T
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L
KO
M
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A
, V
ol
.
17
,
No.
4,
A
ug
us
t
20
1
9,
p
p.2
12
5
~
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13
8
IS
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K
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18
DOI:
10.12928/TE
LK
OM
N
IK
A
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©
2
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1
9
Uni
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a
s
Ahm
a
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D
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hl
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All
rig
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s
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s
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d
.
1.
Int
r
o
d
u
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T
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s
arti
c
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pres
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a
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w
m
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ti
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op
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na
l
r
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c
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f
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na
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d
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(
m
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m
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[
1
-
10
]
.
T
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ex
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g
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[
11
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1
2],
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[13
-
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the
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[18
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th
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[19
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on
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[20
,
21
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the
f
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[4
-
6,
22
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25
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[
11
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,
26
-
28
].
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r
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Evaluation Warning : The document was created with Spire.PDF for Python.
◼
IS
S
N: 16
93
-
6
93
0
T
E
L
KO
M
NIK
A
V
ol
.
17
,
No
.
4
,
A
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20
19
:
21
2
5
-
213
8
2126
=
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Unf
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3,
26
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9]
c
an
no
t
s
o
l
v
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t
he
gl
o
ba
l
op
ti
m
i
z
at
i
o
n p
r
o
bl
em
s
s
i
nc
e:
a.
c
an
no
t
as
s
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e e
x
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s
te
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f
a
be
tte
r
l
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m
i
ni
m
i
z
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a l
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as
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[29
,
30
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;
b.
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whi
c
h
c
on
ta
i
ns
al
l
gl
ob
a
l
m
i
ni
m
i
z
er
s
,
an
d
t
he
oth
er
on
L
i
ps
c
hi
t
z
c
on
s
tan
t
of
r
es
pe
c
ti
v
e
l
y
.
T
he
pa
r
am
ete
r
fr
ee
f
i
l
l
ed
f
un
c
ti
o
n
(
P
F
F
F
)
w
as
i
ni
t
i
a
l
l
y
i
ntrodu
c
ed
i
n [
30
-
35
]
. A
P
F
F
F
pro
po
s
e
d b
y
Ma
et
al
.
[34
]
i
s
(
,
∗
)
=
−
(
(
)
−
(
∗
)
)
a
r
c
ta
n
(
‖
−
∗
‖
2
)
where
(
)
=
{
1
,
(
≥
0
)
−
1
,
(
<
0
)
(
6)
where t
hi
s
m
eth
od
a
l
s
o
ha
s
weak
ne
s
s
es
as
th
e o
the
r
s
.
O
ur
ne
w
P
F
F
F
m
eth
od
(
or
s
i
m
pl
y
I
Y
RH’
s
m
eth
od
)
i
s
ba
s
ed
on
P
F
F
F
[
30
-
35
]
f
or
gl
o
ba
l
op
ti
m
i
z
at
i
o
n o
f
:
⊆
→
w
h
ere
s
ati
s
f
i
es
t
he
f
ol
l
o
wi
n
g s
e
v
en
as
s
u
m
pti
on
s
:
A
1.
i
s
a
tr
i
c
e
c
on
t
i
nu
ou
s
l
y
d
i
f
f
erenti
a
bl
e o
n
(
or
∈
3
(
)
)
A
2.
ha
s
on
l
y
a
f
i
n
i
te
n
um
be
r
of
ex
tr
e
m
e
an
d
i
nf
l
ec
ti
on
p
oi
nts
i
n
,
an
d
(
)
≠
0
f
or
≥
3
where
i
s
an
i
nf
l
ec
t
i
on
po
i
nt
of
.
A
3.
(
1
)
,
(
2
)
an
d
(
3
)
of
are
Li
ps
c
h
i
t
z
-
c
o
n
ti
nu
ou
s
wi
th
c
om
pu
tab
l
e c
o
ns
tan
ts
.
A
4.
(
)
→
∞
as
|
|
→
∞
.
A
5.
F
or
∗
(
[36
])
,
(
)
−
(
∗
)
=
0
y
i
el
ds
a
t
m
os
t
two
ne
ar
es
t
po
i
nts
−
and
+
l
oc
ate
d
o
n
the
l
ef
t
an
d
t
he
r
i
gh
t
ha
n
d
s
i
de
s
of
∗
,
r
es
pe
c
ti
v
e
l
y
s
uc
h
tha
t
(
−
)
=
(
∗
)
=
(
+
)
and
∈
[
−
,
+
]
.
A
6.
(
)
>
(
∗
)
f
or
∈
(
−
,
+
)
and
(
)
=
(
∗
)
i
f
=
−
or
=
+
.
A
7.
T
he
r
e e
x
i
s
ts
on
l
y
o
ne
be
t
ween
t
wo c
on
s
ec
u
ti
v
e m
i
ni
m
i
z
er a
nd
m
ax
i
m
i
z
er of
(
)
.
T
he
r
ea
s
on
w
h
y
we
ne
ed
t
o
s
ol
v
e
o
ne
-
d
i
m
en
s
i
on
a
l
m
ul
t
i
m
od
al
f
un
c
ti
on
i
s
de
s
c
r
i
be
d
i
n
m
an
y
r
ef
erenc
es
c
i
te
d
i
n
[2
0].
T
he
ne
e
de
d
i
s
ap
pe
are
d
i
n
s
c
i
en
ti
f
i
c
an
d
en
g
i
n
ee
r
i
n
g
ap
p
l
i
c
a
ti
o
ns
es
pe
c
i
a
l
l
y
i
n
e
l
ec
tr
i
c
a
l
en
gi
ne
eri
ng
op
ti
m
i
z
at
i
on
pro
bl
e
m
.
O
ne
of
the
i
m
po
r
tan
t
i
s
s
ue
s
i
n
gl
o
ba
l
op
ti
m
i
z
at
i
o
n i
s
“
th
e reg
i
o
n o
f
at
tr
ac
ti
on
”
where
i
ts
de
tai
l
ex
pl
a
na
t
i
on
c
a
n b
e s
ee
n i
n
[3]
.
T
hi
s
pa
pe
r
i
s
organ
i
z
ed
as
f
ol
l
o
w
s
.
S
ec
ti
on
2
d
es
c
r
i
be
s
the
I
Y
RH
’
s
f
un
c
ti
o
n.
S
e
c
ti
on
3
de
s
c
r
i
be
s
ho
w
t
o
f
i
nd
a
l
l
ex
tr
em
e
an
d
i
nf
l
ec
ti
o
n
po
i
nts
us
i
ng
t
he
I
Y
RH’
s
f
un
c
t
i
on
.
S
ec
ti
on
4
di
s
c
us
s
es
the
r
e
l
at
i
o
ns
hi
p
be
t
w
e
en
an
d
I
Y
RH’
s
f
un
c
t
i
on
.
I
n
S
ec
ti
o
n
5
,
th
e
i
de
a
o
f
c
urv
atu
r
e
i
s
de
s
c
r
i
be
d
.
S
ec
t
i
on
6
c
on
t
ai
ns
the
c
o
nv
ergenc
e
t
he
orem
.
T
he
nu
m
eric
al
r
es
ul
ts
of
IY
RH
’
s
al
g
orit
hm
wi
l
l
be
pres
en
ted
i
n
s
ec
ti
o
n
7.
C
om
pa
r
i
s
on
an
d
di
s
c
us
s
i
o
n
wi
l
l
be
gi
v
e
n
i
n
s
ec
ti
on
8.
S
ec
ti
on
9
c
on
ta
i
ns
th
e
c
o
n
c
l
us
i
on
a
nd
t
he
brie
f
ex
p
l
a
na
ti
on
on
ho
w
th
i
s
o
ne
-
d
i
m
en
s
i
on
al
c
as
e
c
an
be
ex
ten
de
d
to
n
-
di
m
e
ns
i
on
al
c
as
e
.
2.
A
R
ela
t
iv
el
y
N
ew
P
ar
a
met
er
Fr
ee
Fi
lle
d
Fu
n
ctio
n
In
thi
s
s
ec
t
i
on
,
th
e
I
Y
RH’
s
w
i
l
l
b
e
de
r
i
v
ed
.
Def
i
n
i
t
i
on
1
(
O
ne
-
Di
m
en
s
i
on
al
P
F
F
F
)
:
S
up
po
s
e
t
ha
t
:
[
,
]
⊂
→
s
ati
s
f
i
es
A
1
–
A
7.
