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Science
Vo
l.
24
,
No
.
2
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N
o
v
em
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e
r
2
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p
p
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0
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5
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ttp
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cs.ia
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M
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De
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a
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ti
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s,
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o
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ters
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c
ien
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s a
n
d
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a
th
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m
a
ti
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s,
M
o
su
l
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i
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e
rsity
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ra
q
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icle
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nfo
AB
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T
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r
ticle
his
to
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y:
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ed
Dec
6
,
2
0
2
0
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ev
is
ed
Sep
17
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0
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1
Acc
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ted
Sep
21
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2
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1
Th
e
u
se
o
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th
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h
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n
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o
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m
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y
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t
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ti
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iza
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ro
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th
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a
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r,
we
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e
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t
a
n
e
w
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m
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n
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m
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th
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lf
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li
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g
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a
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rit
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m
.
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,
b
a
se
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o
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ice
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le
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m
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to
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e
li
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se
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ro
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d
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o
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re
d
a
n
d
e
m
p
l
o
y
e
d
a
n
e
w
n
o
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m
o
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o
to
n
e
i
d
e
a
.
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e
re
a
fter
first,
a
n
u
p
d
a
te
d
fo
r
m
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la
is
e
x
h
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rted
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o
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e
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o
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rg
e
n
t
He
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ian
m
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n
d
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v
e
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h
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e
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th
e
se
c
a
n
t
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o
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it
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c
o
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d
,
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e
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li
sh
e
d
t
h
e
g
lo
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a
l
c
o
n
v
e
rg
e
n
c
e
p
ro
p
e
rti
e
s
o
f
th
e
a
lg
o
rit
h
m
u
n
d
e
r
so
m
e
m
il
d
c
o
n
d
i
ti
o
n
s
a
n
d
t
h
e
o
b
jec
ti
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e
fu
n
c
ti
o
n
is
n
o
t
c
o
n
v
e
x
it
y
h
y
p
o
th
e
sis.
A
p
r
o
m
isin
g
b
e
h
a
v
i
o
r
is ac
h
iev
e
d
a
n
d
t
h
e
n
u
m
e
rica
l
re
s
u
lt
s a
re
a
lso
re
p
o
rted
o
f
t
h
e
n
e
w a
lg
o
rit
h
m
.
K
ey
w
o
r
d
s
:
B
FGS alg
o
r
ith
m
g
lo
b
al
co
n
v
er
g
en
ce
p
r
o
p
e
r
ty
No
n
m
o
n
o
to
n
e
lin
e
s
ea
r
ch
Self
-
s
ca
lin
g
Un
co
n
s
tr
ain
ed
o
p
tim
izatio
n
T
h
is i
s
a
n
o
p
e
n
a
c
c
e
ss
a
rticle
u
n
d
e
r th
e
CC B
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SA
li
c
e
n
se
.
C
o
r
r
e
s
p
o
nd
ing
A
uth
o
r
:
Mu
n
a
M.
M.
Ali
Dep
ar
tm
en
t o
f
Ma
th
em
atics
C
o
lleg
e
o
f
C
o
m
p
u
ter
s
Scien
ce
s
an
d
Ma
th
em
atics
Mo
s
u
l U
n
iv
er
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ity
,
A
l
-
M
ajm
o
a
a
Stre
et,
Mo
s
u
l,
I
r
aq
E
m
ail: m
u
n
am
o
h
7
4
@
u
o
m
o
s
u
l.e
d
u
.
iq
1.
I
NT
RO
D
UCT
I
O
N
C
o
n
s
id
er
th
e
u
n
c
o
n
s
tr
ain
ed
o
p
tim
izatio
n
p
r
o
b
le
m
:
∈
(
)
(
1
)
w
h
er
e
:
|
→
is
a
co
n
tin
u
o
u
s
ly
d
if
f
e
r
en
tiab
le
f
u
n
ctio
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o
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e
p
r
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1
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n
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s
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alg
o
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it
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m
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at
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ates a
s
eq
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en
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o
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s
ac
co
r
d
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g
to
:
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1
=
+
(
2
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f
o
r
≥
0
,
wh
er
e
is
a
s
ea
r
ch
d
ir
ec
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n
,
>
0
is
s
tep
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th
an
d
0
is
g
iv
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n
th
e
in
itial
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o
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asic
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s
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th
ese
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o
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ith
m
s
ar
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ch
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o
s
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g
s
u
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ir
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a
n
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ize.
T
o
s
atis
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y
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e
d
escen
t
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n
d
itio
n
∇
(
)
<
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,
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ally
,
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n
o
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d
er
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r
it
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a
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f
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icien
t
r
ed
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ctio
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t
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alu
e
o
f
f
u
n
ctio
n
we
r
e
q
u
ir
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d
th
e
s
ea
r
ch
d
ir
ec
tio
n
an
d
is
s
p
ec
if
ied
,
th
er
e
ar
e
v
a
r
io
u
s
ex
am
p
les
f
o
r
p
r
o
ce
d
u
r
es
to
ch
o
o
s
e
th
e
s
ea
r
ch
d
ir
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tio
n
,
c
o
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g
ate
g
r
a
d
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n
t
(
C
G)
,
s
teep
est
d
escen
t
(
S
D)
,
New
to
n
,
q
u
asi
-
New
to
n
,
an
d
tr
u
s
t
-
r
eg
io
n
m
eth
o
d
s
s
ee
[
1
]
.
New
to
n
h
as
th
e
h
ig
h
est
r
ate
o
f
co
n
v
er
g
en
ce
an
d
th
e
d
ir
ec
tio
n
is
ac
co
u
n
ted
b
y
s
o
lv
in
g
th
e
s
y
s
tem
=
−
wh
er
e
=
∇
2
(
)
an
d
=
∇
(
)
.
Qu
asi
-
New
to
n
cr
iter
io
n
m
eth
o
d
s
co
n
v
e
n
tio
n
th
e
f
o
llo
win
g
s
ec
an
t
eq
u
atio
n
:
+
1
=
w
h
er
e
=
+
1
−
,
=
+
1
−
,
at
th
e
f
ir
s
t
i
ter
atio
n
,
0
is
an
ar
b
itra
r
y
n
o
n
s
in
g
u
lar
p
o
s
itiv
e
d
ef
in
ite
Evaluation Warning : The document was created with Spire.PDF for Python.
I
SS
N
:
2502
-
4
7
5
2
I
n
d
o
n
esian
J
E
lec
E
n
g
&
C
o
m
p
Sci,
Vo
l.
24
,
No
.
2
,
No
v
em
b
er
2
0
2
1
:
1
0
2
7
-
1
0
3
5
1028
m
atr
ix
an
d
+
1
is
an
ap
p
r
o
x
im
atio
n
o
f
.
T
h
e
m
o
s
t
ef
f
icien
t
o
f
Qu
as
-
New
to
n
m
eth
o
d
s
ar
e
p
e
r
h
ap
s
to
s
elf
-
s
ca
lin
g
B
FG
S
m
eth
o
d
wh
ich
was
u
p
d
ated
s
u
g
g
ested
b
y
[
2
]
,
[
3
]
an
d
th
is
m
eth
o
d
is
o
v
er
all
n
u
m
er
ical
co
m
p
u
tatio
n
t
h
an
th
e
o
t
h
er
m
eth
o
d
.
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h
e
m
atr
ix
+
1
in
th
e
s
elf
-
s
ca
lin
g
B
FG
S
m
eth
o
d
ca
n
b
e
u
p
d
ated
b
y
th
e
f
o
llo
win
g
f
o
r
m
u
la:
+
1
=
[
−
]
+
(
3
)
w
h
er
e:
μ
k
=
s
k
T
y
k
y
k
y
k
T
⁄
(
4
)
I
f
th
e
cu
r
v
atu
r
e
c
o
n
d
itio
n
>
0
h
o
ld
s
,
th
e
m
eth
o
d
o
f
s
elf
-
s
ca
lin
g
B
FGS
m
ain
tain
s
th
e
p
o
s
itiv
en
ess
o
f
th
e
m
atr
ices
{
}
.
