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1.
I
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Mo
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ap
p
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m
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s
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lv
e
th
is
t
y
p
e
o
f
p
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b
le
m
as
in
[
1
]
,
[
2
]
.
I
n
th
is
ar
ticle,
w
e
w
ill
d
is
cu
s
s
h
o
w
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s
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y
:
(
)
=
0
(
1
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So
,
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k
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f
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:
→
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[
8
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an
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-
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[
9
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-
[
11
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.
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(
1
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u
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m
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[
1
2
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as th
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o
b
tain
ed
f
r
o
m
(
2
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.
(
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=
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Evaluation Warning : The document was created with Spire.PDF for Python.
I
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6930
T
E
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elec
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C
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m
p
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l
C
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n
tr
o
l
,
Vo
l
.
19
,
No
.
6
,
Dec
em
b
er
2
0
2
1
:
1
8
4
7
-
1856
1848
W
h
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e
(
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is
th
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atr
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in
[
1
3
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(
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No
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in
f
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ltip
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s
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s
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w
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ter
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.
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s
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ial
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P
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4
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to
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tr
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ch
e
m
e
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s
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es
te
d
in
[
15
]
,
[
16
]
to
b
o
o
s
t
th
e
n
u
m
er
ical
e
f
f
icie
n
c
y
a
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n
ce
p
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ties
o
f
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m
et
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d
s
.
I
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is
also
p
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co
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s
in
g
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s
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r
ch
es
.
T
h
u
s
,
i
n
ex
a
ct
lin
e
s
ea
r
ch
[1
7
],
[
1
8
]
is
t
h
e
m
o
s
t
co
m
m
o
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l
y
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s
ed
ap
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e
f
u
n
cti
o
n
v
al
u
es (
7
)
.
‖
(
+
1
)
‖
≤
‖
(
)
‖
(
7
)
W
e
o
r
g
an
ized
th
e
ar
ticle
in
th
e
f
o
llo
w
i
n
g
o
r
d
er
:
s
ec
tio
n
2
,
d
ea
ls
w
i
th
t
h
e
t
w
o
n
e
w
al
g
o
r
ith
m
s
(
S
-
R
A
an
d
D
-
R
A
)
.
Sectio
n
3
d
ea
ls
w
it
h
i
n
tr
o
d
u
ci
n
g
s
o
m
e
n
e
w
t
h
eo
r
e
m
s
t
h
at
p
r
o
v
e
th
e
co
n
v
e
r
g
en
ce
o
f
t
h
e
n
e
w
l
y
p
r
o
p
o
s
ed
alg
o
r
ith
m
s
(
S
-
R
A
a
n
d
D
-
R
A
)
.
Sectio
n
4
,
co
n
ce
r
n
s
th
e
n
u
m
er
ical
r
esu
lts
w
h
ic
h
d
e
m
o
n
s
tr
ate
th
e
ef
f
icien
c
y
o
f
t
h
e
n
e
w
l
y
p
r
o
p
o
s
ed
alg
o
r
ith
m
s
w
h
e
n
co
m
p
a
r
ed
to
th
e
s
tan
d
ar
d
(
H
WY
)
a
lg
o
r
ith
m
.
Sectio
n
5
d
ea
ls
w
i
th
g
en
er
al
co
n
c
lu
s
io
n
s
.
2.
T
WO
NE
W
AL
G
O
R
I
T
H
M
S
(S
-
RA
AND
D
-
RA)
I
n
th
i
s
s
ec
tio
n
,
w
e
s
u
g
g
es
t
r
ed
u
cin
g
t
h
e
t
w
o
v
ec
to
r
d
ir
ec
tio
n
s
(
3
)
in
to
a
s
in
g
le
t
h
at
w
i
th
r
ely
in
g
o
n
th
e
p
r
o
j
ec
tio
n
tech
n
iq
u
e
to
f
in
d
th
at
d
ir
ec
tio
n
o
f
r
esear
c
h
.
T
h
is
is
m
ad
e
p
o
s
s
ib
le
b
y
allo
w
i
n
g
t
h
e
t
w
o
d
ir
ec
tio
n
s
to
b
e
id
en
t
ical,
i
.
e
.
=
.
W
e
p
r
o
p
o
s
e
th
at
t
h
e
an
d
in
(
3
)
,
th
e
u
n
iq
u
e
s
ea
r
ch
d
ir
ec
tio
n
is
d
escr
ib
ed
as (
8
)
=
=
−
−
1
(
)
(
8
)
No
w
,
p
u
t (
8
)
in
to
(
3
)
,
w
e
g
et
(
9
)
.
+
1
=
+
(
−
−
1
(
1
+
)
)
(
)
(
9
)
T
h
r
o
u
g
h
(
9
)
,
w
e
ca
n
co
n
clu
d
e
th
at
t
h
e
d
o
u
b
le
-
d
ir
ec
tio
n
w
i
ll
b
ec
o
m
e
(
1
0)
=
−
−
1
(
1
+
)
(
)
(
1
0
)
Set th
e
ac
ce
ler
atio
n
p
ar
a
m
eter
u
s
ed
in
(
1
0
)
as (
1
1
)
,
Evaluation Warning : The document was created with Spire.PDF for Python.
T
E
L
KOM
NI
K
A
T
elec
o
m
m
u
n
C
o
m
p
u
t E
l
C
o
n
tr
o
l
A
n
effec
tive
n
ew iter
a
tive
C
G
-
meth
o
d
to
s
o
lve
u
n
co
n
s
tr
a
in
ed
n
o
n
-
lin
e
a
r
…
(
R
a
n
a
Z
.