A
n
e
w
(
,
∗
)
(
∈
[
−
,
+
]
)
c
al
l
ed
I
Y
R
H’
s
f
un
c
ti
on
of
a
t
∗
, i
s
de
f
i
ne
d b
y
:
(
,
∗
)
=
{
−
∫
(
(
)
−
(
∗
)
)
∗
(
−
≤
≤
∗
)
−
∫
(
(
)
−
(
∗
)
)
∗
(
∗
≤
≤
+
)
(
7)
i
f
(
,
∗
)
s
ati
s
f
i
es
the
f
ol
l
o
wi
n
g
3
c
o
nd
i
ti
o
ns
.
C1
.
∗
i
s
a
l
oc
al
i
s
o
l
ate
d
m
ax
i
m
i
z
er
of
(
,
∗
)
,
C2.
(
,
∗
)
ha
s
no
s
tat
i
on
ar
y
p
oi
nt
i
n
the
i
nte
r
v
al
(
−
,
∗
)
∪
(
∗
,
+
)
,
a
nd
C
3.
Evaluation Warning : The document was created with Spire.PDF for Python.
T
E
L
KO
M
NIK
A
IS
S
N: 1
69
3
-
6
93
0
◼
S
ol
v
i
ng
on
e
-
di
me
ns
i
on
al
u
nc
on
s
tr
ai
ne
d
gl
ob
a
l
o
pti
mi
z
ati
o
n p
r
o
bl
em
...
(
Is
ma
i
l
B
i
n
Mo
hd
)
2127
If
∗
i
s
no
t
a
gl
ob
a
l
m
i
ni
m
i
z
e
r
of
,
the
n
−
and
+
are
the
m
i
ni
m
i
z
er
or
s
tat
i
o
na
r
y
po
i
nts
of
(
,
∗
)
.
It i
s
e
no
ug
h
to
c
on
s
i
de
r
t
h
e s
ec
on
d
i
nt
eg
r
at
i
o
n o
f
(
7) whi
c
h
c
a
n b
e
r
e
w
r
i
tt
en
as
f
ol
l
o
w
s
:
(
,
∗
)
=
−
∫
(
(
)
−
(
∗
)
)
∗
,
(
∗
≤
≤
+
)
(
8)
b
y
us
i
ng
A
5
,
t
he
f
ol
l
o
w
i
n
g
r
es
u
l
ts
c
an
b
e
prov
ed
.
T
he
orem
1:
I
f
1)
∈
3
(
,
)
;
2)
∗
∈
[
−
,
+
]
⊆
[
,
]
an
d;
3)
(
,
∗
)
i
s
de
f
i
n
ed
b
y
(
6),
the
n
∗
m
us
t
be
a
l
oc
al
i
s
ol
at
ed
m
ax
i
m
i
z
er
of
(
,
∗
)
.
T
he
orem
2:
If
th
e
h
y
po
t
he
s
es
of
T
he
orem
1
are
v
a
l
i
d,
th
en
(
,
∗
)
do
es
n
ot
ha
v
e
an
y
s
tat
i
o
n
ar
y
p
oi
n
t
i
n
th
e
i
nte
r
v
al
1
=
{
:
(
)
>
(
∗
)
,
∈
(
−
,
+
)
\
{
∗
}
}
=
(
−
,
∗
)
∪
(
∗
,
+
)
.
B
y
(
8),
f
or
1
,
2
∈
[
∗
,
+
]
wi
th
1
<
2
,
(
1
,
∗
)
−
(
2
,
∗
)
=
∫
(
(
)
−
(
∗
)
)
2
1
≥
0
.
T
hu
s
,
(
,
∗
)
de
c
r
ea
s
es
o
v
er
[
∗
,
+
]
.
B
y
s
i
m
i
l
ar
argum
en
t,
(
,
∗
)
i
nc
r
ea
s
es
ov
er
[
−
,
∗
]
.
T
he
orem
3:
If
th
e
h
y
p
oth
es
es
of
T
he
orem
1
are
v
a
l
i
d
a
nd
∗
i
s
n
ot
a
gl
o
ba
l
m
i
ni
m
i
z
er
of
(
)
, th
en
−
and
+
a
r
e t
he
m
i
ni
m
i
z
er
or s
ta
ti
on
ar
y
po
i
nt
of
(
,
∗
)
.
T
he
ob
tai
n
i
n
g
of
al
l
t
he
ex
tr
em
e
an
d
i
nf
l
ec
ti
on
po
i
nts
i
n
ev
er
y
[
∗
,
+
]
(
=
0
,
1
,
2
,
…
,
<
∞
)
i
s
an
i
nd
i
c
ato
r
t
ha
t
t
hi
s
a
l
g
orit
hm
ne
v
er
f
ai
l
to
ob
ta
i
n
t
he
g
l
ob
al
on
e.
T
ha
t
i
s
wh
y
i
t
m
a
k
es
thi
s
m
eth
od
ex
pl
ores
a
l
o
ng
t
he
e
nti
r
e
d
om
ai
n
whi
c
h
v
er
y
m
uc
h
di
f
f
er
en
t
to
ot
he
r
m
eth
od
s
[4
-
6
,
11
-
21].
W
e
are
no
t
a
war
e
wi
th
t
he
m
eth
od
i
n
[
37
],
whi
c
h
qu
i
et
s
i
m
i
l
ar
wi
th
ou
r
m
eth
o
d.
F
ortuna
t
el
y
,
ou
r
m
eth
od
ha
s
be
e
n
pu
b
l
i
s
h
ed
f
i
r
s
t
as
m
en
ti
on
e
d
i
n
[3
0
-
35]
.
H
o
w
e
v
er,
w
e
d
i
d
no
t
k
no
w
h
o
w
t
he
au
t
ho
r
s
[37
]
c
om
pu
te
the
i
r
i
nte
grati
on
.
In
I
Y
RH
’
s
a
l
g
orit
hm
,
the
i
nt
eg
r
at
i
o
n
i
s
ne
v
er
be
e
n
c
om
pu
ted
as
ha
d
b
ee
n
do
n
e
i
n
[
30
-
3
5].
T
he
r
ef
ore,
IY
RH
’
s
al
go
r
i
t
hm
v
er
y
m
uc
h
di
f
f
erent c
om
pa
r
ed
wi
th
oth
ers
.
3.
S
equ
ence
s of
E
xtrem
e
and
Inf
lec
t
ion
P
o
int
s
T
he
I
Y
RH’
s
f
un
c
ti
on
(
,
∗
)
(
∈
[
−
,
+
]
)
ha
s
th
e f
ol
l
o
wi
ng
prop
erti
es
:
P
1.
(
,
∗
)
i
s
c
on
c
a
v
e
do
wn
w
ard
at
∗
an
d c
on
c
a
v
e
up
war
d a
t
bo
th
−
an
d
+
.
P
2.
(
,
∗
)
,
(
1
)
(
,
∗
)
,
(
2
)
(
,
∗
)
and
(
3
)
(
,
∗
)
are c
on
ti
n
uo
us
.
P
3.
(
,
∗
)
<
0
f
or
[
−
,
∗
]
∪
[
∗
,
+
]
and
(
∗
,
∗
)
=
0
.
P
4.
(
,
∗
)
are
i
nc
r
ea
s
i
ng
a
nd
de
c
r
ea
s
i
ng
o
v
er
[
−
,
∗
]
and
[
∗
,
+
]
r
es
pe
c
ti
v
e
l
y
.
P
5.
(
1
)
(
,
∗
)
>
0
(
∈
(
−
,
∗
)
)
an
d
(
1
)
(
,
∗
)
<
0
)
(
∈
(
∗
,
+
)
)
ex
c
ep
t a
t
i
nf
l
ec
t
i
on
po
i
nts
P
6.
(
,
∗
)
ha
s
i
s
o
l
at
ed
m
i
ni
m
i
z
er
or s
tat
i
o
na
r
y
po
i
nt
at
−
or
+
.
A
s
an
ex
am
pl
e,
th
e
gra
p
h
of
ou
r
P
F
F
F
f
or
s
in
+
s
in
(
2
/
3
)
c
an
be
p
l
ott
ed
as
i
n
F
i
gu
r
e
1
.
