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AdvancesinAppliedMathematics
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,2022,11(12),9081-9095
PublishedOnlineDecember2022inHans.http://www.hanspub.org/journal/aam
https://doi.org/10.12677/aam.2022.1112958
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ParameterEstimationand
OutliersDetectionBasedon
NonconvexPenalized
Regression
ZunhaoZhang,DingtaoPeng
∗
,YanyanSu
SchoolofMathematicsandStatistics,GuizhouUniversity,GuiyangGuizhou
Received:Nov.26
th
,2022;accepted:Dec.21
st
,2022;published:Dec.30
th
,2022
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9081-9095.DOI:10.12677/aam.2022.1112958
Ü
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Abstract
Thispaperpresentsamethodtoimplementparameterestimationandoutliersde-
tectionformultiplelinearregressionmodelsbasedonoptimizationtheory.First,a
regressionmodelbasedontheHuberlossfunctionandthe
`
0
penaltytermisdevel-
oped,andthe
`
0
penaltyterminthismodelisfurtherrelaxedtotheCapped-
`
1
penalty
tofacilitatethesolution;andsecond,thedirectionalstabilityp ointoftherelaxation
problemischaracterized,andtheequivalenceoftheoriginalandrelaxationproblems
isestablished.Finally,weproposeasmoothmodelfortherelaxationproblemand
provetheconsistencyofthestablepointofsmoothmodelandtherelaxationproblem.
Keywords
HuberFunction,RobustParameterEstimation,OutliersDetection,Nonconvex
Penalty,SmoothingMethod
Copyright
c
2022byauthor(s)andHansPublishersInc.
This work is licensed undertheCreative Commons Attribution InternationalLicense(CCBY4.0).
http://creativecommons.org/licenses/by/4.0/
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•
•
(
t
−
t
∗
)
•
•
ê
:
ϕ
0
(
t
∗
;
t
−
t
∗
) =
|
t
|
v
,t
∗
= 0
,
sgn(
t
∗
)(
t
−
t
∗
)
v
,
|
t
∗
|∈
(0
,v
)
,
min
{
0
,
sgn(
t
∗
)(
t
−
t
∗
)
v
}
,
|
t
∗
|
=
v,
0
,
otherwise
,
∀
t
∈
R
.
y
²
.
Ï
•
Huber
¼
ê´
Œ
‡
¼
ê
,
¤
±
H
0
(
x
∗
,e
∗
;
x
−
x
∗
,e
−
e
∗
)=
h∇
x
∗
H
(
x
∗
,e
∗
)
,x
−
x
∗
i
+
h∇
e
∗
H
(
x
∗
,e
∗
)
,e
−
e
∗
i
=
1
n
n
X
i
=1
h
0
α
(
a
T
i
x
∗
+
e
∗
i
−
y
i
)
a
T
i
(
x
−
x
∗
)+
1
n
n
X
i
=1
h
0
α
(
a
T
i
x
∗
+
e
∗
i
−
y
i
)(
e
i
−
e
∗
i
)
=
1
n
n
X
i
=1
h
0
α
(
a
T
i
x
∗
+
e
∗
i
−
y
i
)[
a
T
i
(
x
−
x
∗
)+(
e
i
−
e
∗
i
)]
.
(11)
DOI:10.12677/aam.2022.11129589085
A^
ê
Æ
?
Ð
Ü
ƒ
Ê
Š
â
•
•
ê
½
Â
Ú¼
ê
Φ
1
(
x
)
,
Φ
2
(
e
)
Œ
©
5
,
Φ
0
1
(
x
∗
;
x
−
x
∗
) =
p
X
j
=1
ϕ
0
(
x
∗
j
;
x
j
−
x
∗
j
)
,
Φ
0
2
(
e
∗
;
e
−
e
∗
) =
n
X
i
=1
ϕ
0
(
e
∗
i
;
e
i
−
e
∗
i
)
,
Ù
¥
,
Š
â
•
•
ê
½
Â
Ú
ϕ
(
t
) = min
{
1
,
|
t
|
v
}
=
|
t
|
v
,
|
t
|≤
v,
1
,
|
t
|
>v,
Œ
±
ϕ
0
(
t,t
−
t
∗
)
L
ˆ
ª
.
y
.