Al
-
K
a
w
a
z
)
1849
+
1
=
[
‖
‖
2
−
‖
‖
2
]
‖
(
+
1
)
‖
2
(
1
1
)
w
h
er
e
=
+
1
−
an
d
th
e
d
if
f
er
en
ce
b
etw
ee
n
th
e
t
w
o
-
p
o
in
t
is
=
+
1
−
.
T
h
er
ef
o
r
e,
th
r
o
u
g
h
t
h
e
(
9
)
an
d
(
1
0
)
,
w
e
ca
n
g
et
(
1
2
)
.
+
1
=
+
(
1
2
)
T
h
e
p
r
o
j
ec
ti
o
n
ap
p
r
o
ac
h
r
elie
s
o
n
th
e
u
s
e
o
f
a
m
o
n
o
to
n
e
ca
s
e
F
to
ac
ce
ler
ate
an
d
ch
an
g
e
th
e
n
e
w
p
o
in
t u
s
i
n
g
r
ep
etitio
n
.
A
s
i
n
t
h
e
(
1
3
)
.
=
+
(
1
3
)
T
h
e
h
y
p
er
p
lan
e,
as a
n
o
r
ig
i
n
al
iter
ativ
e,
is
(
1
4
)
.
=
{
∈
|
(
)
(
−
)
=
0
}
(
1
4
)
T
o
s
tar
t
u
s
in
g
t
h
e
p
r
o
j
ec
tio
n
te
ch
n
iq
u
e,
w
e
u
s
e
th
e
u
p
d
ate
o
f
th
e
n
e
w
p
o
i
n
t
+
1
as
g
iv
en
i
n
t
h
e
[
19
]
,
[
20
]
to
b
e
th
e
p
r
o
j
ec
tio
n
o
f
o
n
to
th
e
h
y
p
er
p
la
n
e
.
So
,
ca
n
b
e
ev
alu
a
ted
as:
+
1
=
[
−
(
)
]
(
1
5
)
=
(
)
(
−
)
‖
(
)
‖
2
(
1
6
)
I
n
th
e
n
e
x
t
p
ar
ag
r
ap
h
,
w
e
w
il
l
p
r
esen
t
t
h
e
s
tan
d
ar
d
(
H
WY
)
alg
o
r
ith
m
[
1
2
]
an
d
t
h
e
n
e
w
l
y
p
r
o
p
o
s
ed
alg
o
r
ith
m
s
w
h
ic
h
ar
e
d
iv
id
ed
in
to
t
w
o
p
ar
ts
:
f
ir
s
t
o
f
o
n
e
(S
-
R
A
)
alg
o
r
it
h
m
w
h
ic
h
u
s
e
s
(
1
2
)
,
(
8
)
,
an
d
(
1
1
)
an
d
th
e
s
ec
o
n
d
(D
-
R
A
)
al
g
o
r
ith
m
w
h
ic
h
u
s
es
(
1
2
)
,
(
1
0
)
,
an
d
(
1
1
)
.
T
o
clar
if
y
t
h
e
id
ea
o
f
th
e
n
u
m
er
ical
alg
o
r
it
h
m
s
u
s
ed
in
t
h
i
s
r
esear
ch
,
w
e
p
r
ese
n
t th
e
s
tep
s
o
f
ea
c
h
o
f
t
h
ese
al
g
o
r
ith
m
s
i
n
d
etail
.
2
.
1
.
Alg
o
rit
h
m
(
H
WY
)
[
12
]
I
n
p
u
t: Gi
v
e
n
0
,
0
∈
(
0
,
1
)
,
>
0
,
=
10
−
4
,
1
an
d
2
>
0
,
s
et
k
=0
.
C
o
m
p
u
te
=
(
)
.
T
est th
e
s
to
p
p
in
g
cr
iter
io
n
.
I
f
y
es,
t
h
en
s
to
p
; o
th
er
w
is
e,
co
n
t
in
u
e
to
t
h
e
n
e
x
t s
tep
.
C
o
m
p
u
te
s
ea
r
ch
d
ir
ec
tio
n
u
s
i
n
g
(
1
0
)
.
C
o
m
p
u
te
s
tep
len
g
t
h
u
s
i
n
g
t
h
i
s
lin
e
-
s
ea
r
c
h
:
(
+
)
−
(
)
≤
−
1
‖
‖
2
−
2
‖
‖
2
+
(
)
Set
+
1
=
+
.
C
o
m
p
u
te
(
+
1
)
.
Dete
r
m
i
n
e
+
1
u
s
in
g
+
1
=
‖
‖
2
Set k
=
k
+1
,
an
d
g
o
to
2
.
2
.
2
.
Ne
w
s
ing
le
s
ea
rc
h direc
t
io
n
a
lg
o
rit
h
m
(
S
-
RA)
I
n
p
u
t: Gi
v
e
n
0
∈
Ω
,
0
,
r
,
,
∈
(
0
,
1
)
,
>
0
,
>
0
,
s
et
k
=0
.
C
o
m
p
u
te
=
(
)
an
d
test
I
f
‖
‖
≤
y
es,
t
h
e
n
s
to
p
; o
th
e
r
w
i
s
e,
co
n
ti
n
u
e
to
th
e
n
ex
t
s
tep
.
C
o
m
p
u
te
s
ea
r
ch
d
ir
ec
tio
n
(
u
s
i
n
g
(
8
)
)
.
Set
f
r
o
m
(
1
3
)
an
d
co
m
p
u
te
s
t
ep
len
g
t
h
u
s
in
g
t
h
is
li
n
e
-
s
ea
r
c
h
:
−
(
+
)
≥
‖
(
+
)
‖
‖
‖
2
(
1
7
)
I
f
∈
Ω
an
d
‖
(
)
‖
≤
s
to
p
,
else c
o
m
p
u
te
+
1
f
r
o
m
(
1
2
)
.