B
y
P
1,
t
he
r
e
ex
i
s
ts
at
l
ea
s
t
t
w
o
i
nf
l
ec
ti
on
po
i
nts
of
(
,
∗
)
ea
c
h
l
i
es
i
n
(
−
,
∗
)
an
d
(
∗
,
+
)
.
B
y
P
1
-
P
6,
F
i
gu
r
e
1
a
nd
[
∗
,
+
]
(
[
−
,
∗
]
)
,
the
I
Y
RH’
s
f
un
c
ti
on
g
en
era
tes
the
s
eq
ue
nc
e
of
:
0
∗
,
1
∗
,
2
∗
, …
,
∗
,
…
(
0
∗
,
−
1
∗
,
−
2
∗
, …
,
−
∗
,…)
(
9)
us
i
ng
an
y
s
ui
tab
l
e
op
t
i
m
i
z
a
ti
on
to
o
l
s
ex
c
ep
t
0
∗
, s
tarti
ng
a
t
:
0
,
0
+
,
1
+
,
2
+
…
,
+
,
… (
0
,
0
−
,
−
1
−
,
−
2
−
…
,
−
−
,
…)
(
10
)
r
es
pe
c
ti
v
el
y
where
as
be
g
i
nn
i
ng
0
i
s
an
y
gi
v
en
p
oi
n
t
f
or
ob
t
ai
ni
n
g
0
∗
.
T
he
r
ef
ore,
we
ha
v
e
2
ph
as
es
as
f
ol
l
o
w
s
:
P
ha
s
e
I :
S
tart
i
ng
at
(
=
0
,
1
,
2
,
…
)
. M
i
ni
m
i
z
e
(
)
to
o
bta
i
n
i
s
ol
ate
d
m
i
ni
m
i
z
er
∗
.
P
ha
s
e
II :
(
,
∗
)
i
s
c
on
s
tr
uc
te
d t
o
f
i
nd
+
. Rep
l
ac
e
wi
th
+
.
Res
tar
t P
h
as
e 1
.
Ho
w
e
v
er,
i
t
i
s
no
t
ea
s
y
to
o
bta
i
n
(
9)
a
nd
(
1
0)
s
i
nc
e
(
,
∗
)
c
o
nta
i
ns
at
l
e
as
t
1
i
nf
l
ec
ti
on
po
i
nt
i
n
[
∗
,
+
]
an
d
al
s
o
i
n
[
−
,
∗
]
.
T
he
an
a
l
y
t
i
c
al
ex
i
s
te
nc
e
of
i
nf
l
e
c
ti
on
po
i
nts
i
s
s
ho
wn
a
s
f
ol
l
o
w
s
.
a.
(
1
)
(
,
∗
)
=
−
(
(
)
−
(
∗
)
)
(
∗
≤
≤
+
)
an
d
(
2
)
(
,
∗
)
=
−
(
1
)
(
)
(
∗
≤
≤
+
)
.
If
(
2
)
(
,
∗
)
=
0
,
th
en
(
1
)
(
)
=
0
.
T
he
r
ef
ore,
th
e
s
o
l
ut
i
on
of
(
2
)
(
,
∗
)
=
0
be
c
om
es
the
c
r
i
t
i
c
al
po
i
nt
of
(
)
.
b.
(
3
)
(
,
∗
)
=
−
(
2
)
(
)
(
∗
≤
≤
+
)
,
i
f
(
3
)
(
,
∗
)
=
0
,
the
n
(
2
)
(
)
=
0
.
T
he
r
ef
ore,
b
y
A
2,
the
s
o
l
ut
i
on
of
(
3
)
(
,
∗
)
=
0
(
the
c
r
i
ti
c
a
l
po
i
nt
of
(
2
)
(
,
∗
)
)
be
c
om
es
the
i
nf
l
ec
ti
on
po
i
nt
of
(
)
bu
t
t
hi
s
s
o
l
ut
i
on
be
c
om
es
the
c
r
i
ti
c
a
l
po
i
nt
(
m
ax
i
m
i
z
er
or
m
i
ni
m
i
z
er)
of
(
2
)
(
,
∗
)
.
Evaluation Warning : The document was created with Spire.PDF for Python.
◼
IS
S
N: 16
93
-
6
93
0
T
E
L
KO
M
NIK
A
V
ol
.
17
,
No
.
4
,
A
ug
us
t
20
19
:
21
2
5
-
213
8
2128
T
he
r
ef
ore, th
e
i
s
o
l
ate
d
ex
tr
em
e p
oi
nts
i
.
e.
,
(
1
)
(
l
oc
al
m
ax
i
m
i
z
er)
a
nd
,
(
1
)
(
l
oc
al
m
i
ni
m
i
z
er)
of
(
)
as
s
ho
wn
i
n
F
i
gu
r
e
2,
are
i
nf
l
ec
t
i
on
po
i
nts
of
(
,
∗
)
w
h
ere
(1)
th
e
s
u
pe
r
s
c
r
i
pt
of
,
(
1
)
and
,
(
1
)
,
de
no
tes
t
he
f
i
r
s
t i
nn
er
i
terat
i
o
n i
n t
h
e i
nte
r
v
al
[
∗
,
+
]
.
F
i
gu
r
e
1
.
(
,
∗
)
∈
[
−
,
+
]
)
f
or
s
in
+
s
in
(
2
/
3
)
F
i
gu
r
e
2
.
R
el
a
ti
o
ns
hi
p b
et
ween
(
)
,
(
,
∗
)
,
(
1
)
(
,
∗
)
,
(
2
)
(
,
∗
)
and
(
3
)
(
,
∗
)
Note
tha
t
th
ere
m
i
gh
t
ex
i
s
t
m
ore
tha
n
o
ne
i
n
ne
r
i
tera
ti
on
i
n
the
i
nte
r
v
al
[
∗
,
+
]
an
d
th
i
s
wi
l
l
h
ap
p
en
when
m
ore
tha
n
t
w
o
i
nf
l
ec
ti
on
po
i
nts
oc
c
urr
ed
i
n
[
∗
,
+
]
.
T
he
r
ef
ore
f
or
P
h
as
e
II,
we
ne
ed
t
o
an
al
y
s
e
t
he
b
eh
a
v
i
ou
r
of
(
,
∗
)
.
S
i
nc
e
∗
c
an
n
ot
be
us
ed
bl
i
nd
l
y
to
m
i
ni
m
i
z
e
the
f
i
l
l
ed
f
un
c
ti
on
,
t
he
n
i
n
t
he
ph
as
e
II,
a
>
0
m
us
t
be
c
ho
s
en
s
uc
h
th
at
∗
+
c
an
be
s
af
el
y
uti
l
i
z
ed
to
m
i
ni
m
i
z
e
(
,
∗
)
.
F
or
h
an
d
l
i
n
g
t
he
s
e
d
i
f
f
i
c
ul
ti
es
,
c
on
s
i
d
er
the
r
el
ati
on
s
h
i
p
be
t
ween
,
and
(
2
)
as
i
l
l
us
tr
ate
d
i
n
F
i
gu
r
e
2.
S
i
nc
e
Ne
w
t
on
’
s
m
eth
od
[38
]
s
om
eti
m
es
f
ai
l
s
to
c
on
v
erg
e
to
−
or
+
,
w
e
ne
ed
I
Y
RH’
s
f
un
c
ti
on
m
eth
od
to
h
an
d
l
e
i
t.
F
r
o
m
F
i
gu
r
e
2
an
d
t
he
ab
ov
e
di
s
c
us
s
i
on
,
i
t
i
s
c
l
e
ar
t
ha
t
al
l
m
i
ni
m
i
z
ers
an
d
m
ax
i
m
i
z
ers
of
(
2
)
(
,
∗
)
be
c
om
e
th
e
i
nf
l
ec
ti
on
po
i
nts
of
(
)
,
an
d
a
l
l
th
e
r
oo
ts
of
(
2
)
(
,
∗
)
be
c
om
e
m
i
ni
m
i
z
ers
or
m
a
x
i
m
i
z
ers
of
(
)
.
T
he
s
e
s
pe
c
i
al
prop
erti
es
are o
nl
y
po
s
s
es
s
ed
b
y
I
Y
RH’
s
f
un
c
ti
on
.
4.