Fo
r
th
is
r
ea
s
o
n
,
th
e
d
escen
t
d
ir
ec
tio
n
o
f
f
at
is
s
atis
f
y
in
th
e
d
ir
ec
tio
n
o
f
th
e
s
elf
-
s
ca
li
n
g
B
FGS
n
o
t
p
r
o
b
lem
if
is
p
o
s
itiv
e
d
ef
in
ite
o
r
n
o
t.
Ma
n
y
m
o
d
i
f
icatio
n
s
h
av
e
b
ee
n
p
r
o
p
o
s
ed
m
ad
e
to
a
f
f
licted
th
e
g
lo
b
a
l
co
n
v
er
g
en
ce
p
r
o
p
e
r
ty
o
f
th
e
(
B
r
o
y
d
en
-
Fletch
er
-
G
o
ld
f
a
r
b
-
Sh
an
n
o
)
B
FGS
m
eth
o
d
,
f
o
r
in
s
tan
ce
,
s
o
m
e
m
o
d
u
latio
n
s
i
n
th
e
cr
iter
io
n
B
FGS
m
eth
o
d
ar
e
m
ad
e,
an
d
s
u
b
m
itted
ed
a
m
o
d
i
f
ied
B
FGS
(
M
B
FGS
)
alg
o
r
ith
m
s
[
4
]
-
[
6
]
.
T
h
e
s
u
p
er
lin
ea
r
co
n
v
er
g
en
ce
an
d
th
e
g
lo
b
al
o
f
t
h
eir
m
eth
o
d
s
h
a
v
e
b
ee
n
p
r
o
v
e
d
u
n
d
er
ap
p
r
o
p
r
iate
co
n
d
itio
n
s
f
o
r
n
o
n
-
c
o
n
v
e
x
p
r
o
b
le
m
s
.
A
s
u
f
f
icien
t
r
ed
u
ctio
n
p
r
o
d
u
c
es
f
r
o
m
s
u
itab
le
lin
e
s
ea
r
ch
is
an
o
th
er
m
a
k
in
g
a
g
o
o
d
iter
ativ
e
p
r
o
ce
s
s
in
f
u
n
ctio
n
v
alu
e,
as we
s
ay
.
A
p
u
b
lic
s
itu
atio
n
to
ac
ce
p
t a
s
tep
len
g
th
m
en
tio
n
s
ed
Ar
m
ij
o
r
u
le
as
(
5
)
:
(
+
)
≤
+
(
5
)
a
n
d
th
e
lar
g
est m
em
b
er
in
{1
,
,
2
,
…
…
}
s
atis
f
y
in
g
(
4
)
s
u
ch
th
at
∈
(
0
,
1
)
∈
(
0
,
1
)
.
I
t
is
clea
r
th
at
d
en
o
tes
(
)
an
d
+
1
<
f
o
r
ev
er
y
d
escen
t
d
ir
ec
tio
n
,
an
d
ca
lled
m
o
n
o
to
n
e
lin
e
s
ea
r
ch
.
T
h
e
f
ir
s
t
n
o
n
-
m
o
n
o
to
n
e
lin
e
s
ea
r
ch
tech
n
iq
u
e
wer
e
p
r
o
p
o
s
ed
b
y
[
7
]
,
New
to
n
'
s
m
eth
o
d
u
s
in
g
th
e
Ar
m
ijo
co
n
d
itio
n
was d
ef
in
ed
b
y
(
6
)
:
(
+
)
≤
{
−
}
0
≤
≤
(
)
+
(
6
)
wh
er
e
0
≤
(
)
≤
min
{
(
+
1
)
+
1
,
}
,
N
is
a
n
o
n
-
n
eg
ativ
e
in
teg
er
co
n
s
tan
t,
Ma
n
y
k
in
d
s
o
f
r
esear
ch
er
s
,
f
o
r
e
x
am
p
le
[
8
]
-
[
1
2
]
.
A
n
o
n
-
m
o
n
o
to
n
e
s
ch
em
a
ca
n
p
r
o
m
o
te
o
f
f
in
d
in
g
a
g
lo
b
al
o
p
tim
u
m
an
d
also
d
ev
elo
p
e
d
a
s
p
ee
d
o
f
co
n
v
er
g
en
ce
.
On
e
o
f
th
e
ef
f
icien
t
n
o
n
-
mo
n
o
to
n
e
lin
e
s
ea
r
ch
m
eth
o
d
s
h
av
e
b
ee
n
p
r
o
p
o
s
ed
b
y
[
1
3
]
to
o
v
e
r
co
m
e
s
o
m
e
d
r
awb
ac
k
s
in
th
e
n
o
n
-
m
o
n
o
to
n
e
in
(
6
)
th
o
u
g
h
h
av
e
f
ea
tu
r
es
an
d
w
ell
wo
r
k
f
o
r
m
an
y
s
itu
atio
n
s
[
1
4
]
,
an
d
h
av
e
th
e
s
am
e
g
en
er
al
p
lan
n
er
w
h
ile
t
h
e
s
tatem
en
t
"m
a
x
"
is
s
u
b
s
titu
te
av
er
a
g
e
weig
h
ts
f
o
r
v
al
u
es o
f
f
u
n
ctio
n
with
s
eq
u
en
tial iter
atio
n
s
.
2.
M
O
DIFI
E
D
A
N
E
W
NO
N
-
M
O
NO
T
O
N
E
SE
L
F
-
SCA
L
I
NG
B
F
G
S M
E
T
H
O
D
A
n
o
n
-
m
o
n
o
to
n
e
B
FGS
m
eth
o
d
s
wer
e
p
r
o
p
o
s
ed
f
o
r
s
o
lv
in
g
(
1
)
in
[
1
5
]
-
[
1
7
]
.
T
h
ese
alg
o
r
i
th
m
s
wer
e
p
r
o
v
e
d
th
e
co
n
v
e
r
g
en
ce
an
al
y
s
is
u
n
d
er
th
e
co
n
v
ex
h
y
p
o
th
es
is
o
n
th
e
o
b
jectiv
e
f
u
n
ctio
n
.
I
n
th
is
wo
r
k
,
a
n
ew
non
-
m
o
n
o
t
o
n
e
m
o
d
i
f
ied
s
elf
-
s
ca
lin
g
B
F
GS
m
eth
o
d
is
in
s
e
r
ted
an
d
ev
id
en
ce
th
e
g
lo
b
al
co
n
v
er
g
en
ce
o
f
th
e
m
eth
o
d
with
o
u
t
co
n
v
ex
ity
ass
u
m
p
tio
n
.
T
h
is
wo
r
k
is
ar
r
an
g
e
d
as
f
o
llo
ws.
T
h
e
New
1
n
o
n
-
monot
o
n
e
p
r
o
p
o
s
ed
an
d
d
ef
in
e
d
in
lin
e
s
ea
r
ch
(
7
)
-
(
9
)
an
d
we
n
o
te
th
at
th
e
n
u
m
er
ical
r
esu
lts
o
f
th
e
New
1
n
o
n
-
m
o
n
o
t
o
n
e
lin
e
s
ea
r
ch
(
7
)
-
(
9
)
h
av
e
b
ee
n
m
o
r
e
ef
f
ec
tiv
e
th
an
th
e
[
1
8
]
.
T
h
e
New
2
m
eth
o
d
is
ex
p
r
ess
ed
in
th
is
p
ar
t.
Als
o
,
we
r
em
em
b
er
th
e
p
r
o
p
e
r
ties
co
n
v
er
g
en
ce
o
f
th
e
n
ew
al
g
o
r
ith
m
in
p
a
r
t
3
.
Nu
m
er
ical
e
x
p
er
ie
n
ce
s
s
h
o
w
th
at
th
e
n
ew
m
eth
o
d
is
v
er
y
f
av
o
r
ab
l
e
an
d
in
v
esti
g
ated
b
o
th
th
ea
tr
ically
an
d
n
u
m
er
ically
ag
ain
s
t
s
o
m
e
well
-
k
n
o
wn
alg
o
r
ith
m
s
.
I
n
th
e
last
p
ar
t,
s
o
m
e
co
n
clu
s
io
n
s
ar
e
lis
t.
No
w
we
ex
p
lain
th
e
n
ew
n
o
n
-
m
o
n
o
to
n
e
lin
e
s
ea
r
ch
m
eth
o
d
(
New1
)
wh
ich
is
d
escr
ib
ed
as f
o
llo
ws:
(
+
)
−
{
−
}
≤
−
‖
‖
2
0
≤
≤
(
)
(
7
)
w
h
er
e:
=
1
(
8
)
Evaluation Warning : The document was created with Spire.PDF for Python.