Dete
r
m
i
n
e
+
1
u
s
in
g
(
1
1
)
.
Set k
=
k
+1
,
an
d
g
o
to
2
.
Evaluation Warning : The document was created with Spire.PDF for Python.
I
SS
N
:
1
6
9
3
-
6930
T
E
L
KOM
NI
K
A
T
elec
o
m
m
u
n
C
o
m
p
u
t E
l
C
o
n
tr
o
l
,
Vo
l
.
19
,
No
.
6
,
Dec
em
b
er
2
0
2
1
:
1
8
4
7
-
1856
1850
2
.
3
.
Ne
w
do
ub
le
s
ea
rc
h direc
t
io
n
a
lg
o
rit
h
m
(
D
-
RA)
I
n
p
u
t: Gi
v
e
n
0
∈
Ω
,
0
,
r
,
,
∈
(
0
,
1
)
,
>
0
,
>
0
,
s
et
k
=0
.
C
o
m
p
u
te
=
(
)
an
d
test
I
f
‖
‖
≤
y
es,
t
h
e
n
s
to
p
; o
th
er
w
i
s
e,
co
n
ti
n
u
e
to
th
e
n
ex
t
s
tep
.
C
o
m
p
u
te
s
ea
r
ch
d
ir
ec
tio
n
(
u
s
i
n
g
(
1
0
)
)
.
Set
f
r
o
m
(
1
3
)
an
d
co
m
p
u
te
s
t
ep
len
g
t
h
u
s
in
g
t
h
is
li
n
e
-
s
ea
r
c
h
f
r
o
m
(
1
7
)
.
I
f
∈
Ω
an
d
‖
(
)
‖
≤
s
to
p
,
else c
o
m
p
u
te
+
1
f
r
o
m
(
1
2
)
.
Dete
r
m
i
n
e
+
1
u
s
in
g
(
1
1
)
.
Set k
=
k
+1
,
an
d
g
o
to
2
.
3.
C
O
NVER
G
E
NC
E
ANA
L
YS
I
S
I
n
th
e
p
r
ev
io
u
s
s
ec
tio
n
,
w
e
p
r
o
p
o
s
ed
tw
o
n
e
w
al
g
o
r
ith
m
s
(
S
-
R
A
a
n
d
D
-
R
A
)
d
ep
en
d
in
g
o
n
t
h
e
p
ar
am
eter
+
1
.
No
w
in
t
h
is
s
ec
t
io
n
,
w
e
w
ill
p
r
esen
t
an
af
f
i
n
it
y
an
al
y
s
is
f
o
r
th
e
s
ec
o
n
d
alg
o
r
it
h
m
,
w
h
ic
h
is
m
o
r
e
g
e
n
er
al
t
h
a
n
t
h
e
f
ir
s
t
as
in
t
h
e
co
m
i
n
g
t
h
eo
r
e
m
s
,
b
u
t
b
ef
o
r
e
th
a
t,
w
e
m
u
s
t
g
iv
e
t
h
e
b
asic
as
s
u
m
p
tio
n
s
a
s
p
ac
e
o
f
atten
tio
n
w
h
ich
i
s
:
3
.
1
.
Ass
u
m
ptio
n A
Ass
u
m
p
t
io
n
A
m
ea
n
s
t
h
at
t
h
e
s
p
ec
ial
s
o
lu
tio
n
o
f
(
1
)
in
s
tan
d
s
f
o
r
∗
.
Sin
ce
′
(
)
is
ap
p
r
o
x
im
a
ted
b
y
alo
n
g
t
h
e
d
i
r
ec
tio
n
,
w
e
m
i
g
h
t
m
e
n
tio
n
an
o
th
er
ass
u
m
p
tio
n
o
f
t
h
e
s
a
m
e
id
ea
.
Su
p
p
o
s
e
th
er
e
is
a
s
e
t le
v
el
d
e
f
i
n
e
d
b
y
:
=
{
|
‖
(
)
‖
≤
‖
(
0
)
‖
}
T
h
er
e
is
an
∗
th
at
b
elo
n
g
s
to
,
w
h
er
e
(
∗
)
=
0
is
tr
u
e
.
L
et
t
h
e
f
u
n
ctio
n
b
e
d
if
f
er
en
ti
ab
le
an
d
co
n
tin
u
o
u
s
i
n
s
o
m
e
n
eig
h
b
o
r
h
o
o
d
,
th
at
is
,
N
o
f
∗
co
n
tain
ed
i
n
.
On
N,
i
.
e
.
,
th
er
e
is
a
J
ac
o
b
ian
o
f
r
estricte
d
an
d
p
o
s
itiv
e
d
ef
in
ite,
i
.
e
.
th
er
e
ar
e
a
p
o
s
itiv
e
co
n
s
ta
n
ts
M
>
m
>
0
ar
e
s
u
ch
t
h
at:
‖
′
(
)
‖
≤
,
∀
∈
.
(
1
8
)
A
nd
‖
‖
2
≤
′
(
)
,
∀
∈
,
∈
.
(
1
9
)
3
.
2
.
Ass
u
m
ptio
n B
I
f
w
e
co
n
s
id
er
t
h
at
is
a
g
o
o
d
ap
p
r
o
x
im
a
tio
n
o
f
′
(
)
,
w
h
ich
m
ea
n
s
th
at:
‖
(
′
(
)
−
)
‖
≤
‖
(
)
‖
(
2
0
)
w
h
er
e
∈
(
0
,
1
)
is
a
s
m
all
q
u
an
tity
[
6
]
.
3
.
3
.