Co
mp
u
t
atio
n
of
t
h
e I
n
f
l
ec
t
ion
P
o
int
s
B
as
ed
on
the
d
i
s
c
us
s
i
on
i
n
S
ec
t
i
on
3,
we
ha
v
e
prov
ed
the
f
ol
l
o
wi
n
g
t
h
eo
r
em
s
:
T
he
ore
m
4:
If
the
h
y
p
oth
es
es
of
T
he
orem
1
are
v
al
i
d,
th
en
t
he
s
ol
uti
on
of
(
2
)
(
,
∗
)
=
0
be
c
om
es
the
c
r
i
ti
c
a
l
p
oi
n
t
of
(
)
.
T
he
orem
5:
If
the
h
y
p
oth
es
es
of
T
he
orem
1
are
v
a
l
i
d
,
th
en
the
c
r
i
t
i
c
al
po
i
nt
of
(
2
)
(
,
∗
)
be
c
om
e
s
th
e i
nf
l
ec
ti
o
n p
oi
n
ts
of
(
)
.
B
y
A
5
a
n
d
F
i
g
u
r
e
2
,
,
(
1
)
,
,
(
1
)
and
,
(
1
)
a
r
e
t
h
e
fi
r
s
t
i
s
o
l
a
t
e
d
i
n
fl
e
c
t
i
o
n
,
ma
x
i
mu
m
a
n
d
mi
n
i
mu
m
p
o
i
n
t
s
o
f
(
)
r
e
s
p
e
c
t
i
v
e
l
y
,
,
(
2
)
i
s
t
h
e
s
e
c
o
n
d
i
n
f
l
e
c
t
i
o
n
p
o
i
n
t
o
f
(
)
f
o
u
n
d
a
f
t
e
r
∗
w
h
e
r
e
,
(
1
)
<
,
(
1
)
<
,
(
2
)
<
,
(
1
)
,
a
n
d
i
t
mi
g
h
t
c
o
n
t
i
n
u
e
w
i
t
h
a
n
o
t
h
e
r
s
e
q
u
e
n
c
e
o
f
e
x
t
r
e
me
a
n
d
i
n
f
l
e
c
t
i
o
n
p
o
i
n
t
s
u
n
t
i
l
+
s
u
c
h
t
h
a
t
(
+
)
=
(
∗
)
a
n
d
(
∗
)
<
(
)
(
∈
(
∗
,
+
)
)
.
I
n
o
r
d
e
r
t
o
g
u
a
r
a
n
t
e
e
n
o
e
x
t
r
e
me
o
r
i
n
f
l
e
c
t
i
o
n
p
o
i
n
t
s
o
f
(
)
mi
s
s
e
d
d
u
r
i
n
g
t
h
e
c
o
mp
u
t
a
t
i
o
n
,
t
h
e
o
u
t
e
r
a
n
d
i
n
n
e
r
i
t
e
r
a
t
i
o
n
s
a
r
e
u
s
e
d
o
v
e
r
[
∗
,
+
]
.
I
n
i
n
n
e
r
i
t
e
r
a
t
i
o
n
,
(
3
)
(
,
∗
)
a
n
d
(
2
)
(
,
∗
)
a
r
e
u
s
e
d
to
c
o
mp
u
t
e
i
n
f
l
e
c
t
i
o
n
a
n
d
e
x
t
r
e
me
p
o
i
n
t
s
o
f
(
)
r
e
s
p
e
c
t
i
v
e
l
y
w
h
e
r
e
a
s
i
n
o
u
t
e
r
i
t
e
r
a
t
i
o
n
,
(
)
i
s
mi
n
i
mi
z
e
d
o
r
s
o
l
v
e
(
1
)
(
,
∗
)
=
0
t
o
o
b
t
a
i
n
+
.
T
h
e
f
o
l
l
o
w
i
n
g
s
t
e
p
s
i
mp
l
e
me
n
t
t
h
o
s
e
b
o
t
h
i
n
n
e
r
a
n
d
o
u
t
e
r
i
t
e
r
a
t
i
o
n
s
:
O
ute
r
Ite
r
a
ti
o
n
S
tep
1
:
c
o
ns
tr
uc
t
(
,
∗
)
at
∗
.
Inn
er
It
erat
i
on
S
tep
2
:
S
o
l
v
e
(
3
)
(
,
∗
)
=
0
b
y
Ne
w
t
on
’
s
m
eth
od
f
or
i
nf
l
ec
ti
o
n p
o
i
nt
of
(
)
ne
ares
t to
∗
.
S
tep
3
:
S
o
l
v
e
(
2
)
(
,
∗
)
=
0
by
Ne
w
t
on
’
s
m
eth
od
f
or
i
s
ol
at
ed
m
ax
i
m
i
z
er of
(
)
.
S
tep
4
:
S
o
l
v
e
(
3
)
(
,
∗
)
=
0
b
y
Ne
w
t
on
’
s
m
eth
od
f
or nex
t
i
nf
l
ec
t
i
on
po
i
nt
of
(
)
.
S
tep
5
:
S
o
l
v
e
(
2
)
(
,
∗
)
=
0
by
Ne
w
t
on
’
s
m
eth
od
f
or
i
s
ol
at
ed
m
i
ni
m
i
z
er of
(
)
.
Evaluation Warning : The document was created with Spire.PDF for Python.
T
E
L
KO
M
NIK
A
IS
S
N: 1
69
3
-
6
93
0
◼
S
ol
v
i
ng
on
e
-
di
me
ns
i
on
al
u
nc
on
s
tr
ai
ne
d
gl
ob
a
l
o
pti
mi
z
ati
o
n p
r
o
bl
em
...
(
Is
ma
i
l
B
i
n
Mo
hd
)
2129
S
tep
6
:
If
(
)
>
(
∗
)
an
d
<
+
t
hen
r
ep
e
at
S
tep
2
-
S
te
p
5
el
s
e
s
o
l
v
e
(
1
)
(
,
∗
)
=
0
by
Ne
w
to
n’
s
m
eth
od
f
or
+
s
uc
h t
ha
t
(
+
)
=
(
∗
)
.
S
tep
7
:
Us
e
+
to
y
i
e
l
d
+
1
∗
.
≔
+
1
. G
o
to
S
t
ep
1 i
f
+
1
∗
<
.
5.
Co
n
v
er
g
enc
e w
it
h
Cu
r
va
t
u
r
e
T
he
c
urv
atu
r
e [
39
]
an
d rad
i
us
of
c
urv
atu
r
e
are d
ef
i
ne
d
b
y
:
(
)
=
|
|
=
|
2
/
2
|
[
1
+
(
/
)
2
]
3
/
2
and
(
)
=
1
(
)
r
es
pe
c
ti
v
el
y
.
B
as
i
c
a
l
l
y
,
to
m
a
k
e
New
ton
’
s
m
eth
od
c
on
v
erg
es
to
∗
,
the
s
ol
uti
on
of
(
)
=
0
,
we
ne
ed
an
i
ni
ti
a
l
es
t
i
m
ati
on
whi
c
h
c
l
os
es
e
no
u
gh
t
o
∗
.
A
s
s
i
gn
th
e
r
ad
i
us
of
c
urv
atu
r
e
of
(
)
to
,
the
r
ef
ore
∗
+
be
c
om
es
the
i
ni
t
i
al
be
s
t
es
ti
m
ato
r
f
or
New
t
on
’
s
m
eth
od
to
s
ol
v
e
(
)
=
0
.
W
e
wi
l
l
pro
v
e
th
at
=
|
∗
+
−
∗
|
i
s
the
r
a
di
us
of
the
l
arges
t
i
nte
r
v
al
aro
un
d
∗
s
uc
h
tha
t
the
Ne
wton
’
s
m
eth
od
c
on
v
erges
t
o
∗
∈
(
∗
−
,
∗
+
)
.
Howev
er,
w
e
wi
l
l
ne
ed
t
he
f
ol
l
o
w
i
ng
de
f
i
ni
t
i
o
n.
Def
i
ni
t
i
on
2
[4
0]:
T
he
f
un
c
ti
on
:
⊂
→
i
s
Li
ps
c
h
i
t
z
c
on
t
i
nu
ou
s
f
un
c
ti
on
wi
th
c
o
ns
tan
t
i
n
,
w
r
i
tte
n
∈
(
)
,
i
f
f
or ev
er
y
,
∈
,
|
(
)
−
(
)
|
≤
|
−
|
.
F
or
the
c
on
v
erg
en
c
e
of
N
e
w
ton
’
s
m
eth
od
,
w
e
ne
ed
(
1
)
∈
(
)
w
h
i
c
h
ha
d
b
ee
n
s
ho
w
n
i
n [
40
].