I
n
d
o
n
esian
J
E
lec
E
n
g
&
C
o
m
p
Sci
I
SS
N:
2502
-
4
7
5
2
Mo
d
ified
limited
-
mem
o
r
y
B
r
o
yd
en
-
F
letch
er
-
Go
ld
fa
r
b
-
S
h
a
n
n
o
a
lg
o
r
ith
m
fo
r
…
(
Mu
n
a
M.
M.
A
li
)
1029
=
−
1
−
1
−
1
−
1
(
9
)
1
=
0
.
0001
,
≥
1
,
wi
th
∈
(
0
,
1
)
T
wo
r
ea
s
o
n
s
m
ad
e
th
e
B
FGS
alg
o
r
ith
m
h
ad
im
p
o
r
tan
t
d
is
ad
v
an
tag
es
d
esp
ite
t
h
is
m
e
th
o
d
is
a
s
u
cc
ess
f
u
l a
lg
o
r
ith
m
f
o
r
u
n
c
o
n
s
tr
ain
ed
n
o
n
lin
ea
r
o
p
tim
izatio
n
.
On
ce
,
th
e
d
ir
ec
tio
n
s
o
f
th
e
m
eth
o
d
m
ay
n
o
t b
e
d
escen
t
esp
ec
ially
wh
en
>
0
is
n
o
t
s
atis
f
ied
an
d
ca
n
n
o
t
g
u
ar
an
t
ee
p
o
s
itiv
e
d
ef
in
iten
ess
o
f
th
e
m
atr
ix
.
Seco
n
d
,
in
g
en
er
al
is
s
u
ess
,
T
h
e
B
FGS
m
eth
o
d
m
ay
n
o
t
b
e
c
o
n
v
e
r
g
en
t
f
o
r
n
o
n
-
c
o
n
v
ex
o
b
jectiv
e
f
u
n
ctio
n
s
,
d
esp
ite
estab
lis
h
ed
s
u
p
er
lin
ea
r
co
n
v
er
g
en
ce
an
d
t
h
e
g
lo
b
al
f
o
r
c
o
n
v
e
x
p
r
o
b
lem
s
.
A
New2
non
-
m
o
n
o
to
n
e
m
o
d
if
ied
s
elf
-
s
ca
lin
g
B
FG
S a
lg
o
r
ith
m
is
p
r
esen
ted
g
u
ar
an
teein
g
th
e
p
o
s
itiv
e
d
ef
in
iten
ess
o
f
th
e
m
atr
ix
f
o
r
n
o
n
-
co
n
v
ex
o
b
jectiv
e
f
u
n
cti
o
n
s
.
I
n
th
is
p
ar
t,
th
e
n
ew
m
e
th
o
d
is
in
s
er
ted
af
ter
d
escr
ib
in
g
s
o
m
e
in
s
p
ir
ati
o
n
.
W
e
d
ef
in
e
d
th
e
m
o
d
if
ied
s
ec
an
t e
q
u
atio
n
s
:
+
1
=
∗
(
10
)
w
h
er
e
:
∗
≜
+
∗
(
11
)
an
d
d
ef
i
n
ed
b
y
th
r
ee
f
o
r
m
s
:
∗
(
1
)
=
2
‖
∗
‖
2
∗
(
12
)
∗
(
2
)
=
1
+
2
‖
∗
‖
2
∗
(
13
)
∗
(
3
)
=
‖
‖
+
ma
x
{
‖
∗
‖
2
∗
,
0
}
≥
0
(
14
)
w
h
er
e
is
a
p
o
s
it
iv
e
co
n
s
tan
t,
s
ee
[
1
9
]
,
[
2
0
]
.
T
h
en
we
h
av
e
r
ef
o
r
m
e
d
th
e
s
elf
-
s
ca
lin
g
B
F
GS
u
p
d
ate
f
o
r
m
u
la
b
ased
o
n
(
10
)
as f
o
llo
ws:
+
1
=
[
−
]
∗
+
∗
∗
∗
(
15
)
w
h
er
e
:
∗
=
∗
∗
∗
⁄
(
16
)
an
d
d
ef
i
n
ed
an
ef
f
icien
t a
lg
o
r
i
th
m
th
at
is
ca
lled
m
o
d
if
ie
d
s
elf
-
s
ca
lin
g
B
FGS.
I
t is cle
ar
th
at
(
17
).
‖
‖
‖
∗
‖
2
≥
∗
>
0
,
f
o
r
all
∈
(
17
)
T
h
is
p
r
o
p
er
ty
is
g
u
ar
an
tees
p
o
s
itiv
e
d
ef
in
iten
ess
o
f
th
e
m
atr
i
x
an
d
s
ep
ar
ate
o
n
th
e
co
n
v
ex
i
ty
o
f
f
,
as
s
u
ch
th
e
u
s
ed
lin
e
s
ea
r
ch
.
T
h
e
n
ew
MBF
GS
m
eth
o
d
c
o
m
b
in
ed
with
th
e
n
ew
n
o
n
-
m
o
n
o
t
o
n
e
lin
e
s
ea
r
ch
an
d
s
atis
f
ies
th
e
g
lo
b
al
co
n
v
er
g
en
ce
.
Fo
r
u
n
co
n
s
tr
ain
ed
o
p
tim
iz
atio
n
in
wh
ich
is
u
p
d
ated
in
[
2
1
]
,
p
r
o
p
o
s
ed
th
e
r
elat
io
n
:
+
1
=
−
+
̃
(
)
(
18
)
an
d
:
̃
=
2
(
−
+
1
+
+
1
)
(1
9
)
s
o
,
th
e
l
o
ca
l
s
u
p
er
lin
ea
r
co
n
v
er
g
en
ce
an
d
g
lo
b
al
p
r
o
p
er
ties
f
o
r
co
n
v
ex
o
b
jectiv
e
f
u
n
ctio
n
s
p
r
eser
v
es
in
th
is
al
g
o
r
ith
m
to
o
.
Evaluation Warning : The document was created with Spire.PDF for Python.
I
SS
N
:
2502
-
4
7
5
2
I
n
d
o
n
esian
J
E
lec
E
n
g
&
C
o
m
p
Sci,
Vo
l.
24
,
No
.
2
,
No
v
em
b
er
2
0
2
1
:
1
0
2
7
-
1
0
3
5
1030
No
w,
th
e
New2
alg
o
r
ith
m
is
s
u
g
g
ested
wh
ich
th
e
s
elf
-
s
ca
lin
g
B
FGS
m
eth
o
d
u
p
d
ate
f
o
r
m
u
la
u
s
in
g
∗
in
(
11
)
,
an
d
co
m
p
u
te
th
e
u
p
d
a
te
f
o
r
m
u
la
as f
o
llo
ws:
+
1
=
[
−
]
∗
+
∗
∗
∗
(
20
)
w
h
er
e
:
∗
=
∗
∗
∗
⁄
(
21
)
an
d
s
atis
f
ies th
e
s
ec
an
t c
o
n
d
itio
n
as f
o
llo
ws:
+
1
=
∗
∗
(
22
)
o
u
tlin
e
o
f
t
h
e
n
ew
n
o
n
-
m
o
n
o
to
n
e
s
elf
-
s
ca
lin
g
MBF
GS
d
esc
r
ib
ed
in
A
lg
o
r
ith
m
1.
Algorithm
1
.
New
-
self
-
scaling
BFGS (
new
-
non
-
monotone modified self
-
scaling
BFGS)
A start an initial point
0
∈
, a symmetric positive definite matrix
0
∈
∗
,
,
∈
(
0
,
1
)
.
Step
1
: set
1
=
0
.
0001
,
=
1
Step
2
: if
‖
‖
<
∈
,
Step
3
: compute search direction
by solving
=
−
Step
4:
set
=
−
1
−
1
−
1
−
1
where
the smallest positive integer and
satisfies (7),(8),
(9)
Step
5
: compute
+
1
=
+
Step
6
: compute
∗
in (
11
) and
∗
in (
21
). then
, update
in (
20
)
Step
7
: set
=
+
1
and go to step
1.
3.