T
heo
re
m
(
des
ce
nt
direc
t
io
n
)
Su
p
p
o
s
e
ass
u
m
p
tio
n
B
h
o
ld
s
a
n
d
th
at
n
e
w
al
g
o
r
ith
m
(
S
-
R
A
)
an
d
(
D
-
R
A
)
p
r
o
d
u
ce
s
{
}
.
T
h
en
,
in
(
8
)
in
th
e
d
ir
ec
tio
n
o
f
th
e
d
esc
en
t o
f
(
)
at
i
.
e
.
∇
(
)
<
0
(
2
1
)
P
r
o
o
f
:
W
e
w
ill
d
iv
id
e
th
e
p
r
o
o
f
in
to
tw
o
p
ar
ts
,
ea
ch
p
ar
t
co
n
ce
r
n
ed
w
ith
an
alg
o
r
i
th
m
to
ch
an
g
e
th
e
s
ea
r
ch
d
ir
ec
tio
n
in
ea
ch
o
f
th
em
as f
o
llo
w
s
:
P
ar
t 1
: w
h
en
d
ea
lin
g
w
ith
th
e
f
ir
s
t
alg
o
r
ith
m
,
w
e
w
ill
n
ee
d
a
s
ea
r
ch
d
ir
ec
tio
n
f
r
o
m
(
8
)
,
as:
∇
(
)
=
(
)
′
(
)
=
(
)
[
(
′
(
)
−
)
−
(
)
]
∇
(
)
=
(
)
(
′
(
)
−
)
−
‖
(
)
‖
2
(
2
2
)
b
y
C
au
ch
y
-
Sc
h
w
ar
z
in
eq
u
ality
,
w
e
h
av
e:
Evaluation Warning : The document was created with Spire.PDF for Python.
T
E
L
KOM
NI
K
A
T
elec
o
m
m
u
n
C
o
m
p
u
t E
l
C
o
n
tr
o
l
A
n
effec
tive
n
ew iter
a
tive
C
G
-
meth
o
d
to
s
o
lve
u
n
co
n
s
tr
a
in
ed
n
o
n
-
lin
e
a
r
…
(
R
a
n
a
Z
.
Al
-
K
a
w
a
z
)
1851
∇
(
)
≤
‖
(
)
‖
‖
(
′
(
)
−
)
‖
−
‖
(
)
‖
2
(
2
3
)
I
f
(
2
0
)
s
atis
f
y
th
en
,
∇
(
)
≤
‖
(
)
‖
2
−
‖
(
)
‖
2
≤
−
(
1
−
)
‖
(
)
‖
2
(
2
4
)
Hen
ce
f
o
r
∈
(
0
,
1
)
,
th
is
p
r
o
v
es o
f
p
ar
t 1
is
tr
u
e
.
P
ar
t 2
: w
h
en
d
ea
lin
g
w
ith
th
e
s
ec
o
n
d
alg
o
r
ith
m
,
w
e
w
ill
n
ee
d
a
s
ea
r
ch
d
ir
ec
tio
n
f
r
o
m
(
8
)
an
d
(
1
0
)
,
as:
∇
(
)
=
(
)
′
(
)
=
(
)
(
′
(
)
−
)
−
(
1
+
)
‖
(
)
‖
2
(
2
5
)
b
y
C
au
ch
y
-
Sc
h
w
ar
z
in
eq
u
ality
,
w
e
h
av
e:
∇
(
)
≤
‖
(
)
‖
‖
(
′
(
)
−
)
‖
−
(
1
+
)
‖
(
)
‖
2
I
f
(
2
0
)
s
atis
f
y
th
en
,
∇
(
)
≤
‖
(
)
‖
2
−
(
1
+
)
‖
(
)
‖
2
≤
−
(
1
−
+
)
‖
(
)
‖
2
Hen
ce
th
is
p
r
o
v
es
p
ar
t
2
.
T
h
is
m
ea
n
s
th
a
t
th
e
tw
o
n
ew
p
r
o
p
o
s
ed
alg
o
r
ith
m
s
h
av
e
d
escen
t
s
ea
r
ch
d
ir
ec
tio
n
s
.
W
e
ca
n
d
ed
u
ce
f
r
o
m
th
e
th
eo
r
em
(
d
escen
t
d
ir
ec
tio
n
)
th
at
th
e
n
o
r
m
f
u
n
ctio
n
(
)
is
a
d
ec
lin
e
f
o
r
,
w
h
ich
im
p
lies
th
at
‖
(
+
1
)
‖
≤
‖
(
)
‖
≤
.
.
.
≤
‖
(
0
)
‖
.
T
h
is
im
p
lies
th
at
∈
Ω
.
3
.
4
.
L
e
m
m
a
(
bo
un
d
e
d
+
1
)
Su
p
p
o
s
e
th
at
a
s
s
u
m
p
tio
n
A
h
o
ld
s
an
d
{
}
is
g
en
er
ated
b
y
a
n
al
g
o
r
ith
m
(
S
-
R
A
)
a
n
d
(
D
-
R
A
)
.
T
h
en
th
er
e
ex
i
s
ts
a
co
n
s
tan
ts
M >
m
> 0
s
u
ch
t
h
at
f
o
r
all
k
:
[
‖
‖
2
−
‖
‖
2
]
‖
(
+
1
)
‖
2
≤
−
3
2
(
2
6
)
P
r
o
o
f
:
Fro
m
ass
u
m
p
tio
n
A
w
e
g
et:
≥
‖
‖
2
(
2
7
)
f
r
o
m
[
1
8
]
,
w
e
h
av
e:
2
≥
‖
‖
2
(
2
8
)
th
en
,
[
‖
‖
2
−
‖
‖
2
]
≤
[
−
2
]
⇒
[
‖
‖
2
−
‖
‖
2
]
≤
[
1
−
2
]
f
r
o
m
th
e
th
eo
r
em
w
e
h
av
e:
‖
‖
≤
‖
+
1
‖
≤
‖
+
1
−
‖
≤
‖
‖
(
2
9
)
h
en
ce
,
[
‖
‖
2
−
‖
‖
2
]
‖
(
+
1
)
‖
2
≤
[
1
−
2
]
2
‖
‖
2
≤
[
1
−
2
]
2
(
3
0
)
T
h
e
in
eq
u
ality
(
2
6
)
is
tr
u
e
.