Le
m
m
a
1
[40
]
:
If
1)
:
⊂
→
f
or
an
op
en
i
nt
erv
a
l
;
2)
(
1
)
∈
(
)
,
the
n
f
or
an
y
,
∈
,
|
(
)
−
(
)
−
(
1
)
(
)
(
−
)
|
≤
(
−
)
2
/
2
.
F
or m
os
t p
r
ob
l
em
s
, New
to
n’
s
m
eth
od
wi
l
l
c
on
v
erg
e
-
qu
ad
r
at
i
c
al
l
y
to
i
ts
r
oo
t
[4
0
].
T
he
ore
m
6
[40
]:
If
1)
:
⊂
→
f
or
a
n
op
e
n
i
n
terv
al
D
;
2)
(
1
)
∈
(
)
3)
f
or
s
o
m
e
>
0
,
|
(
1
)
(
)
|
≥
(
∈
)
;
4)
(
)
=
0
ha
s
a
s
ol
uti
on
∗
∈
,
the
n
the
r
e
i
s
s
om
e
>
0
s
uc
h
th
at
i
f
|
0
−
∗
|
<
,
th
en
{
}
ge
n
erate
d
b
y
+
1
=
−
(
(
)
/
(
1
)
(
)
)
(
=
0
,
1
,
2
,
…
)
ex
i
s
ts
an
d
c
on
v
erg
es
to
∗
. Fur
the
r
m
ore,
|
+
1
−
∗
|
≤
(
/
2
)
|
−
∗
|
2
(
=
0
,
1
,
2
,
…
)
.
No
w
,
we
pro
v
e
tha
t
̂
=
|
1
∗
+
̂
−
∗
|
,
th
e
r
ad
i
us
of
t
he
l
arg
es
t
i
nt
erv
a
l
aroun
d
t
he
s
ol
ut
i
on
of
(
1
)
(
)
=
0
ho
l
ds
T
he
orem
6
.
T
he
s
i
m
i
l
arit
y
proof
i
s
ap
pl
i
ed
f
or
2
(
,
∗
)
.
T
he
ore
m
7:
If
1)
:
⊂
→
i
s
a
n
ob
j
ec
ti
v
e
f
un
c
ti
o
n;
2)
1
∗
i
s
a
l
o
c
al
i
s
ol
ate
d
m
i
ni
m
i
z
er
o
f
(
)
;
3)
(
1
)
:
⊂
→
and
(
2
)
∈
(
)
f
or
⊆
;
4)
f
or
s
om
e
>
0
,
|
(
2
)
(
)
|
≥
f
or
ev
er
y
∈
;
5)
(
1
)
(
)
=
0
ha
s
a
s
o
l
ut
i
o
n
∗
∈
,
the
n
t
he
r
e
i
s
s
om
e
>
0
s
uc
h
tha
t
i
f
|
0
−
∗
|
<
,
the
n
the
s
e
qu
e
nc
e
{
}
ge
n
era
ted
b
y
+
1
=
−
(
(
1
)
(
)
/
(
2
)
(
)
)
(
=
0
,
1
,
2
,
…
)
ex
i
s
ts
a
nd
c
on
v
erg
es
to
∗
. Fur
the
r
m
ore,
|
+
1
−
∗
|
≤
(
/
2
)
|
−
∗
|
2
(
=
0
,
1
,
2
,
…
)
.
6.
Co
n
v
er
g
enc
e of
t
h
e I
Y
RH’s
A
lgo
r
it
h
m
B
y
A2
,
I
Y
RH
’
s
al
g
orit
hm
ac
tua
l
l
y
g
en
era
tes
(
9)
an
d
(
10
)
ac
c
ordi
ng
t
o
the
f
ol
l
o
wi
ng
pa
tt
ern
:
+
−
+
−
−
+
−
+
−
−
+
+
Z
n
n
Z
n
n
Z
k
k
Z
k
k
Z
Z
x
x
x
x
x
x
x
x
x
x
x
x
x
x
1
*
2
*
1
1
*
2
*
1
1
*
2
0
*
1
0
*
0
...
...
whi
c
h
s
at
i
s
f
y
(
0
∗
)
≥
(
1
∗
)
≥
.
.
.
≥
(
∗
)
≥
.
.
.
≥
(
−
1
∗
)
≥
(
∗
)
w
h
ere
0
i
s
an
y
g
i
v
en
p
oi
nt
i
n
the
c
on
s
i
de
r
ed
i
nt
erv
a
l
.
T
he
r
ef
ore,
I
Y
RH
’
s
al
go
r
i
t
h
m
ge
ne
r
ate
s
a
f
i
ni
te
s
eq
ue
nc
e
[
0
∗
,
0
+
]
,
[
1
∗
,
1
+
]
,
.
.
.
,
[
∗
,
+
]
,
.
.
.
,
[
−
1
∗
,
−
1
+
]
.
T
hu
s
,
b
y
A2
,
I
Y
RH
’
s
a
l
go
r
i
thm
c
on
v
erg
es
to
∗
as
a
g
l
o
ba
l
m
i
ni
m
i
z
er.
I
Y
RH
’
s
a
l
g
orit
h
m
al
s
o
au
t
om
ati
c
al
l
y
ge
n
e
r
ate
s
at
l
ea
s
t
a
s
et
of
f
i
ni
t
e
s
eq
ue
nc
e
of
i
nf
l
ec
ti
on
,
l
oc
a
l
i
s
o
l
ate
d
m
a
x
i
m
i
z
ers
an
d
i
s
ol
ate
d
m
i
ni
m
i
z
ers
,
1
=
{
,
(
1
)
,
,
(
1
)
,
,
(
2
)
}
i
n
ev
er
y
s
ub
i
nt
erv
al
[
∗
,
+
]
(
=
0
,
1
,
.
.
.
,
)
i
f
ex
i
s
t
w
he
r
e
the
s
up
ers
c
r
i
pt
(1)
on
,
(
1
)
de
no
tes
the
f
i
r
s
t
nu
m
be
r
of
l
oc
al
m
ax
i
m
i
z
er
an
d
s
ub
s
c
r
i
pt
of
de
no
tes
t
he
nu
m
be
r
of
l
oc
al
m
ax
i
m
i
z
er
i
n
[
∗
,
+
]
.
I
f
i
t
c
on
ta
i
ns
two
l
oc
al
i
s
ol
ate
d
m
ax
i
m
i
z
ers
,
th
en
i
t
ge
n
erates
Evaluation Warning : The document was created with Spire.PDF for Python.
◼
IS
S
N: 16
93
-
6
93
0
T
E
L
KO
M
NIK
A
V
ol
.
17
,
No
.
4
,
A
ug
us
t
20
19
:
21
2
5
-
213
8
2130
2
=
{
,
(
1
)
,
,
(
1
)
,
,
(
2
)
,
,
(
1
)
,
,
(
3
)
,
,
(
2
)
,
,
(
4
)
}
an
d
s
o
f
orth.
Howe
v
er
f
or
2
,
the
i
nf
l
ec
ti
on
po
i
nt
,
(
4
)
is
op
ti
on
. T
hu
s
, th
e f
ol
l
o
wi
ng
t
he
orem
i
s
prov
ed
.
T
he
ore
m
8
(
Con
v
erg
en
c
e
T
he
orem
)
:
If
1)
al
l
t
he
h
y
p
oth
es
i
s
of
T
he
orem
6
an
d
T
he
or
em
7
are
v
al
i
d
f
or
,
(
1
)
,
(
2
)
,
(
3
)
an
d;
2)
(
,
∗
)
i
s
I
Y
RH
’
s
f
un
c
ti
on
at
∗
,
the
l
oc
a
l
i
s
ol
a
ted
m
i
ni
m
i
z
er of
, the
n
I
Y
RH
’
s
al
go
r
i
t
hm
c
on
v
erge
s
to
the
r
i
gh
t s
ol
uti
on
.
7.
Nu
me
r
ica
l Re
sult
s
T
he
tes
t
ex
am
pl
es
are
l
i
s
t
ed
i
n
T
ab
l
e
s
1
–
3.