.
Peng
<
[21–23]
y
²
•
•
-
½:
ä
k
X
e
Û
Ü
•
`
5
Ÿ
:
Ú
n
2.1
¼
ê
f
:
R
n
→
R
3
:
x
∗
∈
R
n
?
´
Û
Ü
Lipschitz
ë
Y
…
•
•
Œ
‡
,
K
(1)
e
x
∗
´
¼
ê
f
Û
Ü
4
Š
:
,
K
x
∗
´
¼
ê
f
•
•
-
½:
¶
(2)
x
∗
´
î
‚
Û
Ü
4
Š
:
¿
÷
v
˜
O
•
5
^
‡
,
=
•
3
x
∗
•
W
Ú
δ>
0
,
¦
f
(
x
)
≥
f
(
x
∗
)+
δ
k
x
−
x
∗
k
,
∀
x
∈W
,
…
=
x
∗
÷
v
f
0
(
x
∗
;
x
−
x
∗
)
>
0
,
∀
x
∈
R
n
\{
x
∗
}
.
Š
â
Ú
n
2.1,
Ï
•
t
µ
¯
K
(8)
8
I
¼
ê´
Û
Ü
Lipschitz
ë
Y
…
•
•
Œ
‡
,
Ï
d
x
∗
´
¯
K
(8)
Û
Ü
4
Š
7
‡
^
‡
´
x
∗
´
¯
K
(8)
•
•
-
½:
.
3.
e
.
5
Ÿ
9
)
d
5
É
©
[21–24]
é
u
,
!
‰
Ñ
t
µ
¯
K
(8)
•
•
-
½:
e
.
5
Ÿ
,
¿
y
²
3
˜
½
^
‡
e
¯
K
(7)
Ú
t
µ
¯
K
(8)
)
d
.
3.1.
)
e
.
5
Ÿ
Ä
k
½
Â
X
e
•
I
8
:
P
1
(
x
) =
{
j
: 0
<
|
x
j
|
<v
}
,
P
2
(
x
) =
{
j
:
|
x
j
|≥
v
}
,
P
1
(
x
)
∪
P
2
(
x
) =
{
j
∈{
1
,...,p
}
:
x
j
6
= 0
}
,
P
3
(
e
) =
{
i
: 0
<
|
e
i
|
<v
}
,
P
4
(
e
) =
{
i
:
|
e
i
|≥
v
}
,
P
3
(
e
)
∪
P
4
(
e
) =
{
i
∈{
1
,...,n
}
:
e
i
6
= 0
}
.
½
Â
¼
ê
M
:
R
p
+
n
→
R
X
e
:
M
(
x,e
) = max
(
1
λ
1
n
n
X
i
=1
h
0
α
(
a
T
i
x
+
e
i
−
y
i
)
a
i
1
,
1
λ
2
n
n
X
i
=1
h
0
α
(
a
T
i
x
+
e
i
−
y
i
)
)
.
(12)
DOI:10.12677/aam.2022.11129589086
A^
ê
Æ
?
Ð
Ü
ƒ
Ê
e
¡
½
n
L
²
:
3
˜
½
^
‡
e
,
t
µ
¯
K
(8)
•
•
-
½:
š
"
©
þ
ä
k
˜
—
e
.
.
½
n
3.1
e
(
x
∗
,e
∗
)
∈
R
p
+
n
´
t
µ
¯
K
(8)
•
•
-
½:
,
…
÷
v
M
(
x
∗
,e
∗
)
<
1
v
,
K
(i)
‡
o
|
x
∗
j
|≥
v
,
‡
o
|
x
∗
j
|
= 0
,j
∈{
1
,...,p
}
;
(ii)
‡
o
|
e
∗
i
|≥
v
,
‡
o
|
e
∗
i
|
= 0
,i
∈{
1
,...,n
}
.
y
²
.
(
x
∗
,e
∗
)
∈
R
p
+
n
(
x
∗
j
6
=0,
∀
j
;
e
∗
i
6
=0,
∀
i
)
´
t
µ
¯
K
(8)
•
•
-
½:
.