Usi
n
g
(
2
6
)
,
+
1
is
g
en
er
ated
b
y
th
e
u
p
d
ate
o
f
(
1
1
)
an
d
w
e
ca
n
d
ed
u
ce
th
at
+
1
I
in
h
er
it
th
e
p
o
s
itiv
e
d
ef
in
iten
ess
o
f
I
.
3
.
5
.
L
e
m
m
a
(
bo
un
de
d
)
Su
p
p
o
s
e
th
at
ass
u
m
p
tio
n
A
an
d
B
h
o
ld
s
an
d
{
}
is
g
en
er
ated
b
y
an
alg
o
r
ith
m
(
S
-
R
A
)
an
d
(
D
-
R
A
)
.
T
h
en
th
er
e
ex
i
s
ts
a
co
n
s
ta
n
t b
>0
s
u
ch
t
h
at
∀
>
0
,
Evaluation Warning : The document was created with Spire.PDF for Python.
I
SS
N
:
1
6
9
3
-
6930
T
E
L
KOM
NI
K
A
T
elec
o
m
m
u
n
C
o
m
p
u
t E
l
C
o
n
tr
o
l
,
Vo
l
.
19
,
No
.
6
,
Dec
em
b
er
2
0
2
1
:
1
8
4
7
-
1856
1852
‖
‖
≤
(
3
1
)
w
h
er
e
i=1
,
2
.
P
r
o
o
f
:
W
e
w
ill
p
r
e
s
en
t
tw
o
p
ar
ts
in
th
is
lem
m
a,
ea
ch
o
f
w
h
ich
d
ep
en
d
s
o
n
th
e
s
ea
r
ch
d
ir
ec
tio
n
r
esu
ltin
g
f
r
o
m
an
alg
o
r
ith
m
as
in
:
P
ar
t 1
: Fr
o
m
(
8
)
,
(
1
1
)
,
an
d
ass
u
m
p
tio
n
A
w
e
h
av
e:
‖
‖
=
‖
−
(
)
‖
‖
2
−
‖
‖
2
‖
(
+
1
)
‖
2
‖
(
3
2
)
an
d
u
s
i
n
g
th
e
r
es
u
lt o
f
L
e
m
m
a
(
bou
n
d
ed
+
1
)
,
‖
‖
≤
[
1
−
2
]
2
‖
(
)
‖
≤
[
1
‖
(
0
)
‖
]
≤
1
(
3
3
)
w
h
er
e
1
>0
an
d
1
=
1
‖
(
0
)
‖
.
P
ar
t 2
: Fr
o
m
(
1
0
)
,
(
1
1
)
,
an
d
ass
u
m
p
tio
n
A
w
e
h
av
e:
‖
‖
=
‖
−
(
1
+
)
(
)
‖
‖
2
−
‖
‖
2
‖
(
+
1
)
‖
2
‖
(
3
4
)
an
d
u
s
in
g
th
e
r
esu
lt
o
f
L
em
m
a
(
b
o
u
n
d
ed
ψ
k
+
1
)
,
‖
‖
≤
(
1
+
)
[
1
−
2
]
2
‖
(
)
‖
≤
[
(
‖
(
0
)
‖
+
‖
(
)
‖
)
[
1
−
2
]
2
]
(
3
5
)
‖
‖
≤
[
(
‖
(
0
)
‖
+
2
)
[
1
−
2
]
2
]
≤
2
(
3
6
)
w
h
er
e
2
>0
an
d
2
=
(
‖
(
0
)
‖
+
2
)
[
1
−
2
]
2
.
T
h
e
f
o
llo
w
in
g
th
eo
r
em
d
ea
ls
w
ith
th
e
g
lo
b
al
co
n
v
er
g
en
ce
p
r
o
p
er
ty
.
T
o
p
r
o
v
e
th
a
t
u
n
d
er
a
f
ew
s
u
itab
le
co
n
d
itio
n
s
,
th
er
e
ex
is
t
an
ac
cu
m
u
latio
n
p
o
in
t
o
f
w
h
ich
is
a
s
o
lu
tio
n
to
th
e
p
r
o
b
lem
(
1
)
.
3
.
6
.
T
heo
re
m
(
g
lo
ba
l c
o
nv
er
g
ence
)
Su
p
p
o
s
e
th
at
ass
u
m
p
tio
n
B
h
o
ld
s
,
{
}
is
g
en
er
ated
b
y
an
alg
o
r
ith
m
(
S
-
R
A
)
an
d
(
D
-
R
A
)
.
Ass
u
m
e
fu
r
t
h
er
∀
>
0
,
≥
|
(
)
|
‖
‖
2
(
3
7
)
w
h
er
e
is
s
o
m
e
p
o
s
itiv
e
co
n
s
tan
t
.
T
h
en
→
∞
‖
(
)
‖
=
0
(
3
8
)
P
r
o
o
f
:
Fro
m
(
3
1
)
,
an
d
(
Descen
t D
ir
ec
tio
n
T
h
eo
r
em
)
w
e
h
av
e:
→
∞
‖
‖
=
0
(
3
9
)
an
d
th
e
b
o
u
n
d
ed
o
f
‖
‖
,
w
e
h
av
e:
→
∞
‖
‖
2
=
0
(
4
0
)
Fro
m
(
3
7
)
an
d
(
4
0
)
it f
o
llo
w
s
th
at:
Evaluation Warning : The document was created with Spire.PDF for Python.