I
n
T
ab
l
e
1
w
he
r
e
,
(
)
,
,
∗
and
∗
de
no
te
t
he
nu
m
be
r
of
f
un
c
ti
on
,
the
ex
pres
s
i
o
n
of
th
e
ob
j
ec
t
i
v
e
f
un
c
ti
on
,
the
d
om
ai
n,
gl
ob
a
l
m
i
ni
m
u
m
v
al
ue
an
d
gl
ob
a
l
m
i
ni
m
i
z
er
r
es
pe
c
t
i
v
el
y
.
T
he
nu
m
eric
al
r
es
ul
ts
w
i
l
l
b
e
pres
en
te
d
to
c
o
m
pa
r
e
the
c
ap
a
bi
l
i
t
y
of
the
IY
RH
’
s
m
eth
od
w
i
t
h
t
w
o
-
pa
r
am
ete
r
f
i
l
l
ed
f
un
c
ti
o
n
m
eth
od
s
[4,
26
,
29
]
,
on
e
-
pa
r
am
ete
r
f
i
l
l
ed
f
un
c
ti
o
n
m
eth
od
s
[6,
22
,
28
],
an
d
th
e
P
F
F
F
m
eth
od
[34
].
W
e
al
s
o
pres
en
t
t
he
r
es
u
l
ts
f
or
ob
s
erv
i
n
g
the
s
en
s
i
t
i
v
i
t
y
of
I
Y
R
H’
s
m
eth
od
du
e
t
o
di
f
f
erent i
n
i
ti
al
po
i
nts
. T
he
r
ef
ore, th
e
pres
en
t
ati
on
i
s
ar
r
an
gi
ng
i
nt
o f
ou
r
c
ate
g
orie
s
.
T
ab
l
e 1
.
20
T
es
t Func
ti
on
s
Ci
te
d
f
r
o
m
[4
1
]
f
or Mi
n
i
m
i
z
a
ti
on
P
r
o
bl
em
(
O
r
i
gi
na
l
R
es
ul
ts
)
(
)
∗
∗
1.
1
6
6
−
52
25
5
+
39
80
4
+
71
10
3
−
79
20
2
−
+
1
10
[
−
1
.
5
,
11
]
−
29763
.
233
10
2.
si
n
+
si
n
(
10
/
3
)
[
2
.
7
,
7
.
5
]
−
1
.
899599
5
.
145735
3.
−
∑
si
n
(
(
+
1
)
+
)
5
=
1
[
−
9
.
4
,
10
]
[
−
9
.
4
,
10
]
[
−
9
.
4
,
10
]
−
12
.
03124
−
12
.
03124
−
12
.
03124
−
6
.
77
45
76
1
−
0
.
491391
5
.
791785
4.
−
(
16
2
−
24
+
5
)
−
[
1
.
9
,
3
.
9
]
−
3
.
85045
2
.
868034
5.
−
(
−
3
+
1
.
4
)
si
n
(
18
)
[
0
,
1
.
2
]
−
1
.
48907
0
.
96609
6.
(
−
+
si
n
(
)
)
−
2
[
−
10
,
10
]
−
0
.
824239
−
0
.
679579
7.
si
n
(
)
+
si
n
(
(
10
/
3
)
)
+
ln
−
0
.
84
+
3
[
2
.
7
,
7
.
5
]
−
1
.
6013
5
.
19978
8.
−
∑
cos
(
(
+
1
)
+
)
5
=
1
[
−
9
.
7
,
10
]
[
−
9
.
7
,
10
]
[
−
9
.
7
,
10
]
−
14
.
508
−
14
.
508
−
14
.
508
−
7
.
083506
−
0
.
800321
5
.
48286
9.
si
n
+
si
n
(
2
/
3
)
[
3
,
20
]
−
1
.
90596
17
.
039
10.
−
si
n
[
0
,
10
]
−
7
.
91673
7
.
9787
11.
−
2
cos
−
cos
2
[
−
1
.
57
,
6
.
28
]
−
3
4
.
76837
−
009
12.
si
n
3
+
c
o
s
3
[
0
,
6
.
28
]
[
0
,
6
.
28
]
−
1
−
1
4
.
712389
13.
−
2
/
3
−
(
1
−
2
)
1
/
3
[
0
.
001
,
0
.
99
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−
1
.
5874
1
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14.
−
−
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n
2
[
0
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4
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−
0
.
788685
0
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224885
15.
(
2
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÷
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2
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)
[
−
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5
]
−
7
.
03553
−
0
.
41422
16.
2
(
−
3
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2
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−
2
/
2
[
−
3
,
3
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0
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0111090
3
17.
6
−
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+
27
2
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−
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]
[
−
4
,
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]
7
7
−
3
−
3
18.
{
(
−
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≤
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ln
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1
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o
t
h
e
r
w
i
s
e
)
0
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19.
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467511
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.
(
−
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n
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−
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19514
F
or
f
i
r
s
t
c
ate
go
r
y
,
the
r
e
s
ul
ts
of
T
ab
l
e
4
s
h
o
w
s
t
ha
t
I
Y
RH
’
s
al
go
r
i
t
hm
c
an
s
ol
v
e
the
g
l
o
ba
l
op
ti
m
i
z
at
i
on
pro
bl
em
s
l
i
s
ted
i
n
T
ab
l
e
1.
In
T
ab
l
e
4,
(
≥
0
)
i
s
the
n
um
be
r
of
ou
te
r
i
terat
i
on
,
(
≥
0
)
i
s
t
he
nu
m
be
r
of
i
nf
l
ec
ti
on
po
i
nts
,
(
)
an
d
,
(
+
1
)
w
h
ere
r
ef
ers
to
the
wor
d
“
i
nf
l
ec
ti
o
n”,
i
s
the
nu
m
be
r
of
l
oc
a
l
i
s
o
l
at
ed
m
ax
i
m
i
z
er
,
(
)
∈
[
∗
,
+
]
an
d
m
i
ni
m
i
z
er
,
(
)
∈
[
∗
,
+
]
where
and
de
n
ote
m
ax
i
m
i
z
er
an
d
m
i
ni
m
i
z
er
r
es
p
ec
ti
v
el
y
,
∗
(
≥
0
)
i
s
i
s
ol
a
ted
m
i
ni
m
i
z
er a
nd
+
and
−
(
≥
0
)
are p
oi
n
ts
s
uc
h t
h
at
(
−
)
=
(
∗
)
=
(
+
)
.
Evaluation Warning : The document was created with Spire.PDF for Python.
T
E
L
KO
M
NIK
A
IS
S
N: 1
69
3
-
6
93
0
◼
S
ol
v
i
ng
on
e
-
di
me
ns
i
on
al
u
nc
on
s
tr
ai
ne
d
gl
ob
a
l
o
pti
mi
z
ati
o
n p
r
o
bl
em
...
(
Is
ma
i
l
B
i
n
Mo
hd
)
2131
In
i
nn
er
i
tera
ti
o
n,
t
h
ere
are
s
ev
eral
c
as
es
tha
t
,
(
)
eq
u
al
s
to
+
and
+
1
∗
as
s
ho
w
n
i
n
T
ab
l
e
4
f
or
=
2
,
3
o
f
e
x
a
m
pl
e
3
(
=
3
),
=
3
of
e
x
a
m
pl
e
8
(
=
8
),
=
1
of
e
x
am
pl
e
12
(
=
12
)
an
d
=
0
of
e
x
am
pl
e
17
(
=
17
).
F
or
s
ec
on
d
c
a
teg
or
y
,
T
ab
l
e
s
5
–
8
c
om
pa
r
e
I
Y
RH
’
s
al
g
orit
hm
wi
th
N
e
w
a
l
g
orit
h
m
[42
],
the
d
i
r
ec
t
m
eth
od
[
42
]
a
nd
La
gra
ng
e
i
n
terpo
l
a
ti
on
[43
],
us
i
ng
tes
t
f
un
c
ti
on
s
i
n
T
ab
l
e
1
[4
4]
an
d
1
00
on
e
-
d
i
m
en
s
i
on
al
r
a
nd
om
i
z
e
d
tes
t
f
un
c
t
i
on
s
[
45
].
T
ab
l
e
5
s
ho
w
s
the
r
e
l
at
i
v
e
err
ors
[42
]
of
gl
o
ba
l
m
i
ni
m
u
m
v
al
ue
s
an
d
gl
o
ba
l
m
i
ni
m
i
z
ers
of
f
un
c
ti
on
s
as
s
ho
w
n
i
n
T
ab
l
e
1
ob
tai
ne
d
b
y
I
Y
RH
’
s
al
g
orit
hm
i
s
be
tte
r
th
an
L
ag
r
an
ge
i
nte
r
po
l
an
t
o
n
81
Cheb
y
s
he
v
n
od
es
[
43
].