P
x
∗
=
(
x
∗
1
,...,x
∗
p
)
>
,
e
∗
= (
e
∗
1
,...,e
∗
n
)
>
.
•
I
y
²
P
1
(
x
∗
) =
∅
Ú
P
3
(
e
∗
) =
∅
.
e
¡
^
‡
y
{
y
²
.
b
P
1
(
x
∗
)
6
=
∅
½
P
3
(
e
∗
)
6
=
∅
.
(i)
b
P
1
(
x
∗
)
6
=
∅
.
d
½
n
2.1,
e
=
e
∗
,
K
1
n
n
X
i
=1
h
0
α
(
a
T
i
x
∗
+
e
∗
i
−
y
i
)
a
T
i
(
x
−
x
∗
)+
λ
1
p
X
j
=1
ϕ
0
(
x
∗
j
;
x
j
−
x
∗
j
)
≥
0
,
∀
x
∈
R
p
.
(13)
∀
i
= 1
,...,p
,
P
a
i
= (
a
∗
i
1
,...,a
∗
ip
)
>
,
K
1
n
n
X
i
=1
h
0
α
(
a
T
i
x
∗
+
e
∗
i
−
y
i
)
a
T
i
(
x
−
x
∗
) =
1
n
n
X
i
=1
p
X
j
=1
h
0
α
(
a
T
i
x
∗
+
e
∗
i
−
y
i
)
a
ij
(
x
j
−
x
∗
j
)
.
(14)
∀
ε
1
∈
(0
,
1),
b
x
= (
b
x
1
,...,
b
x
p
)
>
∈
R
p
÷
v
b
x
j
=
(1
−
ε
1
)
x
∗
j
,j
∈
P
1
(
x
∗
)
,
x
∗
j
,j
∈
P
2
(
x
∗
)
.
du
P
1
(
x
∗
)
6
=
∅
,
(
b
x
−
x
∗
)
Ø
´
"
•
þ
.
du
(13)
é
?
¿
x
∈
R
p
þ
¤
á
,
K
n
X
i
=1
h
0
α
(
a
T
i
x
∗
+
e
∗
i
−
y
i
)
a
T
i
(
b
x
−
x
∗
)=
n
X
i
=1
p
X
j
=1
h
0
α
(
a
T
i
x
∗
+
e
∗
i
−
y
i
)
a
ij
(
b
x
j
−
x
∗
j
)
=
ε
1
n
X
i
=1
X
j
∈
P
1
(
x
∗
)
h
0
α
(
a
T
i
x
∗
+
e
∗
i
−
y
i
)
a
ij
(
−
x
∗
j
)
≤
ε
1
n
X
i
=1
h
0
α
(
a
T
i
x
∗
+
e
∗
i
−
y
i
)
a
i
1
·
X
j
∈
P
1
(
b
x
)
|
x
∗
j
|
.
(15)
,
˜
•
¡
,
p
X
j
=1
ϕ
0
(
x
∗
j
;
b
x
j
−
x
∗
j
) =
X
j
∈
P
1
(
b
x
)
sgn
(
x
∗
j
)(
−
ε
1
x
∗
j
)
v
=
−
ε
1
X
j
∈
P
1
(
b
x
)
sgn
(
x
∗
j
)(
x
∗
j
)
v
=
−
ε
1
v
X
j
∈
P
1
(
b
x
)
|
x
∗
j
|
.
(16)
DOI:10.12677/aam.2022.11129589087
A^
ê
Æ
?
Ð
Ü
ƒ
Ê
d
(13)-(16),
λ
1
ε
1
v
X
j
∈
P
1
(
b
x
)
|
x
∗
j
|≤
ε
1
n
n
X
i
=1
h
0
α
(
a
T
i
x
∗
+
e
∗
i
−
y
i
)
a
i
1
X
j
∈
P
1
(
b
x
)
|
x
∗
j
|
,
du
P
j
∈
P
1
(
x
∗
)
|
x
∗
j
|6
= 0,
ò
Ù
ž
1
v
≤
1
λ
1
n
n
X
i
=1
h
0
α
(
a
T
i
x
∗
+
e
∗
i
−
y
i
)
a
i
1
≤
M
(
x
∗
,e
∗
)
,
ù
†
M
(
x
∗
,e
∗
)
<
1
v
g
ñ
,
l
P
1
(
x
∗
) =
∅
.