T
E
L
KOM
NI
K
A
T
elec
o
m
m
u
n
C
o
m
p
u
t E
l
C
o
n
tr
o
l
A
n
effec
tive
n
ew iter
a
tive
C
G
-
meth
o
d
to
s
o
lve
u
n
co
n
s
tr
a
in
ed
n
o
n
-
lin
e
a
r
…
(
R
a
n
a
Z
.
Al
-
K
a
w
a
z
)
1853
→
∞
|
(
)
|
=
0
(
4
1
)
I
n
th
is
s
tag
e
o
f
th
e
p
r
o
o
f
,
w
e
w
ill
tak
e
tw
o
p
ar
ts
ac
c
o
r
d
in
g
to
th
e
tw
o
n
ew
alg
o
r
ith
m
s
as in
:
P
ar
t 1
: A
cc
o
r
d
in
g
to
th
e
(
S
-
R
A
)
alg
o
r
ith
m
an
d
f
r
o
m
(
8
)
,
w
e
h
av
e:
(
)
=
−
−
1
‖
(
)
‖
2
⟹
|
|
|
(
)
|
=
‖
(
)
‖
2
an
d
as w
e
im
p
o
s
ed
in
th
e
th
eo
r
em
:
|
|
1
‖
‖
2
≥
‖
(
)
‖
2
(
4
2
)
W
h
ile
|
|
=
[
‖
−
1
‖
2
−
‖
−
1
‖
2
]
‖
(
)
‖
2
≤
[
1
−
2
]
2
≤
s
o
,
f
r
o
m
th
e
(
4
2
)
,
th
en
0
←
‖
‖
2
≥
‖
(
)
‖
2
≥
0
(
4
3
)
th
er
ef
o
r
e,
in
(
4
2
)
is
tr
u
e
an
d
th
e
p
r
o
o
f
f
o
r
p
ar
t 1
is
co
m
p
leted
.
P
ar
t 2
: A
cc
o
r
d
in
g
to
th
e
(
D
-
R
A
)
alg
o
r
ith
m
an
d
u
s
in
g
(
1
0
)
,
w
e
h
av
e:
(
)
=
−
−
1
(
1
+
)
‖
(
)
‖
2
‖
(
)
‖
2
=
‖
−
(
)
‖
−
‖
(
)
‖
2
≤
|
|
|
(
)
|
(
4
4
)
w
h
ile
|
|
≤
as in
p
ar
t 1
,
s
o
f
r
o
m
th
e
(
5
0
)
,
th
en
0
←
|
(
)
|
≥
‖
(
)
‖
2
≥
0
(
4
5
)
th
er
ef
o
r
e,
th
e
(
3
8
)
is
tr
u
e
an
d
th
e
p
r
o
o
f
f
o
r
p
ar
t
2
is
co
m
p
leted
.
4.
NUM
E
RICAL
P
E
RF
O
RM
ANCE
I
n
th
is
s
ec
tio
n
,
w
e
w
ill
p
r
esen
t
o
u
r
n
u
m
er
ical
r
esu
lts
f
o
r
co
m
p
ar
is
o
n
s
b
etw
ee
n
th
e
tw
o
n
ew
p
r
o
p
o
s
ed
alg
o
r
ith
m
s
(
S
-
R
A
)
an
d
(
D
-
R
A
)
an
d
th
e
s
tan
d
ar
d
(
H
WY
)
alg
o
r
ith
m
w
h
ich
is
d
ev
o
id
o
f
th
e
d
er
iv
ativ
e
to
s
o
lv
e
ce
r
tain
n
o
n
lin
ea
r
test
p
r
o
b
lem
s
.
I
n
o
u
r
im
p
lem
en
tin
g
all
th
r
ee
alg
o
r
ith
m
s
,
w
e
u
s
ed
th
e
Ma
tlab
R
2
0
1
8
b
p
r
o
g
r
am
in
a
lap
to
p
ca
lcu
lato
r
w
ith
its
C
o
r
ei5
s
p
ec
if
icatio
n
s
.
A
s
f
o
r
th
e
to
o
ls
u
s
ed
in
th
e
tw
o
alg
o
r
ith
m
s
,
th
ey
ar
e
as
f
o
llo
w
s
:
0
=
0
.
6
1
,
=
0
.
9
,
=
0
.
02
,
=
1
,
1
=
2
=
10
−
4
,
‖
(
)
‖
<
10
−
8
.
T
h
e
p
r
o
g
r
am
f
in
d
s
th
e
r
esu
lts
o
n
s
ev
er
al
n
o
n
-
d
er
iv
ativ
e
f
u
n
ctio
n
s
th
r
o
u
g
h
s
ev
er
al
tw
o
in
itial
p
o
in
ts
in
d
icate
d
in
th
e
T
ab
les 1
an
d
2
.
T
ab
le
1
.
T
h
e
in
itial p
o
in
ts
N
a
me
o
f
V
a
r
i
a
b
l
e
I
n
i
t
i
a
l
p
o
i
n
t
1
(
1
,
1
,
1
,
.
.
,
1
)
2
(
0
.
2
,
0
.
2
,
0
.
2
,
.
.
,
0
.
2
)
3
(
20
,
20
,
20
,
.
.
,
20
)
4
(
,
,
,
.
.
,
)
T
h
ese
alg
o
r
ith
m
s
w
e
im
p
lem
en
ted
w
ith
in
d
im
en
s
io
n
s
n
(
1
0
0
0
,
2
0
0
0
,
5
0
0
0
,
7
0
0
0
,
1
2
0
0
0
)
.