T
ab
l
e
6
s
ho
w
s
t
ha
t
a
“
f
ortun
e
ef
f
ec
t”
do
es
no
t
h
ap
pe
n
ed
to
I
Y
R
H’
s
al
g
orit
hm
w
he
n
i
t
i
s
ap
pl
i
ed
to
the
ex
am
pl
e
g
i
v
en
i
n
T
ab
l
e
3
f
or
=
67
and
∗
i
s
c
ho
s
en
r
an
do
m
l
y
a
nd
d
i
f
f
erentl
y
wh
ere i
ts
gr
ap
h
i
s
s
ho
wn i
n Fi
gu
r
e
3
.
T
ab
l
e 2
.
7 T
es
t Func
ti
o
ns
f
or Com
pa
r
i
s
on
w
i
th
E
x
i
s
ted
Fil
l
ed
F
un
c
ti
on
Me
t
ho
ds
(
)
,
,
0
1
(
)
=
si
n
(
)
+
si
n
(
10
/
3
)
+
ln
(
)
−
0
.
84
2
(
)
=
−
∑
si
n
(
(
+
1
)
+
)
5
=
1
3
(
)
=
{
si
n
2
(
1
)
+
∑
[
(
1
−
)
2
(
1
+
si
n
2
(
+
1
)
)
]
−
1
=
1
+
(
−
)
2
}
1
=
1
+
−
0
.
25
(
−
1
)
,
=
10
,
=
1
and
d
e
n
o
t
e
s
t
h
e
d
i
men
s
ion
a
l
it
y
o
f
t
h
e
p
r
o
b
le
m
4
(
)
=
{
si
n
2
(
1
)
+
∑
[
(
1
−
)
2
(
1
+
si
n
2
(
+
1
)
)
]
−
1
=
1
+
(
−
)
2
}
=
10
,
=
1
and
d
e
n
o
t
e
s
t
h
e
d
i
men
s
ion
a
l
it
y
o
f
t
h
e
p
r
o
b
le
m
5
(
)
=
si
n
2
0
1
+
1
∑
[
(
1
−
)
2
(
1
+
si
n
2
0
1
)
]
−
1
=
1
+
1
(
−
)
2
(
1
−
0
si
n
2
0
1
)
w
h
e
r
e
t
h
e
c
o
n
s
t
a
n
t
s
in
t
h
i
s
e
q
u
a
t
i
o
n
h
a
v
e
b
e
e
n
f
ix
e
d
a
s
f
o
ll
o
w
s
:
0
=
1
,
1
=
0
.
1
,
=
1
,
0
=
3
T
ab
l
e 3
.
T
es
t Func
ti
on
s
f
or “
F
ortune
E
f
f
ec
t”
of
IY
RH’
s
F
un
c
ti
o
n [
4
4]
1
,
…
,
100
0
.
025
(
−
∗
)
2
+
si
n
2
[
(
−
∗
)
+
(
−
∗
)
2
]
+
si
n
2
(
−
∗
)
[
−
5
,
5
]
F
or
thi
r
d
c
ate
go
r
y
,
T
ab
l
e
7
c
om
pa
r
es
the
r
es
ul
ts
of
IY
RH’
s
al
go
r
i
t
hm
w
i
t
h
M
a
et
a
l
.’
s
f
i
l
l
ed
f
un
c
ti
on
a
nd
Lu
c
i
di
a
nd
P
i
c
c
i
al
l
y
’
s
f
i
l
l
ed
f
un
c
ti
o
n
f
or
a
s
et
o
f
5
tes
t
ex
a
m
pl
es
i
n
T
ab
l
e
2.
F
or
the
l
as
t
c
at
eg
or
y
,
t
he
r
es
ul
ts
pres
e
nte
d
i
n
T
ab
l
e
8
i
s
us
ed
to
o
bs
erv
e
th
e
s
en
s
i
t
i
v
i
t
y
of
I
Y
RH
’
s
al
go
r
i
t
hm
du
e
to
thr
ee
i
ni
t
i
a
l
po
i
nts
us
i
ng
e
x
am
pl
e
3
f
r
o
m
T
ab
l
e
2.
I
t
i
s
c
l
e
a
r
tha
t
I
Y
RH
’
s
f
un
c
ti
on
c
an
be
us
ed
t
o s
ol
v
e t
he
gl
o
ba
l
op
ti
m
i
z
at
i
on
p
r
ob
l
em
s
fro
m
an
y
i
ni
t
i
a
l
p
oi
nt.
F
i
gu
r
e
3.
G
r
ap
h o
f
67
(
)
on
e o
f
t
he
1
00
o
ne
-
d
i
m
en
s
i
on
a
l
r
a
nd
om
i
z
ed
te
s
t f
un
c
ti
on
s
(
)
Evaluation Warning : The document was created with Spire.PDF for Python.
◼
IS
S
N: 16
93
-
6
93
0
T
E
L
KO
M
NIK
A
V
ol
.
17
,
No
.
4
,
A
ug
us
t
20
19
:
21
2
5
-
213
8
2132
T
ab
l
e 4
.
N
um
eric
al
Res
u
l
ts
of
a
S
et
of
20
T
es
t Func
ti
o
ns
b
y
I
Y
RH
’
s
A
l
g
orit
hm
/
0
∗
,
(
)
,
(
)
,
(
+
1
)
,
(
)
+
1
0
1
/1
-
1
.
5
-
1
.
4
1
4
2
1
-
0
.
9
8
2
4
4
8
-
0
.
1
0
.1
8
6
8
8
0
.5
3
/2
1
.0
6
9
5
5
1
.4
1
4
2
1
8
.0
4
6
0
1
2
9
2
.0
4
4
9
7
1
1
/1
10
2
0
1
/1
2
.7
3
.3
8
7
2
5
3
.7
8
6
1
4
4
.1
9
6
6
4
.6
8
5
3
6
4
.7
7
3
3
4
1
1
/1
5
.1
4
5
7
4
5
.6
7
0
4
6
.2
1
7
3
1
6
.6
0
5
9
1
7
.0
0
0
1
5
3
0
1
/1
-
9
.
4
-
9
.
0
3
7
4
4
-
8
.
7
8
0
9
9
-
8
.
5
4
9
7
7
-
8
.
2
5
9
7
6
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8
.
0
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8
6
8
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8
.
2
4
1
4
9
1
1
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-
8
.
0
0
8
6
8
-
7
.
6
8
9
7
-
7
.
3
9
7
2
8
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7
.
0
9
2
5
7
-
6
.
7
7
4
5
8
-
6
.
9
0
2
1
8
2
1
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6
.
7
7
4
5
8
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6
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5
0
8
3
8
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6
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2
0
2
9
7
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5
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9
6
9
9
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5
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7
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6
2
4
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5
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6
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5
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4
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4
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3
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9
8
4
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3
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2
5
2
6
3
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2
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9
9
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4
9
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2
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2
6
6
5
8
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1
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9
7
6
5
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4
9
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6
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0
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1
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8
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1
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3
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1
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0
5
5
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2
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5
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5
2
2
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9
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5
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2
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3
3
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3
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8
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2
8
9
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3
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6
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4
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4
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6
6
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6
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0
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8
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7
9
6
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9
9
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7
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4
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8
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8
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8
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9
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4
9
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9
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7
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5
4
8
0
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4
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9
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0
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Evaluation Warning : The document was created with Spire.PDF for Python.
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2133
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-
6
T
ab
l
e 6
.
N
um
eric
al
Res
u
l
ts
of
67
(
)
b
y
I
Y
RH
’
s
A
l
go
r
i
thm
/
0
∗
,
(
)
,
(
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+
1
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5
Evaluation Warning : The document was created with Spire.PDF for Python.
◼
IS
S
N: 16
93
-
6
93
0
T
E
L
KO
M
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A
V
ol
.
17
,
No
.
4
,
A
ug
us
t
20
19
:
21
2
5
-
213
8
2134
T
ab
l
e 7
.
