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AdvancesinAppliedMathematicsA^êÆ?Ð,2023,12(4),1732-1743
PublishedOnlineApril2023inHans.https://www.hanspub.org/journal/aam
https://doi.org/10.12677/aam.2023.124180
&Ò-`zŽ{9Ù3ã¡¡E¥A^
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Á‡
©?˜Ú•Ä&Ò-† ã”D¯K`z•{"•d§JÑ˜«ÄuaqArmijo‚|¢
#.Ž{§•[y²TŽ{ÛÂñ5ÚO(1/k
2
)g‚5Âñ„Ç"•§ÏLDÕ&Ò¡
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&Ò-ïÚã”D¯K§Ž{§ÛÂñ§g‚5Âñ„Ý
AnOptimizationAlgorithmonSignal
ReconstructionandItsApplicationin
ImageRestoration
ZunyangWang
1
,ChaoGuo
1
,HongchunSun
2∗
1
Department of Electronic and Communication Engineering, Beijing Institute of Electronic Science
andTechnology,Beijing
2
SchoolofMathematicsandStatistics,LinyiUniversity,LinyiShandong
Received:Mar.24
th
,2023;accepted:Apr.18
th
,2023;published:Apr.27
th
,2023
∗ÏÕŠö"
©ÙÚ^:ƒ,H‡,šöS.&Ò-`zŽ{9Ù3ã¡¡E¥A^[J].A^êÆ?Ð,2023,12(4):
1732-1743.DOI:10.12677/aam.2023.124180
ƒ
Abstract
In thispaper, we furtherconsider anoptimizationmethodfor solvingthesignal recon-
struction and imagedenoising problem.To this end, a new algorithmwith Armijo-like
line search is proposed.Global convergence results of the new algorithmis established
in detail.Furthermore, we also show that the method is sublinearly convergent rate of
O(1/k
2
).Finally,theefficiencyoftheproposedalgorithmisillustratedthroughsome
numericalexamplesonsparsesignalrecoveryandimagedenoising.
Keywords
TheSignalReconstructionandImageDenoisingProblem,Algorithm,Global
Convergence,SublinearlyConvergentRate
Copyright
c
2023byauthor(s)andHansPublishersInc.
This work is licensed undertheCreative Commons Attribution InternationalLicense(CCBY4.0).
http://creativecommons.org/licenses/by/4.0/
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DOI:10.12677/aam.2023.1241801733A^êÆ?Ð
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N
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L
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L
2
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2

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-
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L
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L
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…ïá(2.2)d=†:
p
L
(y) = argmin
(
ρϕ(x)+
L
2




x−

y−
1
L
∇f(y)





2
)
.(2.3)
e5£e¡Ún´C
1,1
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N
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kx−yk
2
],∀x,y∈R
N
.(2.4)
DOI:10.12677/aam.2023.1241801734A^êÆ?Ð
ƒ
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f
,Kéu?¿y∈R
n
,k
F(p
L
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L
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L
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≤
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f(y)+hp
L
(y)−y,∇f(y)i+
L
2
kp
L
(y)−yk
2
i
+ρϕ(p
L
(y))
= Q
L
(p
L
(y),y).
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0
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,y
1
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,
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k
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k
)
i
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−y
k
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k
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−y
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k
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m
β
2


x
k
−y
k

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.(2.8)
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τ
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k+1
= x
k
+
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k
−σ
τ
k+1
(x
k
−x
k−1
).(2.10)
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k
)−F(x
k−1
)k≤,ÊŽ.Kx
k
´¯K(1.1)).ÄK,=Ú1,-k
4
= k+1.
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∂(|t|) =
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





−1ift<0,
[−1,1]ift= 0,
1ift>0.
(Ü(2.6),ke¡(J.
DOI:10.12677/aam.2023.1241801735A^êÆ?Ð
ƒ
e(y
k
−
1
L
k
∇f(y
k
))
i
<−
ρ
L
k
,K(x
k
)
i
= (y
k
−
1
L
k
∇f(y
k
))
i
+
ρ
L
k
<0,=
(x
k
)
i
= (y
k
−
1
L
k
∇f(y
k
))
i
−
ρ
L
k
∂ϕ(x
k
)
∂x
i
,
Ù¥
∂ϕ(x
k
)
∂x
i
= −1.
e(y
k
−
1
L
k
∇f(y
k
))
i
>
ρ
L
k
,K(x
k
)
i
= (y
k
−
1
L
k
∇f(y
k
))
i
−
ρ
L
k
>0,=
(x
k
)
i
= (y
k
−
1
L
k
∇f(y
k
))
i
−
ρ
L
k
∂ϕ(x
k
)
∂x
i
,
Ù¥
∂ϕ(x
k
)
∂x
i
= 1.
e|(y
k
−
1
L
k
∇f(y
k
))
i
|≤
ρ
L
k
,K|(y
k
−
1
L
k
∇f(y
k
))
i
|/
ρ
L
k
≤1.d(x
k
)
i
= 0,
(x
k
)
i
= 0 = (y
k
−
1
L
k
∇f(y
k
))
i
+
ρ
L
k
(y
k
−
1
L
k
∇f(y
k
))
i
/
ρ
L
k
= (y
k
−
1
L
k
∇f(y
k
))
i
−
ρ
L
k
∂ϕ(x
k
)
∂x
i
,
Ù¥
∂ϕ(x
k
)
∂x
i
= (y
k
−
1
L
k
∇f(y
k
))
i
/
ρ
L
k
.
Äu±þ©Û,kx
k
= y
k
−
1
L
k
∇f(y
k
)−
ρ
L
k
ξ
k
,=
ρξ
k
+∇f(y
k
)+L
k
(x
k
−y
k
) = 0.(2.11)
Ù¥ξ
k
∈∂ϕ(x
k
).g•e¡à`z¯K
min
(
ρϕ(x)+
L
k
2




x−

y
k
−
1
L
k
∇f(y
k
)