A
ll
s
u
ch
alg
o
r
ith
m
s
ar
e
r
ec
o
g
n
ized
b
y
th
eir
p
er
f
o
r
m
an
ce
in
(
I
ter
)
th
e
n
u
m
b
er
o
f
iter
atio
n
s
,
(
E
v
al
-
F)
th
e
n
u
m
b
er
o
f
ev
alu
atio
n
s
o
f
f
u
n
ctio
n
s
,
(
T
im
e)
in
s
ec
o
n
d
C
P
U
tim
e,
(
No
r
m
)
ap
p
r
o
x
im
atio
n
s
o
lu
tio
n
n
o
r
m
.
T
h
e
test
p
r
o
b
lem
s
(
)
=
(
1
,
2
,
3
,
.
.
.
,
)
w
h
e
r
e
=
(
1
,
2
,
3
,
…
,
)
,
f
o
r
=
1
,
2
,
.
.
.
,
an
d
=
+
ar
e
f
r
o
m
[
2
1
]
-
[
2
4
]
an
d
lis
ted
as
s
h
o
w
n
in
T
ab
le
2
.
Usi
n
g
Do
lan
an
d
Mo
r
´
e
s
ty
le
[
2
5
]
,
th
e
Fig
u
r
es
1
-
3
ar
e
u
s
ed
f
o
r
co
m
p
ar
is
o
n
b
etw
ee
n
th
e
(
HW
Y)
w
ith
(
S
-
R
A
)
an
d
(
D
-
R
A
)
alg
o
r
ith
m
s
w
h
en
s
w
itch
in
g
th
e
s
ea
r
ch
d
ir
ec
tio
n
.
T
h
e
Fig
u
r
es
1
-
3
ar
e
ab
o
u
t th
e
in
itial
p
o
in
t 1
b
ec
au
s
e
it is
th
e
b
est p
er
f
o
r
m
an
ce
.
Evaluation Warning : The document was created with Spire.PDF for Python.
I
SS
N
:
1
6
9
3
-
6930
T
E
L
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NI
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elec
o
m
m
u
n
C
o
m
p
u
t E
l
C
o
n
tr
o
l
,
Vo
l
.
19
,
No
.
6
,
Dec
em
b
er
2
0
2
1
:
1
8
4
7
-
1856
1854
T
ab
le
2
.
Def
in
e
th
e
p
r
o
b
le
m
s
No
.
P
r
o
b
l
e
ms
1
(
)
=
−
2
(
)
=
−
1
3
(
)
=
√
(
1
−
1
)
,
=
2
,
3
,
.
.
,
−
1
.
(
)
=
1
4
∑
2
=
1
−
1
/
4
,
=
1
∗
10
−
5
4
(
)
=
(
|
|
+
1
)
−
5
(
)
=
(
(
|
|
,
2
)
,
(
|
|
,
3
)
)
6
1
(
)
=
1
−
(
1
+
2
)
+
1
(
)
=
−
(
+
1
+
+
−
1
)
+
1
,
=
2
,
3
,
.
.
,
−
1
(
)
=
−
(
−
1
+
)
+
1
7
(
)
=
−
1
8
1
(
)
=
1
−
1
(
)
=
−
−
1
−
1
9
(
)
=
∑
|
|
=
1
10
(
)
=
∑
|
|
=
1
11
(
)
=
=
1
,
.
.
,
|
|
12
(
)
=
∑
|
|
=
1
−
∑
(
2
)
=
1
13
(
)
=
∑
|
|
+
1
=
1
(
a)
(
b
)
Fig
u
r
e
1
.
P
er
f
o
r
m
a
n
ce
o
f
iter
a
tio
n
s
f
o
r
th
e
(
S
-
R
A
a
n
d
D
-
R
A
v
s
.
HW
Y)
alg
o
r
ith
m
s
:
(
a)
S
-
R
A
an
d
HW
Y
an
d
(
b
)
D
-
R
A
an
d
HW
Y
Evaluation Warning : The document was created with Spire.PDF for Python.
T
E
L
KOM
NI
K
A
T
elec
o
m
m
u
n
C
o
m
p
u
t E
l
C
o
n
tr
o
l
A
n
effec
tive
n
ew iter
a
tive
C
G
-
meth
o
d
to
s
o
lve
u
n
co
n
s
tr
a
in
ed
n
o
n
-
lin
e
a
r
…
(
R
a
n
a
Z
.
Al
-
K
a
w
a
z
)
1855
(
a)
(
b
)
Fig
u
r
e
2
.
P
er
f
o
r
m
a
n
ce
o
f
f
u
n
c
tio
n
ev
al
u
atio
n
s
f
o
r
th
e
(
S
-
R
A
an
d
D
-
R
A
v
s
.
H
WY
)
alg
o
r
ith
m
s
:
(
a)
S
-
R
A
a
n
d
HW
Y
a
n
d
(
b
)
D
-
R
A
an
d
HW
Y
(
a)
(
b
)
Fig
u
r
e
3
.
P
er
f
o
r
m
a
n
ce
o
f
ti
m
e
f
o
r
th
e
(
S
-
R
A
a
n
d
D
-
R
A
v
s
.
HW
Y
)
alg
o
r
ith
m
s
:
(
a)
S
-
R
A
a
n
d
HW
Y
an
d
(
b
)
D
-
R
A
an
d
HW
Y
5.
