
“
A
c
o
n
ju
g
a
te
g
ra
d
ien
t
m
e
th
o
d
w
it
h
su
ff
icie
n
t
d
e
sc
e
n
t
p
ro
p
e
rty
,
”
Nu
me
ric
a
l
Al
g
o
rit
h
ms
,
v
o
l.
7
0
,
n
o
.
2
,
p
p
.
2
6
9
-
2
8
6
,
2
0
1
5
.
[1
6
]
L
iv
ieris,
I.
E.
,
T
a
m
p
a
k
a
s,
V
.
,
a
n
d
P
in
tela
s,
P
.
,
“
A
d
e
sc
e
n
t
h
y
b
rid
c
o
n
j
u
g
a
tes
g
ra
d
ien
t
m
e
th
o
d
b
a
s
e
d
o
n
t
h
e
m
e
m
o
r
y
les
sb
f
g
s u
p
d
a
te,”
Nu
me
ri
c
a
l
Al
g
o
rit
h
ms
,
v
o
l.
7
9
,
n
o
.
4
,
p
p
.
1
1
6
9
-
1
1
8
5
,
2
0
1
8
.
[1
7
]
Ya
o
,
S
.
,
He
,
D.,
a
n
d
S
h
i,
L
.
,
“
A
n
i
m
p
ro
v
e
d
p
e
rr
y
c
o
n
ju
g
a
te
g
r
a
d
ien
t
m
e
th
o
d
w
it
h
a
d
a
p
ti
v
e
p
a
r
a
m
e
ter
c
h
o
ice
,
”
Nu
me
ric
a
l
Al
g
o
rit
h
m
s,
v
o
l.
7
8
,
n
o
.
3
,
p
p
.
1
-
15
,
2
0
1
8
.
[1
8
]
Al
-
Ba
a
li
,
M
.
,
“
De
sc
e
n
t
p
ro
p
e
rt
y
a
n
d
g
lo
b
a
l
c
o
n
v
e
rg
e
n
c
e
o
f
th
e
.
F
letc
h
e
r
-
re
e
v
e
s
m
e
th
o
d
w
it
h
in
e
x
a
c
t
li
n
e
se
a
rc
h
,
”
IM
A
J
o
u
rn
a
l
o
f
N
u
me
ric
a
l
An
a
lys
is
,
v
o
l.
5
,
n
o
.
1
,
p
p
.
1
2
1
-
1
2
4
,
1
9
8
5
.
[1
9
]
Ha
g
e
r,
W
.
W
.
a
n
d
Zh
a
n
g
,
H.,
“
A
lg
o
rit
h
m
8
5
1
:
Cg
d
e
sc
e
n
t,
a
c
o
n
j
u
g
a
te
g
ra
d
ien
t
m
e
th
o
d
w
it
h
g
u
a
ra
n
tee
d
d
e
sc
e
n
t,
”
ACM
T
ra
n
sa
c
ti
o
n
s
o
n
M
a
th
e
ma
ti
c
a
l
S
o
ft
wa
re
(
T
OM
S
)
,
v
o
l.
3
2
,
n
o
.
1
,
p
p
.
1
1
3
-
1
3
7
,
2
0
0
6
.
[2
0
]
Zh
a
n
g
,
L
.
,
Zh
o
u
,
W
.
,
a
n
d
L
i,
D.,
“
G
lo
b
a
l
c
o
n
v
e
rg
e
n
c
e
o
f
a
m
o
d
i
fied
.
F
letc
h
e
r
-
re
e
v
e
s
c
o
n
ju
g
a
te
g
r
a
d
ien
t
m
e
th
o
d
w
it
h
A
r
m
ij
o
-
t
y
p
e
li
n
e
se
a
rc
h
,
”
Nu
me
risc
h
e
M
a
th
e
ma
ti
k
,
v
o
l.
1
0
4
,
n
o
.
4
,
p
p
.
5
6
1
-
5
7
2
,
2
0
0
6
[2
1
]
No
c
e
d
a
l,
J.
a
n
d
W
rig
h
t,
S
.
,
“
Nu
m
e
rica
l
o
p
ti
m
iza
ti
o
n
,
”
S
p
rin
g
e
r S
c
i
e
n
c
e
&
Bu
sin
e
ss
M
e
d
ia
,
2
0
0
6
.
[2
2
]
S
.
S
a
n
m
ti
a
s
a
n
d
E.
V
e
rc
h
e
r
E
.
,
“
A
g
e
n
e
ra
li
z
e
d
c
o
n
ju
g
a
te
g
ra
d
ien
t
a
lg
o
rit
h
m
,
”
J
o
u
rn
a
l
o
f
Op
t
imiza
t
io
n
T
h
e
o
ry
a
n
d
Ap
p
li
c
a
ti
o
n
s
,
v
o
l
.
9
8
,
p
p
.
4
8
9
-
5
0
2
,
1
9
9
8
.
[2
3
]
Z.
J.
S
h
i
,
“
No
n
li
n
e
a
r
c
o
n
j
u
g
a
te
g
ra
d
ien
t
m
e
th
o
d
w
it
h
e
x
a
c
t
li
n
e
se
a
rc
h
(in
Ch
in
e
se
),
”
Acta
M
a
t
h
e
ma
ti
c
a
S
i
n
ica
,
En
g
li
s
h
S
e
rie
s
,
v
o
l.
2
4
,
n
o
.
6
,
p
p
.
675
-
6
8
2
,
2
0
0
4
.
[2
4
]
G
.
Zo
u
ten
d
ij
k
,
“
No
n
li
n
e
a
r
p
ro
g
ra
m
m
in
g
c
o
m
p
u
tatio
n
a
l
m
e
th
o
d
s,
”
in
J
.
Ab
a
d
ie
(
Ed
.
),
In
teg
e
r
a
n
d
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