(ii)
b
P
3
(
e
∗
)
6
=
∅
.
d
½
n
2.1,
x
=
x
∗
,
K
1
n
n
X
i
=1
h
0
α
(
a
T
i
x
∗
+
e
∗
i
−
y
i
)(
e
i
−
e
∗
i
)+
λ
2
n
X
i
=1
ϕ
0
(
e
∗
i
;
e
i
−
e
∗
i
)
≥
0
,
∀
e
∈
R
n
.
(17)
∀
ε
2
∈
(0
,
1),
b
e
= (
b
e
1
,...,
b
e
n
)
>
∈
R
n
÷
v
b
e
i
=
(1
−
ε
2
)
e
∗
i
,i
∈
P
3
(
e
∗
)
,
e
∗
i
,i
∈
P
4
(
e
∗
)
.
du
P
3
(
e
∗
)
š
˜
,
(
b
e
−
e
∗
)
Ø
´
"
•
þ
,
?
n
X
i
=1
h
0
α
(
a
T
i
x
∗
+
e
∗
i
−
y
i
)(
b
e
i
−
e
∗
i
)=
ε
2
X
i
∈
P
3
(
e
∗
)
h
0
α
(
a
T
i
x
∗
+
e
∗
i
−
y
i
)
e
∗
i
≤
ε
2
X
i
∈
P
3
(
e
∗
)
|
h
0
α
(
a
T
i
x
∗
+
e
∗
i
−
y
i
)
|·|
e
∗
i
|
.
(18)
,
˜
•
¡
,
n
X
i
=1
ϕ
0
(
e
∗
i
;
b
e
i
−
e
∗
i
) =
X
i
∈
P
3
(
e
∗
)
ϕ
0
(
e
∗
i
;
b
e
i
−
e
∗
i
) =
−
ε
2
v
X
i
∈
P
3
(
e
∗
)
sgn
(
e
∗
i
)(
e
∗
i
) =
−
ε
2
v
X
i
∈
P
3
(
e
∗
)
|
e
∗
i
|
.
(19)
d
(17)-(19),
λ
2
ε
2
v
X
i
∈
P
3
(
e
∗
)
|
e
∗
i
|≤
ε
2
n
X
i
∈
P
3
(
e
∗
)
|
h
0
α
(
a
T
i
x
∗
+
e
∗
i
−
y
i
)
|·|
e
∗
i
|≤
ε
2
n
n
X
i
=1
|
h
0
α
(
a
T
i
x
∗
+
e
∗
i
−
y
i
)
|
X
i
∈
P
3
(
e
∗
)
|
e
∗
i
|
,
1
v
≤
1
λ
2
n
n
X
i
=1
|
h
0
α
(
a
T
i
x
∗
+
e
∗
i
−
y
i
)
|
.
du
P
i
∈
P
3
(
e
∗
)
|
e
∗
i
|6
= 0,
ò
Ù
ž
1
v
≤
M
(
x
∗
,e
∗
),
†
b
g
ñ
,
P
3
(
e
∗
) =
∅
.
y
.
.
DOI:10.12677/aam.2022.11129589088
A^
ê
Æ
?
Ð
Ü
ƒ
Ê
3.2.
¯
K
(7)
†
(8)
)
d
5
e
¡
/
Ï
½
n
3.1
ï
Ä
¯
K
(7)
†
t
µ
¯
K
(8)
Û
•
`
)
ƒ
m
'
X
.
½
n
3.2
(
x
∗
,e
∗
)
∈
R
p
+
n
÷
v
M
(
x
∗
,e
∗
)
<
1
v
,
K
(
x
∗
,e
∗
)
´
t
µ
¯
K
(8)
Û
•
`
)
…
=
(
x
∗
,e
∗
)
´
¯
K
(7)
Û
•
`
)
.
y
²
.