CO
NVERG
E
NC
E
ANA
L
YS
I
S
Fo
r
th
e
g
en
er
al
n
o
n
lin
ea
r
o
b
j
ec
tiv
e
f
u
n
ctio
n
,
t
h
is
p
ar
t
is
to
ex
p
lain
an
d
p
r
o
v
e
th
e
p
r
o
p
er
ties
o
f
th
e
n
ew
alg
o
r
ith
m
.
An
d
th
e
f
o
llo
win
g
ass
u
m
p
tio
n
s
o
n
t
h
e
o
b
je
ctiv
e
f
u
n
ctio
n
(
f
)
.
3
.
1
.
Ass
um
ptio
n (
H
)
T
h
e
lev
el
s
et
=
{
:
∈
,
(
)
≤
(
1
)
}
is
b
o
u
n
d
ed
,
wh
er
e
1
is
th
e
s
tar
tin
g
p
o
in
t
.
I
n
a
n
eig
h
b
o
r
h
o
o
d
Ω
,
f
is
co
n
tin
u
o
u
s
ly
d
if
f
er
e
n
tiab
le
an
d
its
g
r
ad
i
en
t
g
is
L
ip
ch
itz
c
o
n
tin
u
o
u
s
l
y
,
n
am
ely
,
th
er
e
ex
is
ts
a
co
n
s
tan
t
≥
0
s
u
ch
th
at
‖
(
)
−
(
)
‖
≤
‖
−
‖
,
∀
,
∈
.
I
t
is
clea
r
th
at
f
r
o
m
th
e
ass
u
m
p
tio
n
(
H,
i)
,
th
e
r
e
ex
is
ts
a
p
o
s
itiv
e
co
n
s
tan
t D
s
u
ch
th
at
=
m
ax
{
‖
−
‖
∀
,
∈
}
.
3
.
2
.
So
m
e
re
la
t
ed
pro
pert
is
So
m
e
p
r
o
v
en
m
ath
e
m
atica
l
p
r
o
p
er
ties
to
c
o
m
p
letin
g
th
e
s
tab
ilit
y
s
tu
d
y
o
f
th
e
th
eo
r
etica
l
s
id
e
.
P
r
o
p
er
ty
(
1
)
.
L
et
{
}
is
th
e
s
eq
u
en
ce
g
en
e
r
ated
b
y
A
lg
o
r
ith
m
1
n
ew
-
n
o
n
-
m
o
n
o
to
n
e
s
elf
-
s
ca
lin
g
MBF
G
S,
th
en
{
}
is
a
n
o
n
-
in
cr
ea
s
in
g
s
eq
u
en
ce
an
d
f
o
r
all
∈
∪
{
0
}
,
{
}
⊂
(
0
)
.
Pro
o
f
: S
ee
[
2
2
]
.
Pr
o
p
er
ty
(
2
)
.
I
f
th
e
ass
u
m
p
tio
n
s
(
H,
i)
an
d
(
H,
ii)
ar
e
co
n
ten
ted
an
d
{
}
is
th
e
s
eq
u
en
ce
p
r
o
d
u
ce
d
b
y
th
e
n
ew
A
lg
o
r
ith
m
1
(
n
ew
-
n
o
n
-
s
el
f
-
s
ca
lin
g
MBF
GS)
.
I
f
‖
‖
≥
h
o
ld
s
f
o
r
all
∈
with
a
co
n
s
tan
t
>
0
th
en
th
e
r
e
ex
is
t
p
o
s
itiv
e
co
n
s
tan
ts
1
,
2
,
3
s
u
ch
th
at,
f
o
r
all
∈
,
th
e
in
eq
u
alities
:
‖
‖
≤
1
‖
‖
,
2
‖
‖
2
≤
2
≤
3
‖
‖
2
(
23
)
c
o
n
tr
ac
t f
o
r
f
u
lly
a
h
alf
o
f
th
e
i
n
d
ices
∈
{
1
,
2
,
…
…
,
}
.
Pro
o
f
:
T
o
p
r
o
v
e
th
at,
m
u
s
t o
f
f
er
th
at
th
er
e
s
u
b
s
is
t two
p
o
s
itiv
e
r
an
d
R
s
u
ch
th
at:
∗
‖
‖
2
≥
(
24
)
a
nd
‖
∗
‖
2
∗
≤
(
25
)
Fro
m
ass
u
m
p
tio
n
‖
‖
≥
an
d
f
r
o
m
(
1
7
)
we
h
av
e:
Evaluation Warning : The document was created with Spire.PDF for Python.
I
n
d
o
n
esian
J
E
lec
E
n
g
&
C
o
m
p
Sci
I
SS
N:
2502
-
4
7
5
2
Mo
d
ified
limited
-
mem
o
r
y
B
r
o
yd
en
-
F
letch
er
-
Go
ld
fa
r
b
-
S
h
a
n
n
o
a
lg
o
r
ith
m
fo
r
…
(
Mu
n
a
M.
M.
A
li
)
1031
∗
≥
‖
‖
‖
∗
‖
2
≥
‖
∗
‖
2
≥
⩪
‖
‖
2
(
26
)
s
o
:
∗
‖
‖
2
≥
,
wh
er
e
=
⩪
is
a
p
o
s
itiv
e
c
o
n
s
tan
t.
On
t
h
e
o
t
h
er
h
an
d
,
it
f
o
l
lo
ws
(
11
)
,
(
12
)
an
d
C
au
ch
y
-
Sch
war
tz
in
eq
u
ality
th
at:
‖
∗
‖
≤
‖
‖
+
‖
‖
(
‖
‖
+
‖
‖
‖
‖
)
a
n
d
f
r
o
m
ass
u
m
p
tio
n
s
(
H,
i
)
,
(
H,
ii)
an
d
th
e
r
elatio
n
in
co
r
o
llar
y
(
3
.
3
)
th
e
r
e
ex
is
ts
̂
>
0
s
u
ch
th
at
‖
‖
≤
̂
.
T
h
er
ef
o
r
e,
it c
an
b
e
s
ee
n
th
at:
‖
∗
‖
≤
‖
‖
(
+
̂
+
)
=
‖
‖
(
27
)
L
is
L
ip
ch
itz
co
n
s
tan
t
f
r
o
m
a
h
y
p
o
th
esis
(
H,
ii),
an
d
=
+
̂
+
.
T
h
e
r
elatio
n
(
26
)
al
o
n
g
with
(
27
)
f
o
r
all
∈
,
r
esu
lt:
‖
∗
‖
2
∗
≤
w
h
er
e:
R
=
2
.
Fro
m
(
24
)
,
(
25
)
,
an
d
th
eo
r
em
(
2
.
1
)
in
[
6
]
we
h
av
e
th
e
r
est o
f
th
e
p
r
o
o
f
.
Pro
p
er
ty
(
3
)
.
I
f
th
e
ass
u
m
p
ti
o
n
(
H,
i)
a
n
d
(
H,
ii)
ex
is
t
an
d
{
}
is
th
e
s
eq
u
en
ce
g
e
n
er
ated
b
y
th
e
New1
alg
o
r
ith
m
.
I
f
‖
‖
≥
h
o
ld
s
f
o
r
all
∈
with
a
co
n
s
tan
t
>
0
th
en
th
er
e
is
a
p
o
s
itiv
e
co
n
s
tan
t
̆
s
u
ch
th
at
>
́
f
o
r
all
k
b
elo
n
g
in
g
t
o
J
=
{
∈
ℎ
(
16
)
}
.
P
r
o
o
f
:
s
ee
[
2
2
]
.
Pro
p
e
r
ty
(
4
)
.
Su
p
p
o
s
e
th
at
th
e
ass
u
m
p
tio
n
(
H,
i)
an
d
(
H,
ii)
h
o
ld
,
th
en
:
∑
−
<
∞
∞
=
0
(2
8
)
Pro
o
f
:
Usi
n
g
(
7
)
,
(
8
)
,
(
9
)
we
h
av
e:
+
1
−
≤
=
−
(
‖
+
1
‖
2
+
[
1
+
‖
∗
‖
2
∗
]
(
+
1
)
2
∗
)
≤
0
(2
9
)
t
h
er
ef
o
r
e,
{
}
is
d
ec
r
ea
s
in
g
s
eq
u
en
ce
.