C
om
pa
r
i
s
on
of
th
e
Num
eric
al
Res
ul
ts
b
y
I
Y
R
H’
s
A
l
go
r
i
thm
w
i
th
T
w
o
O
th
er Met
ho
ds
E
x
a
m
p
le
n
f
I
/
n
f
M
/
n
f
L
n
f
*I
/
n
f
*M
/
n
f
*L
n
FI
/
n
FM
/
N
f
l
n
F*I
/
n
F*M
/
n
F*L
1
2
7
/
1
3
6
1
/
f
a
il
e
d
2
7
/
1
3
6
1
/
f
a
il
e
d
5
5
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1
4
1
/
f
a
il
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d
6
2
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1
2
1
1
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f
a
il
e
d
2
1
6
/
1
7
0
/
f
a
il
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d
1
6
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7
0
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f
a
il
e
d
1
6
/
7
2
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f
a
il
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d
2
2
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1
3
2
/
f
a
il
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d
3
1
0
4
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3
3
9
/
f
a
il
e
d
1
0
4
/
3
3
9
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d
3
5
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3
6
/
f
a
il
e
d
5
7
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2
9
6
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f
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4
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8
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d
9
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1
9
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2
9
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f
a
il
e
d
5
1
8
/
5
0
5
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f
a
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e
d
1
8
/
5
0
5
/
f
a
il
e
d
2
0
/
3
9
7
/
f
a
il
e
d
3
1
/
4
5
7
/
f
a
il
e
d
T
he
m
ea
ni
ng
of
th
e
ab
bre
v
i
ati
o
ns
us
ed
i
n T
ab
l
e
8
i
s
as
f
ol
l
o
w
s
:
-
nf
I,
nf
M
an
d
nf
L
are
t
he
nu
m
be
r
o
f
f
un
c
ti
on
ev
al
ua
t
i
on
s
ne
ed
e
d
to
y
i
el
d
th
e
g
l
ob
a
l
m
i
ni
m
u
m
of
IY
RH’
s
,
Ma
et
a
l
.’
s
an
d
L
uc
i
di
an
d
P
i
c
i
a
l
y
’
s
a
l
go
r
i
th
m
s
r
es
pe
c
ti
v
e
l
y
.
-
nf
*
I,
nf
*
M
an
d
nf
*
L
are
t
he
nu
m
be
r
of
f
un
c
ti
on
e
v
a
l
u
ati
o
ns
ne
e
de
d
to
s
at
i
s
f
y
t
h
e
s
top
p
i
ng
c
r
i
teri
on
of
I
Y
RH
’
s
, M
a e
t
al
.’
s
an
d L
uc
i
di
an
d
P
i
c
c
i
a
l
l
y
’
s
al
go
r
i
thm
s
r
es
pe
c
ti
v
e
l
y
.
-
nFI
,
nFM
an
d
nFL
are
the
nu
m
be
r
of
f
i
l
l
e
d
f
u
nc
ti
on
e
v
a
l
ua
t
i
o
ns
ne
e
de
d
to
o
bta
i
n
the
gl
o
ba
l
m
i
ni
m
u
m
of
IY
RH’
s
,
Ma
et
al
.
’
s
an
d
Lu
c
i
di
an
d
P
i
c
c
i
al
l
y
’
s
a
l
g
orit
hm
s
r
es
pe
c
ti
v
e
l
y
.
-
nF*I,
nF*M
an
d
nF
*
L
are
the
nu
m
be
r
of
f
i
l
l
e
d
f
un
c
ti
on
ev
al
ua
t
i
on
s
ne
e
de
d
to
s
at
i
s
f
y
the
s
to
pp
i
ng
c
r
i
t
erio
n o
f
I
Y
RH’
s
, M
a e
t
al
.
’
s
a
nd
L
uc
i
d
i
an
d
P
i
c
c
i
al
l
y
’
s
a
l
go
r
i
thm
s
r
es
pe
c
ti
v
e
l
y
.
-
5.“f
ai
l
ed
”
m
ea
ns
th
e m
eth
od
of
Lu
c
i
d
i
an
d
P
i
c
c
i
a
l
l
y
f
ai
l
s
t
o a
c
h
i
e
v
e t
he
r
es
u
l
ts
.
T
ab
l
e 8
.
N
um
eric
al
Res
u
l
ts
d
ue
to
3
D
i
f
f
ere
nt
In
i
ti
al
P
o
i
nts
f
or
E
x
am
pl
e 3
of
T
ab
l
e 2
0
/
∗
,
(
)
,
(
)
,
(
+
1
)
,
(
)
+
-
12
0
1
/
1
-
1
0
.
8
7
8
9
-
9
.
9
9
3
5
5
-
9
.
1
0
2
8
1
-
8
.
0
0
6
4
5
-
7
.
8
7
0
6
9
1
1
/
1
-
6
.
9
1
9
5
4
-
5
.
9
9
3
5
5
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.
0
6
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4
.
0
0
6
4
5
-
3
.
6
7
9
9
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2
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2
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5
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5
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1
.
9
9
3
5
5
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1
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0
2
0
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0
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2
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3
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/
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1
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5
4
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9
3
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1
-
4
.
0
0
6
4
5
-
3
.
6
7
9
9
6
2
1
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2
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9
9
3
5
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1
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0
2
0
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0
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0
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2
0
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5
9
4
6
6
2
3
1
/
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2
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0
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3
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2
0
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9
3
5
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4
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5
9
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5
3
/
2
6
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0
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4
5
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1
5
1
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9
9
3
5
5
8
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1
9
5
4
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/
3
1
0
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1
1
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2
8
1
1
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9
3
5
-
5
.
6
0
1
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1
-
6
.
9
1
9
5
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-
5
.
9
9
3
5
5
-
5
.
0
6
1
5
1
-
4
.
0
0
6
4
5
-
3
.
6
7
9
9
6
1
1
/
1
-
2
.
9
5
9
8
5
-
1
.
9
9
3
5
5
-
1
.
0
2
0
4
8
-
0
.
0
0
6
4
5
0
4
2
0
.
5
9
4
6
6
2
2
1
/
1
1
2
.
0
0
6
4
5
3
.
0
2
0
4
8
3
.
9
9
3
5
5
4
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9
5
9
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5
3
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0
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5
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0
6
1
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1
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3
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0
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5
1
1
.
1
0
2
8
1
1
.
9
9
3
5
8.
Co
mp
a
r
ison
and
Disc
u
ss
ion
T
he
graphi
c
a
l
c
om
pa
r
i
s
on
be
t
w
e
en
I
Y
RH
’
s
f
i
l
l
ed
f
un
c
t
i
on
m
eth
od
w
i
t
h
oth
er
be
s
t
c
urr
en
t
f
i
l
l
ed
f
un
c
ti
on
m
eth
od
s
(
1)
-
(
4) i
nc
l
ud
e
d t
u
nn
e
l
i
ng
an
d
b
r
i
dg
i
ng
m
eth
od
s
,
wi
l
l
be
pre
s
en
ted
.
8.1
.
Co
mp
ar
i
son
w
it
h
t
h
e
T
u
n
n
eli
n
g
M
eth
o
d
[
8]
T
he
w
e
ak
ne
s
s
of
tun
ne
l
i
ng
m
eth
od
[8]
(
,
Γ
)
=
(
(
)
−
(
1
∗
)
)
/
[
(
−
1
∗
)
Γ
(
−
1
∗
)
]
ap
pe
ared
when
N
e
w
to
n’
s
m
eth
od
i
s
us
ed
s
i
nc
e
th
e
n
on
-
c
on
v
ex
i
t
y
pro
bl
em
.
F
ortuna
t
el
y
,
I
Y
RH’
s
f
i
l
l
ed
f
un
c
ti
on
c
an
be
ut
i
l
i
z
e
d
(
T
he
orem
6
an
d
T
he
ore
m
7
)
us
i
ng
the
r
ad
i
us
of
c
u
r
v
atu
r
e
a
pp
l
i
e
d
to
Ne
wton
’
s
m
eth
od
t
o
f
i
nd
the
r
oo
t
of
no
n
-
c
on
v
ex
probl
em
s
.
F
or
e
x
a
m
pl
e
7
[8]
,
the
tun
ne
l
i
n
g
m
eth
od
c
an
o
nl
y
o
bta
i
n
t
h
e
gl
ob
al
m
i
ni
m
i
z
er,
where
a
s
I
Y
RH
’
s
al
go
r
i
thm
c
an
ob
tai
n
the
en
ti
r
e
ex
tr
em
e a
nd
i
nf
l
ec
t
i
on
po
i
nt
s
i
n c
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