(1)
(
x
∗
,e
∗
)
∈
R
p
+
n
´
t
µ
¯
K
(8)
Û
•
`
)
,
d
Ú
n
2.1,(
x
∗
,e
∗
)
´
t
µ
¯
K
(8)
•
•
-
½:
.
d
½
n
3.1,Φ
1
(
x
∗
)=
k
x
∗
k
0
,Φ
2
(
e
∗
)=
k
e
∗
k
0
.
5
¿
,
∀
(
x,e
)
∈
R
p
+
n
,
o
k
Φ
1
(
x
)
≤
k
x
k
0
,
Φ
2
(
e
)
≤k
e
k
0
.
l
,
∀
(
x,e
)
∈
R
p
+
n
,
k
H
(
x
∗
,e
∗
)+
λ
1
k
x
∗
k
0
+
λ
2
k
e
∗
k
0
=
H
(
x
∗
,e
∗
)+
λ
1
Φ
1
(
x
∗
)+
λ
2
Φ
2
(
e
∗
)
≤
H
(
x,e
)+
λ
1
Φ
1
(
x
)+
λ
2
Φ
2
(
e
)
≤
H
(
x,e
)+
λ
1
k
x
k
0
+
λ
2
k
e
k
0
.
Ï
d
,(
x
∗
,e
∗
)
´
¯
K
(7)
Û
•
`
)
.
(2)
(
x
∗
,e
∗
)
∈
R
p
+
n
´
¯
K
(7)
Û
•
`
)
,
b
§
Ø
´
t
µ
¯
K
(8)
Û
•
`
)
.
(
x
0
,e
0
)
´
t
µ
¯
K
(8)
Û
•
`
)
,
d
½
n
3.1,
Φ(
x
0
) =
k
x
0
k
0
,Φ(
e
0
) =
k
e
0
k
0
.
u
´
,
H
(
x
0
,e
0
)+
λ
1
k
x
0
k
0
+
λ
2
k
e
0
k
0
=
H
(
x
0
,e
0
)+
λ
1
Φ
1
(
x
0
)+
λ
2
Φ
2
(
e
0
)
<H
(
x
∗
,e
∗
)+
λ
1
Φ
1
(
x
∗
)+
λ
2
Φ
2
(
e
∗
)
≤
H
(
x
∗
,e
∗
)+
λ
1
k
x
∗
k
0
+
λ
2
k
e
∗
k
0
.
ù
†
(
x
∗
,e
∗
)
´
¯
K
(7)
Û
•
`
)
g
ñ
,
Ï
d
(
x
∗
,e
∗
)
∈
R
p
+
n
´
t
µ
¯
K
(8)
Û
•
`
)
.
4.
t
µ
¯
K
(8)
1
wz
•{
du
æ
^
š
à
š
1
w
K
,
t
µ
¯
K
(8)
´
š
1
w
`
z
,
¦
)
t
µ
¯
K
(8)
š
~
;
.
˜
a
•{
´
1
wz
•{
.
1
wz
•{
˜
‡
Ø
%
¯
K
´
1
wz
c
ü
‡
.
)
˜
—
5
.
!
k
‰
Ñ
t
µ
¯
K
(8)
1
wz
.
,
,
y
²
1
w
¯
K
-
½:
†
t
µ
¯
K
(8)
•
•
-
½
:
ä
k
˜
—
5
.
4.1.
¯
K
(8)
1
wz
.
Capped-
`
1
¼
ê´
˜
a
š
~
;
.
ò
U
]
¼
ê
,
§
Œ
±
•
ü
‡
à
¼
ê
(DC)[21,25],
Ù
1
wz
¼
ê
E
Ä
u
Ù
à
L
«
:
ϕ
(
t
) = min
1
,
|
t
|
v
= 1
−
max
0
,
1
−
|
t
|
v
=
g
(
t
)
−
h
(
|
t
|
)
,
(20)
DOI:10.12677/aam.2022.11129589089
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ê
Æ
?
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Ü
ƒ
Ê
Ù
¥
g
(
t
) = 1
,h
(
|
t
|
) = max
0
,
1
−
|
t
|
v
.