CO
NCLU
SI
O
NS
T
h
e
r
esu
lts
,
p
r
esen
ted
in
t
h
e
s
ix
f
ig
u
r
es
s
h
o
w
t
h
e
e
f
f
icien
c
y
o
f
th
e
t
w
o
n
e
w
al
g
o
r
ith
m
s
(
S
-
R
A
)
a
n
d
(D
-
R
A
)
w
h
e
n
co
m
p
ar
ed
w
it
h
th
e
p
r
ev
io
u
s
s
tan
d
ar
d
(
H
WY
)
alg
o
r
ith
m
,
a
n
d
th
eir
e
f
f
icie
n
c
y
is
b
etter
b
y
ta
k
i
n
g
th
e
f
ir
s
t
in
itial
p
o
in
t
a
n
d
in
cr
e
ase
w
h
e
n
in
cr
ea
s
in
g
t
h
e
d
i
m
e
n
s
io
n
s
i
n
th
e
v
ar
iab
les
u
s
ed
.
T
h
e
n
e
w
al
g
o
r
ith
m
s
h
av
e
g
i
v
en
a
clea
r
co
n
v
er
g
e
n
c
e
in
r
ea
ch
i
n
g
t
h
e
o
p
ti
m
al
p
o
in
t
f
o
r
s
o
lv
i
n
g
n
o
n
-
li
n
ea
r
f
u
n
ctio
n
s
.
ACK
NO
WL
E
DG
E
M
E
NT
S
T
h
e
r
esear
ch
is
s
u
p
p
o
r
ted
b
y
th
e
C
o
lleg
e
o
f
C
o
m
p
u
ter
Scien
ce
s
an
d
Ma
th
em
atics,
Un
iv
er
s
ity
o
f
Mo
s
u
l,
an
d
C
o
lleg
e
o
f
B
asic
E
d
u
ca
tio
n
,
Un
iv
er
s
ity
o
f
T
elaf
er
,
R
ep
u
b
lic
o
f
I
r
aq
.
Th
e
au
th
o
r
s
d
ec
lar
e
th
at
th
er
e
ar
e
n
o
co
n
f
licts
o
f
in
ter
est
r
eg
ar
d
in
g
th
is
w
o
r
k
.
RE
F
E
R
E
NC
E
S
[
1
]
R
.
Z
.
Al
-
Ka
w
a
z
,
A
.
Y
.
Al
-
Ba
y
a
ti
,
a
n
d
M
.
S
.
Ja
m
e
e
l,
"
In
tera
c
ti
o
n
b
e
tw
e
e
n
u
n
-
u
p
d
a
ted
F
R
-
CG
a
lg
o
rit
h
m
s
w
it
h
a
n
o
p
ti
m
a
l
Cu
c
k
o
o
a
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o
rit
h
m
,
"
In
d
o
n
e
sia
n
J
o
u
rn
a
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o
f
El
e
c
trica
l
En
g
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e
rin
g
a
n
d
Co
mp
u
ter
S
c
ien
c
e
,
v
o
l
.
1
9
,
n
o
.
3
,
pp
.
1497
-
1
5
0
4
,
2
0
2
0
,
d
o
i:
1
0
.
1
1
5
9
1
/i
jee
c
s
.
v
1
9
.
i3
.
pp1497
-
1504
.
[
2
]
A
.
A
.
Al
-
A
rb
o
a
n
d
R
.
Z
.
Al
-
Ka
w
a
z
,
"
Im
p
lem
e
n
tatio
n
o
f
a
c
o
m
b
in
e
d
n
e
w
o
p
ti
m
a
l
c
u
c
k
o
o
a
lg
o
rit
h
m
w
it
h
a
g
ra
y
w
o
lf
a
lg
o
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h
m
to
so
lv
e
u
n
c
o
n
stra
in
e
d
o
p
ti
m
iza
ti
o
n
n
o
n
li
n
e
a
r
p
ro
b
lem
s,"
In
d
o
n
e
sia
n
J
o
u
rn
a
l
o
f
El
e
c
trica
l
En
g
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e
e
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a
n
d
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ter
S
c
ien
c
e
,
v
o
l
.
1
9
,
n
o
3
,
p
p
.
1582
-
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5
8
9
,
2
0
2
0
,
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o
i:
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.
1
1
5
9
1
/i
jee
c
s
.
v
1
9
.
i3
.
pp1582
-
1589
.
Evaluation Warning : The document was created with Spire.PDF for Python.
I
SS
N
:
1
6
9
3
-
6930
T
E
L
KOM
NI
K
A
T
elec
o
m
m
u
n
C
o
m
p
u
t E
l
C
o
n
tr
o
l
,
Vo
l
.
19
,
No
.
6
,
Dec
em
b
er
2
0
2
1
:
1
8
4
7
-
1856
1856
[
3
]
J
.
E
.
De
n
n
is
a
n
d
R
.
B
.
S
c
h
n
a
b
e
l,
Nu
me
ric
a
l
me
th
o
d
s
fo
r
u
n
c
o
n
st
ra
in
e
d
o
p
ti
miza
ti
o
n
a
n
d
n
o
n
li
n
e
a
r
e
q
u
a
ti
o
n
s
,
P
h
il
a
d
e
lp
h
ia:
S
IA
M
,
1996
,
d
o
i:
1
0
.
1
1
3
7
/1
.
9781611971200
.
[
On
li
n
e
]
.
A
v
a
il
a
b
le:
h
tt
p
s://
e
p
u
b
s
.
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m
.
o
rg
/d
o
i/
p
d
f
/1
0
.
1
1
3
7
/1
.
9781611971200
.
fm
[
4
]
M
.
W
.
Yu
su
f
,
L
.
W
.
Ju
n
e
,
a
n
d
M
.
A
.
Ha
ss
a
n
,
"
Ja
c
o
b
ian
-
f
re
e
d
iag
o
n
a
l
Ne
w
to
n
’s
m
e
th
o
d
f
o
r
so
lv
in
g
n
o
n
li
n
e
a
r
sy
ste
m
s
w
it
h
sin
g
u
lar
Ja
c
o
b
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T
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