Sin
ce
f
is
b
o
u
n
d
e
d
b
elo
w,
th
er
e
e
x
is
ts
a
co
n
s
tan
t
̂
s
u
ch
th
at:
l
im
→
∞
=
̂
.
I
t f
o
llo
ws th
at:
∑
(
∞
=
0
−
+
1
)
=
l
im
→
∞
∑
(
−
+
1
)
=
l
im
→
∞
(
0
−
+
1
)
=
0
=
0
−
̂
Hen
ce
,
∑
(
∞
=
0
−
+
1
)
<
+
∞
.
3
.
3
.
T
heo
re
m
I
f
th
e
ass
u
m
p
tio
n
(
H,
i)
a
n
d
(
H,
ii)
ex
is
t
an
d
{
}
is
th
e
s
eq
u
en
ce
g
en
er
ated
b
y
t
h
e
New
A
lg
o
r
ith
m
1
(
s
elf
-
s
ca
lin
g
NB
FGS),
th
en
:
l
im
→
∞
‖
‖
=
0
.
(
30
)
Pro
o
f
:
I
f
we
ass
u
m
e
th
at
l
im
→
∞
‖
‖
≠
0
,
s
o
th
er
e
ex
is
ts
a
co
n
s
tan
t
>
0
s
u
ch
th
at
‖
‖
≥
.
Fo
r
all
k
s
u
f
f
icien
tly
,
s
in
ce
B
k
s
k
=
α
k
B
k
d
k
=
−
α
k
k
,
it
f
o
llo
ws
f
r
o
m
(
2
8
)
th
at
∑
B
k
s
k
‖
B
k
s
k
‖
∞
=
0
‖
‖
2
=
∑
1
B
k
s
k
∞
=
0
=
∑
(
−
∞
=
0
k
d
k
)
<
∞
.
‖
‖
≥
,
Fro
m
th
e
p
r
o
p
er
ty
(
3
)
d
e
f
in
itio
n
o
f
J
ar
e
h
o
ld
s
,
lead
s
u
s
to
:
∑
B
k
S
k
‖
B
k
S
k
‖
2
∞
=
0
‖
‖
2
≥
2
∑
∞
=
0
B
k
S
k
‖
B
k
S
k
‖
2
≥
2
∑
∞
∈
B
k
S
k
‖
B
k
S
k
‖
2
>
2
̅
∑
∞
∈
B
k
S
k
‖
B
k
S
k
‖
2
f
r
o
m
th
e
last
in
eq
u
ity
in
wh
ic
h
co
m
es
f
r
o
m
p
r
o
p
er
t
y
(
4
)
th
is
lead
s
to
:
Evaluation Warning : The document was created with Spire.PDF for Python.
I
SS
N
:
2502
-
4
7
5
2
I
n
d
o
n
esian
J
E
lec
E
n
g
&
C
o
m
p
Sci,
Vo
l.
24
,
No
.
2
,
No
v
em
b
er
2
0
2
1
:
1
0
2
7
-
1
0
3
5
1032
∑
∞
∈
B
k
S
k
‖
B
k
S
k
‖
2
<
∞
(
31
)
b
ec
au
s
e
th
e
s
et
J
i
s
in
f
in
ite,
it
is
lead
to
th
at
B
k
s
k
‖
B
k
s
k
‖
2
→
0
f
o
r
∈
.
T
h
is
im
m
ed
iately
co
n
tr
ad
icts
th
e
f
a
ct:
B
k
s
k
‖
B
k
s
k
‖
2
≥
2
‖
‖
2
1
2
‖
‖
2
=
2
1
2
th
at
is
in
(
31
).
4.
RE
SU
L
T
S AN
D
D
I
SCU
SS
I
O
N
T
h
e
m
a
i
n
w
o
r
k
o
f
t
h
i
s
s
e
c
t
i
o
n
i
s
t
o
c
o
m
p
a
r
e
t
h
e
n
u
m
e
r
i
c
a
l
e
x
p
e
r
i
m
e
n
t
s
o
f
t
h
e
N
e
w
1
non
-
m
o
n
o
t
o
n
e
m
o
d
i
f
i
e
d
M
B
F
G
S
a
l
g
o
r
i
t
h
m
w
i
t
h
t
h
e
(
M
B
F
G
S
-
X
G
)
a
l
g
o
r
i
t
h
m
p
r
o
p
o
s
e
d
b
y
[
2
3
]
.
W
e
p
r
e
s
e
n
t
a
n
e
w
a
l
g
o
r
i
t
h
m
i
n
w
h
i
c
h
t
h
e
n
e
w
n
o
n
-
m
o
n
o
t
o
n
e
l
i
n
e
s
e
a
r
c
h
t
o
a
p
p
r
o
x
i
m
a
t
e
c
o
m
p
a
r
i
s
o
n
i
s
n
a
m
e
d
(
New1
N
M
B
F
G
S
)
.
O
n
t
h
e
o
t
h
e
r
h
a
n
d
,
w
e
c
o
m
p
a
r
e
t
h
e
n
u
m
e
r
i
c
a
l
e
x
p
e
r
i
m
e
n
t
s
o
f
t
h
e
n
e
w
s
e
l
f
-
s
c
a
l
i
n
g
m
o
d
i
f
i
e
d
B
F
G
S
a
l
g
o
r
i
t
h
m
n
a
m
e
d
(
N
e
w
2
s
e
l
f
-
s
c
a
l
i
n
g
M
B
F
G
S
)
w
i
t
h
t
h
e
s
t
a
n
d
a
r
d
s
e
l
f
-
s
c
a
l
i
n
g
B
F
G
S
m
e
t
h
o
d
s
t
r
a
i
g
h
t
w
i
t
h
A
r
m
i
j
o
l
i
n
e
s
e
a
r
c
h
[
7
]
,
[
9
]
.
W
e
w
r
o
t
e
F
O
R
T
R
A
N
l
a
n
g
u
a
g
e
a
n
d
d
o
u
b
l
e
-
p
r
e
c
i
s
i
o
n
a
r
i
t
h
m
e
t
i
c
.
T
h
e
s
e
r
e
s
u
l
t
s
w
e
r
e
p
e
r
f
o
r
m
e
d
o
n
a
P
C
.
O
u
r
a
t
t
e
m
p
ts
w
e
r
e
p
e
r
f
o
r
m
e
d
o
n
s
e
t
o
f
(
5
0
)
n
o
n
l
i
n
e
a
r
u
n
c
o
n
s
t
r
a
i
n
e
d
p
r
o
b
l
e
m
s
t
h
a
t
h
a
v
e
a
s
e
c
o
n
d
d
e
r
i
v
a
t
i
v
e
a
v
a
i
l
a
b
l
e
,
a
n
d
t
h
e
e
x
p
e
r
i
e
n
c
e
p
r
o
b
l
e
m
s
a
r
e
c
o
n
t
r
i
b
u
t
e
d
i
n
C
U
T
E
[
2
4
]
,
[
2
5
]
.
W
e
co
n
s
id
er
ed
n
u
m
er
ical
ex
p
er
im
en
ts
with
s
ev
er
al
v
ar
ia
b
le
=
2
,
4
,
6
,
…
…
1000
,
All
th
ese
m
eth
o
d
s
ter
m
in
ate
wh
en
th
e
f
o
llo
win
g
s
to
p
p
in
g
cr
iter
io
n
is
m
et
‖
‖
∞
≤
10
−
6
.
Ou
r
ex
p
er
ien
ce
s
s
h
o
w
th
e
p
ar
am
e
ter
s
=
0
.
46
,
=
0
.
38
,
1
=
0
.
0001
,
h
av
e
th
e
b
est
co
n
clu
s
io
n
s
f
o
r
all
th
e
alg
o
r
ith
m
s
.
T
ab
les
1
an
d
2
co
m
p
ar
e
s
o
m
e
n
u
m
er
ical
e
x
p
er
im
en
ts
f
o
r
t
h
e
New1
,
New2
o
f
alg
o
r
ith
m
s
ag
ain
s
t
th
e
B
FGS
alg
o
r
ith
m
s
,
an
d
th
e
test
p
r
o
b
lem
s
with
d
if
f
er
en
t
d
im
e
n
s
io
n
s
,
=
2
,
4
,
…
1000
.