P
m
(
t
) =
|
t
|
,
N
´
y
Ù
•
•
ê
•
:
m
0
(
t
;
d
) =
|
d
|
,t
= 0
,
t
|
t
|
d,t
6
= 0
.
du
g
(
t
)
´
1
w
,
·
‚
•
I
ò
h
(
|
t
|
)
1
wz
.
®
•
1
w
¼
ê
E
ܼ
ê
E
´
1
w
,
Ï
d
·
‚
Œ
±
©
O
E
¼
ê
h
(
s
)
†
S
¼
ê
m
(
t
)
1
w
¼
ê
e
h
µ
(
s
)
†
e
m
µ
(
t
),
2
ò
§
‚
E
Ü
,
B
Œ
ë
Y
Œ
‡
1
w
¼
ê
e
h
µ
(
t
) =
e
h
µ
◦
e
m
µ
(
t
).
m
(
t
)
1
wz
:
m
(
t
)
1
w
¼
ê
•
:
e
m
µ
(
t
) =
p
t
2
+
µ,µ>
0
,
(21)
Ù
¥
µ
´
1
wz
ë
ê
.
e
m
µ
(
t
)
ê
Ú
•
•
ê
©
O
•
:
e
m
0
µ
(
t
) =
t
p
t
2
+
µ
,
e
m
0
µ
(
t
;
d
) =
e
m
0
µ
(
t
)
d
=
td
p
t
2
+
µ
.
Œ
±
y
²
1
w
¼
ê
e
m
µ
(
t
)
÷
v
X
e
5
Ÿ
[21]:
(1)
(
1
w
¼
ê
˜
—
Â
ñ
5
):
lim
w
→
t,µ
↓
0
e
m
µ
(
w
) =
m
(
t
)
,
∀
t
∈
R
.
(2)
(
•
•
ê
˜
—
Â
ñ
5
†
f
˜
—
Â
ñ
5
):
lim
w
→
t,µ
↓
0
e
m
0
µ
(
w
;
d
) =lim
w
→
t,µ
↓
0
e
m
0
µ
(
w
)
d
=
m
0
(
t
)
d
=
m
0
(
t,d
)
,
∀
t
∈
R
\{
0
}
,
∀
d
∈
R
,
limsup
w
→
0
,µ
↓
0
e
m
0
µ
(
w
;
d
) =limsup
w
→
0
,µ
↓
0
td
p
t
2
+
µ
=
|
d
|
=
m
0
(0
,d
)
,
∀
d
∈
R
.
h
(
s
)
1
wz
:
é
h
(
s
) = max
{
0
,
1
−
s
v
}
,
Œ
±
æ
^
b
s
(
z,µ
) =
1
2
z
+
p
z
2
+
µ
2
5
1
wz
Ü
¼
ê
max
{
0
,z
}
[21,26].
2
ò
z
= 1
−
s
v
“
\
b
s
(
z,µ
),
h
(
s
)
1
wz
¼
ê
•
e
h
µ
(
s
) =
b
s
(1
−
s
v
,µ
) =
1
2
1
−
s
v
+
r
(1
−
s
v
)
2
+
µ
2
.
(22)
N
´
y
²
1
w
¼
ê
e
h
µ
(
s
)
ä
k
X
e
5
Ÿ
:
(1)
(
1
w
¼
ê
˜
—
Â
ñ
5
):
lim
w
→
s,µ
↓
0
e
h
µ
(
w
) =
1
2
1
−
s
v
+
|
1
−
s
v
|
=
h
(
s
)
,
∀
s
∈
R
.
DOI:10.12677/aam.2022.11129589090
A^
ê
Æ
?
Ð
Ü
ƒ
Ê
(2)
(
•
•
ê
˜
—
Â
ñ
5
†
f
˜
—
Â
ñ
5
):
lim
w
→
s,µ
↓
0
e
h
0
µ
(
w
;
d
) =
h
0
(
s,d
)
,
∀
s
∈
R
\{
0
}
,
∀
d
∈
R
,
limsup
w
→
0
,µ
↓
0
e
h
0
µ
(
w
) =
h
0
+
(0)
.
(
Ü1
w
¼
ê
e
m
µ
(
x
)
Ú1
w
¼
ê
e
h
µ
(
s
),
·
‚
t
µ
¯
K
(8)
1
wz
.
X
e
:
min
x,e
e
F
µ
(
x,e
) :=
H
(
x,e
)+
λ
1
p
+
λ
2
n
−
λ
1
p
X
j
=1
e
h
µ
◦
e
m
µ
(
x
j
)
−
λ
2
n
X
i
=1
e
h
µ
◦
e
m
µ
(
e
i
)
.
(23)
4.2.
1
w
¯
K
(23)
†
t
µ
¯
K
(8)
-
½:
˜
—
5
¯
K
(23)
´
1
w
,
Ù
-
½:
´¦
∇
e
F
µ
(
x,e
) = 0
:
.
!
y
²
1
w
¯
K
(23)
-
½:
†
t
µ
¯
K
(8)
•
•
-
½:
ä
k
˜
—
5
.
½
Â
•
I
8
:
P
1
(
x
) =
{
j
∈{
1
,...,p
}
:
x
j
6
= 0
}
,
P
2
(
x
) =
{
j
∈{
1
,...,p
}
:
x
j
= 0
}
,
Q
2
(
e
) =
{
i
∈{
1
,...,n
}
:
e
i
6
= 0
}
,
Q
2
(
e
) =
{
i
∈{
1
,...,n
}
:
e
i
= 0
}
.
½
n
4.1
3
µ
=
µ
k
ž
,
(
x
∗
µ
k
,e
∗
µ
k
)
´
1
w
¯
K
(23)
-
½:
,
K
1
w
ë
ê
µ
k
↓
0(
k
→∞
)
ž
,
{
(
x
∗
µ
k
,e
∗
µ
k
)
}
∞
k
=1
?
¿
à
:Ñ
´
t
µ
¯
K
(8)
•
•
-
½:
.
y
²
.
(
x
∗
,e
∗
)
•
:
{
(
x
∗
µ
k
,e
∗
µ
k
)
}
∞
k
=1
˜
‡
à
:
,
Ø
”
lim
k
→∞
x
∗
µ
k
=
x
∗
,
lim
k
→∞
e
∗
µ
k
=
e
∗
.
Ï
•
(
x
∗
µ
k
,e
∗
µ
k
)
´
1
w
¯
K
(23)
-
½:
,
∇
e
F
µ
(
x
∗
µ
k
,e
∗
µ
k
) = 0.
K
∀
d
(1)
∈
R
p
,
∀
d
(2)
∈
R
n
,
k
0=
e
F
0
µ
(
x
∗
µ
k
,e
∗
µ
k
;
d
(1)
,d
(2)
) =
h∇
e
F
0
µ
(
x
∗
µ
k
,e
∗
µ
k
)
,d
(1)
,d
(2)
i
=
h∇
H
(
x
∗
µ
k
,e
∗
µ
k
)
,d
(1)
,d
(2)
i−
λ
1
p
X
j
=1
e
h
0
µ
◦
e
m
µ
(
x
∗
µ
k
(
j
)
)
e
m
0
µ
(
x
∗
µ
k
(
j
)
)
d
(1)
j
−
λ
2
n
X
i
=1
e
h
0
µ
◦
e
m
µ
(
e
∗
µ
k
(
i
)
)
e
m
0
µ
(
e
∗
µ
k
(
i
)
)
d
(2)
i
=
h∇
H
(
x
∗
µ
k
,e
∗
µ
k
)
,d
(1)
,d
(2)
i−
λ
1
X
j
∈P
1
(
x
∗
)
+
X
j
∈P
2
(
x
∗
)
e
h
0
µ
◦
e
m
µ
(
x
∗
µ
k
(
j
)
)
e
m
0
µ
(
x
∗
µ
k
(
j
)
)
d
(1)
j
−
λ
2
X
j
∈Q
1
(
e
∗
)
+
X
j
∈Q
2
(
e
∗
)
e
h
0
µ
◦
e
m
µ
(
e
∗
µ
k
(
i
)
)
e
m
0
µ
(
e
∗
µ
k
(
i
)
)
d
(2)
i
.