I
n
all
th
ese
tab
les
:
N
=
Dim
en
s
io
n
o
f
th
e
p
r
o
b
le
m
,
NOI
=
n
u
m
b
er
o
f
iter
atio
n
s
,
NOF
=
Nu
m
b
er
o
f
f
u
n
ctio
n
s
,
C
PU
=
T
o
tal
tim
e
r
eq
u
ir
e
d
to
co
m
p
lete
th
e
ev
alu
atio
n
p
r
o
ce
s
s
f
o
r
ea
c
h
test
p
r
o
b
lem
.
Fig
u
r
e
s
1
to
4
c
o
m
p
a
r
e
o
f
th
e
New1
m
eth
o
d
a
g
ain
s
t
MBF
G
S
-
XG
m
eth
o
d
d
u
e
t
o
NOI
a
n
d
it’s
clea
r
th
at
New1
h
av
e
m
o
r
e
th
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9
Evaluation Warning : The document was created with Spire.PDF for Python.
I
n
d
o
n
esian
J
E
lec
E
n
g
&
C
o
m
p
Sci
I
SS
N:
2502
-
4
7
5
2
Mo
d
ified
limited
-
mem
o
r
y
B
r
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en
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r
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h
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ith
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(
Mu
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M.
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li
)
1033
T
ab
le
2
.
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o
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p
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r
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o
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o
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th
e
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ew1
m
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t M
B
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d
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ain
s
t
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elf
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lin
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eth
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with
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,
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Fig
u
r
e
1
.
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f
o
r
m
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e
d
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e
to
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C
P
U
Fig
u
r
e
2
.
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f
o
r
m
an
c
e
d
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e
to
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P
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Fig
u
r
e
3
.
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f
o
r
m
an
c
e
d
u
e
to
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P
U
Fig
u
r
e
4
.
Per
f
o
r
m
an
c
e
d
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e
to
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P
U
Evaluation Warning : The document was created with Spire.PDF for Python.
I
SS
N
:
2502
-
4
7
5
2
I
n
d
o
n
esian
J
E
lec
E
n
g
&
C
o
m
p
Sci,
Vo
l.
24
,
No
.
2
,
No
v
em
b
er
2
0
2
1
:
1
0
2
7
-
1
0
3
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1034
Fig
u
r
e
5
.
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r
m
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P
U
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r
e
6
.
Per
f
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r
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an
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e
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to
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P
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5.
CO
NCLU
SI
O
N
I
n
th
is
p
ap
er
,
we
h
av
e
p
r
o
p
o
s
ed
a
n
ew
n
o
n
-
m
o
n
o
t
o
n
e
B
FGS
alg
o
r
ith
m
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d
co
m
b
in
ed
it
with
a
n
ew
m
o
d
if
ied
s
elf
-
s
ca
lin
g
B
FGS
u
p
d
ate
to
a
s
ac
r
if
icial
Hess
ian
m
atr
ix
with
a
k
n
o
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e
s
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r
ch
p
lan
n
in
g
f
o
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o
n
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co
n
v
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o
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tim
izatio
n
p
r
o
b
lem
s
.
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t
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ith
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m
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o
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ith
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etitiv
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eth
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o
b
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m
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r
ical
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co
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p
ar
ed
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elf
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lin
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alg
o
r
ith
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ter
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th
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licts
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th
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r
k
.
RE
F
E
R
E
NC
E
S
[1
]
P.
I.
To
i
n
t,
“
An
a
ss
e
ss
m
e
n
t
o
f
n
o
n
-
m
o
n
o
t
o
n
e
li
n
e
a
r
se
a
rc
h
tec
h
n
iq
u
e
fo
r
u
n
c
o
n
stra
i
n
e
d
o
p
ti
m
iza
ti
o
n
,
”
S
I
A
M
J
o
u
rn
a
l
o
n
S
c
ien
ti
fi
c
C
o
mp
u
ti
n
g
,
v
o
l.
1
7
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o
.
3
,
p
p
.
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3
9
,
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9
9
6
,
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o
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0
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1
3
7
/S
1
0
6
4
8
2
7
5
9
4
2
7
0
2
1
X.
[2
]
S
.
S
.
Ore
n
,
“
S
e
lf
-
S
c
a
li
n
g
Va
riab
le
M
e
tri
c
(S
S
V
M
)
Alg
o
rit
h
m
s.
P
a
rt
II:
Im
p
lem
e
n
tatio
n
a
n
d
Ex
p
e
rime
n
ts
,
”
M
a
n
a
g
e
me
n
t
S
c
ien
c
e
,
v
o
l
.
2
0
,
n
o
.
5
,
p
p
.
8
6
2
-
8
7
4
,
1
9
7
4
.
[3
]
S
.
S
.
Ore
n
a
n
d
D
.
G
.
Lu
n
e
rb
e
rg
e
r,
“
S
e
lf
-
S
c
a
li
n
g
Va
riab
le
M
e
tri
c
a
lg
o
rit
h
m
.
P
a
rt
I
:
Crit
e
ria
a
n
d
S
u
fficie
n
t
Co
n
d
it
io
n
s f
o
r
S
c
a
li
n
g
a
Clas
s o
f
Alg
o
rit
h
m
s
,
”
M
a
n
a
g
e
me
n
t
S
c
ien
c
e
,
v
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l
.
2
0
,
n
o
.
5
,
p
p
.
8
4
5
-
8
6
2
,
1
9
7
4
.
[4
]
D.
-
H
.
Li
a
n
d
M
.
F
u
k
u
sh
ima
,
“
A
m
o
d
ifi
e
d
BF
G
S
m
e
th
o
d
a
n
d
it
s
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lo
b
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l
c
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n
v
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rg
e
n
c
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in
n
o
n
-
c
o
n
v
e
x
m
in
imiz
a
ti
o
n
,
”
J
o
u
rn
a
l
o
f
Co
m
p
u
t
a
ti
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n
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l
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n
d
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d
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s
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3
7
7
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4
2
7
(
0
0
)
0
0
5
4
0
-
9
.
[5
]
L.
Zh
a
n
g
,
a
n
d
J.
Li
,
“
A
n
e
w
g
lo
b
a
li
z
a
ti
o
n
tec
h
n
i
q
u
e
fo
r
n
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n
li
n
e
a
r
c
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ju
g
a
te
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ra
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ie
n
t
m
e
th
o
d
s
fo
r
n
o
c
o
n
v
e
x
m
in
imiz
a
ti
o
n
,
”
A
p
p
l
ied
M
a
t
h
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ma
ti
c
s
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n
d
c
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mp
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ti
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n
,
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l
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7
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p
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6
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.
a
m
c
.
2
0
1
1
.
0
5
.
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3
2
.
[6
]
M
.
J.
D
.
P
o
we
ll
,
“
S
o
m
e
g
l
o
b
a
l
c
o
n
v
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rg
e
n
c
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p
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p
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s
o
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a
v
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a
b
le
m
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tri
c
a
lg
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ri
th
m
f
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r
m
in
imiz
a
ti
o
n
wit
h
o
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t
e
x
a
c
t
li
n
e
se
a
rc
h
e
s,”
No
n
li
n
e
a
r
p
ro
g
ra
mm
i
n
g
,
S
l
a
m
-
AM
S
Pr
o
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in
g
s
,
v
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l
.
9
,
p
p
.
5
3
-
7
2
,
1
9
7
6
.
[7
]
J.
J.
M
o
re
,
B.
S
.
G
a
rb
o
w,
a
nd
K.
E.
Hill
stro
m
,
“
Tes
ti
n
g
Un
c
o
n
str
a
in
e
d
o
p
ti
m
iza
ti
o
n
s
o
ftwa
re
,
”
ACM
T
ra
n
sa
c
ti
o
n
s
o
n
M
a
t
h
e
ma
ti
c
a
l
S
o
ft
w
a
re
,
v
o
l.
7
,
n
o
.
1
,
p
p
.
1
7
-
4
1
,
1
9
8
1
,
d
o
i
:
1
0
.
1
1
4
5
/
3
5
5
9
3
4
.
3
5
5
9
3
6
.
[8
]
G
.
Li
u
,
J.
Ha
n
,
a
n
d
D.