(24)
DOI:10.12677/aam.2022.11129589091
A^
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Ð
Ü
ƒ
Ê
Š
â
1
w
¼
ê
e
m
µ
(
x
)
Ú
e
h
µ
(
t
)
5
Ÿ
,
é
ª
(24)
ü
>
4
•
(
k
→∞
),
K
∀
d
(1)
∈
R
p
,
∀
d
(2)
∈
R
n
,
k
0=lim
k
→∞
e
F
0
µ
(
x
∗
µ
k
,e
∗
µ
k
;
d
(1)
,d
(2)
)
=lim
k
→∞
h∇
H
(
x
∗
µ
k
,e
∗
µ
k
)
,d
(1)
,d
(2)
i−
λ
1
X
j
∈P
1
(
x
∗
)
lim
k
→∞
e
h
0
µ
◦
e
m
µ
(
x
∗
µ
k
(
j
)
)
e
m
0
µ
(
x
∗
µ
k
(
j
)
)
d
(1)
j
−
λ
1
X
j
∈P
2
(
x
∗
)
lim
k
→∞
e
h
0
µ
◦
e
m
µ
(
x
∗
µ
k
(
j
)
)
e
m
0
µ
(
x
∗
µ
k
(
j
)
)
d
(1)
j
−
λ
2
X
i
∈Q
1
(
e
∗
)
lim
k
→∞
e
h
0
µ
◦
e
m
µ
(
e
∗
µ
k
(
i
)
)
e
m
0
µ
(
e
∗
µ
k
(
i
)
)
d
(2)
i
−
λ
2
X
i
∈Q
2
(
e
∗
)
lim
k
→∞
e
h
0
µ
◦
e
m
µ
(
e
∗
µ
k
(
i
)
)
e
m
0
µ
(
e
∗
µ
k
(
i
)
)
d
(2)
i
≤
lim
k
→∞
h∇
H
(
x
∗
µ
k
,e
∗
µ
k
)
,d
(1)
,d
(2)
i−
λ
1
X
j
∈P
1
(
x
∗
)
lim
k
→∞
e
h
0
µ
◦
e
m
µ
(
x
∗
µ
k
(
j
)
)
e
m
0
µ
(
x
∗
µ
k
(
j
)
)
d
(1)
j
−
λ
1
X
j
∈P
2
(
x
∗
)
limsup
k
→∞
e
h
0
µ
◦
e
m
µ
(
x
∗
µ
k
(
j
)
)
·
limsup
k
→∞
e
m
0
µ
(
x
∗
µ
k
(
j
)
)
d
(1)
j
−
λ
2
X
i
∈Q
1
(
e
∗
)
lim
k
→∞
e
h
0
µ
◦
e
m
µ
(
e
∗
µ
k
(
i
)
)
e
m
0
µ
(
e
∗
µ
k
(
i
)
)
d
(2)
i
−
λ
2
X
i
∈Q
2
(
e
∗
)
limsup
k
→∞
e
h
0
µ
◦
e
m
µ
(
e
∗
µ
k
(
i
)
)
·
limsup
k
→∞
e
m
0
µ
(
e
∗
µ
k
(
i
)
)
d
(2)
i
=
h∇
H
(
x
∗
,e
∗
)
,d
(1)
,d
(2)
i−
λ
1
p
X
j
=1
h
0
◦
m
(
x
∗
j
)
m
0
(
x
∗
j
)
d
(1)
j
−
λ
2
n
X
i
=1
h
0
◦
m
(
e
∗
i
)
m
0
(
e
∗
i
)
d
(2)
i
=
h∇
H
(
x
∗
,e
∗
)
,d
(1)
,d
(2)
i−
λ
1
p
X
j
=1
h
0
(
m
(
x
∗
j
))
m
0
(
x
∗
j
;
d
(1)
j
)
−
λ
2
n
X
i
=1
h
0
(
m
(
e
∗
i
))
m
0
(
e
∗
i
;
d
(2)
i
)
=
h∇
H
(
x
∗
,e
∗
)
,d
(1)
,d
(2)
i
+
λ
1
p
X
j
=1
[
g
0
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