S
u
n
,
“
G
lo
b
a
l
c
o
n
v
e
rg
e
c
e
o
f
t
h
e
b
f
g
s
a
lg
o
ri
th
m
with
n
o
n
m
o
n
o
t
o
n
e
li
n
e
se
a
rc
h
∗
th
is
w
o
rk
is
su
p
p
o
rted
b
y
n
a
ti
o
n
a
l
n
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tu
ra
l
sc
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e
n
c
e
fo
u
n
d
a
ti
o
n
$
e
f,
”
A
J
o
u
r
n
a
l
o
f
M
a
t
h
e
ma
ti
c
a
l
Pro
g
ra
mm
in
g
a
n
d
O
p
e
ra
ti
o
n
s
Res
e
a
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h
,
v
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l.
3
4
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o
.
2
,
p
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1
4
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5
9
,
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9
9
5
,
d
o
i:
1
0
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1
0
8
0
/
0
2
3
3
1
9
3
9
5
0
8
8
4
4
1
0
1
.
[9
]
L.
Li
u
,
S
.
Ya
o
,
a
n
d
Z.
Wei,
“
Th
e
g
lo
b
a
l
a
n
d
s
u
p
e
rli
n
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a
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c
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w n
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m
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e
M
B
F
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S
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lg
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rit
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m
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n
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x
o
b
jec
ti
v
e
fu
n
c
ti
o
n
,
”
J
o
u
rn
a
l
o
f
C
o
mp
u
ta
ti
o
n
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l
a
n
d
Ap
p
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d
M
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th
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t
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,
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l.
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o
.
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,
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j.
c
a
m
.
2
0
0
7
.
0
8
.
0
1
7
.
[1
0
]
Y.
Xio
,
H.
S
u
n
,
a
n
d
Z.
W
a
n
g
,
“
A
g
lo
b
a
l
ly
c
o
n
v
e
r
g
e
n
t
B
F
G
S
m
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th
o
d
wit
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-
m
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o
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li
n
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se
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rc
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fo
r
n
o
n
-
c
o
n
v
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x
m
in
imiz
a
ti
o
n
,
”
J
o
u
r
n
a
l
o
f
Co
m
p
u
t
a
ti
o
n
a
l
a
n
d
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p
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d
M
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ma
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s
,
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2
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.
c
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.
2
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.
1
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.
0
6
5
.
Evaluation Warning : The document was created with Spire.PDF for Python.
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1035
[1
1
]
H.
C.
Zh
a
n
g
a
n
d
W.
W.
Ha
g
e
r
,
“
A
n
o
n
-
m
o
n
o
to
n
e
l
in
e
se
a
rc
h
tec
h
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iq
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e
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d
it
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p
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to
u
n
c
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n
stra
i
n
e
d
o
p
ti
m
iza
ti
o
n
,
”
S
I
AM
J
o
u
rn
a
l
o
n
Op
ti
miz
a
ti
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n
,
v
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l
.
1
4
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6
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.
[1
2
]
D.
H.
Li
a
n
d
M
.
F
u
k
u
s
h
ima
,
“
On
G
lo
b
a
l
c
o
n
v
e
rg
e
n
c
e
o
f
th
e
BF
G
S
m
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th
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d
fo
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n
-
co
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n
c
o
n
stra
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n
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d
o
p
ti
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iza
ti
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n
p
ro
b
lem
s,”
S
IA
M
J
o
u
rn
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O
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5
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6
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3
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3
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2
.
[1
3
]
Y.
H.
Da
i,
“
On
t
h
e
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o
n
-
m
o
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o
to
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e
li
n
e
se
a
rc
h
,
”
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o
u
rn
a
l
o
f
Op
ti
mi
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T
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3
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5
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9
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3
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.
[1
4
]
I.
Bo
n
g
a
rtz,
A.
R.
C
o
n
n
,
N.
I.
M
.
G
lo
u
d
,
a
n
d
P
.
L.
T
o
in
t
,
“
CUTE:
Co
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stra
in
e
d
a
n
d
Un
c
o
n
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d
Tes
ti
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g
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v
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o
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n
t
,
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T
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re
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4
3
.
[1
5
]
W.
F
.
M
a
sc
a
re
n
h
a
s,
“
Th
e
B
F
GS
m
e
th
o
d
wit
h
e
x
a
c
t
l
in
e
se
a
rc
h
e
s
fa
il
s
fo
r
n
o
n
-
c
o
n
v
e
x
o
b
jec
t
iv
e
fu
n
c
ti
o
n
s,”
M
a
t
h
e
ma
ti
c
a
l
Pro
g
ra
mm
in
g
,
v
o
l
.
9
9
,
p
p
.
4
9
-
6
1
,
2
0
0
4
,
d
o
i
:
1
0
.
1
0
0
7
/s1
0
1
0
7
-
0
0
3
-
0
4
2
1
-
7
.
[1
6
]
M
.
M
il
a
d
i
n
o
v
ic,
P
.
S
tan
imir
o
v
ic,
a
n
d
S
.
M
il
j
k
o
v
ic
,
“
S
c
a
lar
c
o
rre
c
ti
o
n
m
e
th
o
d
f
o
r
s
o
lv
in
g
lar
g
e
sc
a
le
u
n
c
o
n
stra
i
n
e
d
M
in
imiz
a
ti
o
n
P
ro
b
lem
s,”
J
o
u
rn
a
l
o
f
O
p
ti
miza
ti
o
n
T
h
e
o
ry
a
n
d
Ap
p
li
c
a
t
io
n
s
,
v
o
l
.
1
5
1
,
p
p
.
3
0
4
-
3
2
0
,
2
0
1
1
,
d
o
i:
1
0
.
1
0
0
7
/s1
0
9
5
7
-
0
1
1
-
9
8
6
4
-
9
.
[1
7
]
G
.
E.
M
a
n
o
u
ss
a
k
is,
D.
G
.
S
o
ti
r
o
p
o
u
lu
s,
C.
A.
Bo
tsa
ris,
a
n
d
T.
N
.
G
ra
p
sa
,
“
A
n
o
n
-
m
o
n
o
to
n
e
C
o
n
ic
M
e
th
o
d
f
o
r
Un
c
o
n
stra
in
e
d
Op
t
imiz
a
ti
o
n
,
”
in
Pro
c
e
e
d
in
g
s
o
f
4
th
GR
ACM
,
Co
n
g
re
ss
o
n
C
o
mp
u
ta
t
io
n
a
l
M
e
c
h
a
n
ics
,
2
0
0
2
,
p
p
.
2
7
-
2
9
.
[1
8
]
J.
No
c
e
d
a
l
a
n
d
S
.
J.
Wr
i
g
h
t
,
“
Nu
m
e
rica
l
Op
ti
m
iza
ti
o
n
,
”
S
p
rin
g
e
r
,
2
0
0
6
,
d
o
i:
1
0
.
1
0
0
7
/
9
7
8
-
0
-
3
8
7
-
4
0
0
6
5
-
5
.
[1
9
]
M
.
A1
-
Ba
ll
i
a
n
d
H.
K
h
a
lfan
,
“
A
c
o
m
b
in
e
d
c
las
s
o
f
se
lf
-
s
c
a
li
n
g
a
n
d
m
o
d
ifi
e
d
Qu
a
si
-
Ne
wto
n
m
e
th
o
d
s,”
Co
mp
u
t
a
ti
o
n
a
l
O
p
ti
miza
ti
o
n
a
n
d
Ap
p
li
c
a
ti
o
n
s
,
v
o
l
.
5
2
,
n
o
.
2
,
p
p
.
3
9
3
-
4
0
8
,
2
0
1
2
,
d
o
i
:
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0
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1
0
0
7
/s
1
0
5
8
9
-
0
1
1
-
9
4
1
5
-
1
.
[2
0
]
K.
Am
in
i,
S
.
Ba
h
ra
m
i
a
n
d
S
.
Am
iri
,
“
A
N
o
n
-
m
o
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o
to
n
e
M
o
d
if
ied
BF
G
S
Alg
o
rit
h
m
f
o
r
N
o
n
-
c
o
n
v
e
x
Un
c
o
n
stra
in
e
d
Op
ti
m
iza
ti
o
n
p
ro
b
lem
s,”
Fi
l
o
ma
t
,
v
o
l
.
3
0
,
n
o
.
5
,
p
p
.
1
2
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3
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o
i:
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0
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2
2
9
8
/
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IL1
6
0
5
2
8
3
A
.
[
2
1
]
G
.
C
h
a
o
a
n
d
Z
.
D
e
t
o
n
g
,
“
A
non
-
m
o
n
o
t
o
n
e
l
i
n
e
s
e
a
r
c
h
f
i
l
t
e
r
m
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h
o
d
w
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t
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d
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c
e
d
H
e
s
s
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a
n
u
p
d
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t
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g
f
o
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l
i
n
e
a
r
o
p
t
i
m
i
z
a
t
i
o
n
,
”
J
o
u
r
n
a
l
o
f
S
y
s
t
e
m
s
S
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i
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n
c
e
a
n
d
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o
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6
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p
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5
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4
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5
5
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d
o
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:
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0
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1
0
0
7
/
s
1
1
4
2
4
-
012
-
0036
-
2.
[2
2
]
Y.
Z.
Yu
a
n
,
“
A
m
o
d
ifi
e
d
B
F
G
S
a
lg
o
rit
h
m
fo
r
u
n
c
o
n
stra
i
n
e
d
o
p
t
i
m
iza
ti
o
n
,
”
IM
A
J
o
u
r
n
a
l
o
f
Nu
me
ric
a
l
An
a
lys
is
,
v
o
l.
1
1
,
n
o
.
3
,
p
p
.
3
2
5
-
3
3
2
,
1
9
9
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,
d
o
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1
0
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1
0
9
3
/i
m
a
n
u
m
/
1
1
.
3
.
3
2
5
.
[2
3
]
J.
Zh
a
n
g
,
Y.
Xia
o
,
a
n
d
Z.
Wei,
“
No
n
li
n
e
a
r
Co
n
ju
g
a
te
g
ra
d
ien
t
m
e
th
o
d
s
wit
h
su
fficie
n
t
d
e
sc
e
n
t
c
o
n
d
it
io
n
fo
r
larg
e
-
sc
a
le
u
n
c
o
n
stra
in
e
d
o
p
ti
m
iza
ti
o
n
,
”
M
a
th
e
m
a
ti
c
a
l
Pr
o
b
lem
s
i
n
En
g
in
e
e
rin
g
,
v
o
l.
2
0
0
9
,
p
p
.
1
-
1
6
,
2
0
0
9
,
d
o
i:
1
0
.
1
1
5
5
/2
0
0
9
/
2
4
3
2
9
0
.
[2
4
]
R.
Z.
Al
-
Ka
wa
z
,
A.
Y.
A
l
Ba
y
a
ti
,
a
n
d
M
.
Ja
m
e
e
l,
“
In
tera
c
ti
o
n
b
e
twe
e
n
u
n
-
u
p
d
a
ted
F
R
-
CG
a
lg
o
rit
h
m
s
wit
h
a
n
o
p
ti
m
a
l
Cu
c
k
o
o
a
l
g
o
ri
th
m
,
”
I
n
d
o
n
e
sia
n
J
o
u
r
n
a
l
o
f
El
e
c
trica
l
En
g
i
n
e
e
rin
g
a
n
d
Co
mp
u
ter
S
c
i
e
n
c
e
(IJ
EE
CS
)
,
v
o
l.
1
9
,
n
o
.
3
,
p
p
.
1
4
9
7
-
1
5
0
4
,
2
0
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0
,
d
o
i:
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0
.
1
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5
9
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/
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e
c
s.v
1
9
.
i3
.
p
p
1
4
9
7
-
1
5
0
4
.
[2
5
]
A.
Y.
Al
-
Ba
y
a
ti
a
n
d
M
.
S.
Ja
m
e
e
l
,
“
Ne
w
S
c
a
led
P
ro
p
o
se
d
F
o
rm
u
las
fo
r
C
o
n
j
u
g
a
te
G
ra
d
ien
t
M
e
th
o
d
s
i
n
Un
c
o
n
stra
in
e
d
Op
ti
m
iza
ti
o
n
,
”
AL
-
Ra
fi
d
a
i
n
J
o
u
rn
a
l
o
f
Co
m
p
u
te
r
S
c
ien
c
e
s
a
n
d
M
a
th
e
ma
ti
c
s,
v
o
l.
1
1
,
n
o
.
2
,
p
p
.
2
5
-
4
6
,
2
0
1
4
,
d
o
i:
1
0
.
3
3
8
9
9
/cs
m
j.
2
0
1
4
.
1
6
3
7
4
8
.
[2
6
]
S
.
Hu
a
n
g
a
n
d
Z
.
Wan
,
“
A
n
e
w n
o
n
m
o
n
o
to
n
e
sp
e
c
tral
re
sid
u
a
l
m
e
th
o
d
f
o
r
n
o
n
sm
o
o
th
n
o
n
li
n
e
a
r
e
q
u
a
t
io
n
s
,
”
J
o
u
rn
a
l
o
f
c
o
m
p
u
t
a
ti
o
n
a
n
d
a
p
p
li
e
d
m
a
th
e
ma
ti
c
s
,
v
o
l
.
3
1
3
,
n
o
.
C,
p
p
.
8
2
-
1
0
1
,
2
0
1
7
,
d
o
i:
1
0
.
1
0
1
6
/j
.
c
a
m
.
2
0
1
6
.
0
9
.
0
1
4
.
[2
7
]
A.
Z.
M
.
S
o
fi
,
M
.
M
a
m
a
t,
I.
M
o
h
d
,
a
n
d
B.
S
.
P
u
tra,
“
An
Im
p
ro
v
e
d
BF
G
S
S
e
a
rc
h
Dire
c
ti
o
n
u
sin
g
E
x
a
c
t
Li
n
e
S
e
a
rc
h
fo
r
S
o
lv
in
g
Un
c
o
n
stra
i
n
e
d
O
p
ti
m
iza
ti
o
n
P
ro
b
lem
s,
Ap
p
li
e
d
M
a
t
h
e
ma
ti
c
a
l
S
c
ien
c
e
s
,
v
o
l
7
,
n
o
.
2
,
p
p
.
7
3
-
8
5
,
2
0
1
3
,
d
o
i:
1
0
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1
2
9
8
8
/A
M
S
.
2
0
1
3
.
1
3
0
0
7
.
[2
8
]
A
.
Y
.
Al
-
Ba
y
a
ti
a
n
d
M
.
M
.
M
.
Ali,
“
Ne
w
m
u
lt
i
-
ste
p
th
re
e
-
term
c
o
n
ju
g
a
te
g
ra
d
ien
t
a
l
g
o
rit
h
m
s
with
in
e
x
a
c
t
li
n
e
se
a
rc
h
s,”
In
d
o
n
e
sia
n
J
o
u
r
n
a
l
o
f
El
e
c
trica
l
En
g
i
n
e
e
rin
g
a
n
d
Co
mp
u
ter
S
c
ien
c
e
(IJ
EE
C
S
)
,
v
o
l.
1
9
,
n
o
.
3
,
p
p
.
1
5
6
4
-
1
5
7
3
,
2
0
2
0
,
d
o
i:
1
0
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1
1
5
9
1
/i
jee
c
s.v
1
9
.
i3
.
p
p
1
5
6
4
-
1
5
7
3
.
B
I
O
G
RAP
H
Y
O
F
AUTHO
R
Mu
n
a
M.
M.
Ali
,
Tea
c
h
i
n
g
i
n
th
e
De
p
a
rtme
n
t
o
f
M
a
t
h
e
m
a
ti
c
s,
Co
ll
e
g
e
o
f
Co
m
p
u
ters
S
c
ien
c
e
s
a
n
d
M
a
th
e
m
a
ti
c
s,
M
o
s
u
l
U
n
iv
e
rsit
y
,
Al
-
M
a
jmo
a
a
S
tree
t,
M
o
su
l
,
Ira
q
.
I
c
o
m
p
lete
d
m
y
P
h
D
i
n
Nu
m
e
rica
l
o
p
ti
m
iza
ti
o
n
.
I
h
a
v
e
1
2
n
a
ti
o
n
a
l
a
n
d
i
n
ter
n
a
ti
o
n
a
l
p
u
b
li
s
h
e
d
j
o
n
t
a
n
d
sin
g
le res
e
a
rc
h
p
a
p
e
rs
.
Ema
il
:
m
u
n
a
m
o
h
7
4
@u
o
m
o
su
l
.
e
d
u
.
iq
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