2
)
,(2.12)
Äuƒ'•`5nØ([10]),•T¯K)8†§-½:8´˜—.(Ü(2.11),Œ
•x
k
´(2.12)),=
x
k
:= argmin
(
ρϕ(x)+
L
k
2




x−

y
k
−
1
L
k
∇f(y
k
)





2
)
,(2.13)
52.2dL
k
=η
m
k
βÚη>1,kL
k
≥β.,,ÏL
k
/ηØ÷v(2.7)½(2.8),(ÜÚn2.1,
kL
k
<ηkA
>
Ak.Ïd,k
β≤L
k
<ηkA
>
Ak.(2.14)
e5§?ØŽ{2.1Âñ5±9Âñ„Ý.•d,‰Ñ±eÚn.
Ún2.2éu?¿x∈R
n
,k
F(x)−F(x
k
) ≥
L
k
2


x
k
−y
k


2
+L
k

x−y
k
,y
k
−x
k

,∀k≥1.(2.15)
DOI:10.12677/aam.2023.1241801736A^êÆ?Ð
ƒ
y².d(2.5),
F(p
L
k
(y
k
)) ≤Q
L
k
(p
L
k
(y
k
),y
k
).(2.16)
d(2.13),x
k
= p
L
k
(y
k
).(Ü(2.16),Œ
F(x)−F(x
k
) ≥F(x)−Q
L
k
(x
k
,y
k
).(2.17)
ÏfÚϕÑ´à,Œ
F(x)= f(x)+ρϕ(x)
≥[f(y
k
)+hx−y
k
,∇f(y
k
)i]+[ρϕ(x
k
)+ρhx−x
k
,ξ
k
)i],
(2.18)
Ù¥ξ
k
∈∂ϕ(x
k
).A^(2.1)…-x= x
k
,y= y
k
,
Q
L
k
(x
k
,y
k
) = [f(y
k
)+

x
k
−y
k
,∇f(y
k
)

+
L
k
2


x
k
−y
k


2
]+ρϕ(x
k
).(2.19)
A^(2.18),(2.19)Ú(2.17),Œ
F(x)−F(x
k
)≥F(x)−Q
L
k
(x
k
,y
k
)
≥

x−x
k
,∇f(y
k
)+ρξ
k

−
L
k
2


x
k
−y
k


2
= L
k

x−x
k
,y
k
−x
k

−
L
k
2


x
k
−y
k


2
= L
k

(x−y
k
)+(y
k
−x
k
),y
k
−x
k

−
L
k
2


x
k
−y
k


2
= L
k

x−y
k
,y
k
−x
k

+
L
k
2


x
k
−y
k


2
.
(2.20)
Ù¥1˜‡ª¤á´Äu(2.11).
Ún2.3bx
∗
´(1.1)?¿˜),{x
k
}Ú{y
k
}´Ž{2.1)S.K
%ηkA
>
Ak
β
2
L
k
τ
2
k
α
k
1
−
2
L
k+1
τ
2
k+1
α
k+1
1
≥


α
k+1
2


2
−


α
k
2


2
,(2.21)
Ù¥α
k
1
:= F(x
k
)−F(x
∗
),α
k
2
:= τ
k
x
k
−(τ
k
−σ)x
k−1
−σx
∗
.
y².éu?¿x∈R
n
,d(2.15)ke¡Øª¤á
F(x)−F(x
k+1
) ≥
L
k+1
2


x
k+1
−y
k+1


2
+L
k+1

x−y
k+1
,y
k+1
−x
k+1

.(2.22)
3(2.22)¥,©O-x:= x
k
Úx:= x
∗
,Œ
F(x
k
)−F(x
k+1
) ≥
L
k+1
2


x
k+1
−y
k+1


2
+L
k+1

x
k
−y
k+1
,y
k+1
−x
k+1

(2.23)
DOI:10.12677/aam.2023.1241801737A^êÆ?Ð
ƒ
Ú
F(x
∗
)−F(x
k+1
) ≥
L
k+1
2


x
k+1
−y
k+1


2
+L
k+1

x
∗
−y
k+1
,y
k+1
−x
k+1

(2.24)
dα
k
1
½Â,(2.23)Ú(2.24)U•
2
L
k+1
(α
k
1
−α
k+1
1
) ≥


x
k+1
−y
k+1


2
+2

x
k
−y
k+1
,y
k+1
−x
k+1

(2.25)
Ú
−
2
L
k+1
α
k+1
1
≥


x
k+1
−y
k+1


2
+2

x
∗
−y
k+1
,y
k+1
−x
k+1

(2.26)
3(2.25)ü>¦±τ
k+1
−σ
(τ
k+1
−σ)
2
L
k+1
(α
k
1
−α
k+1
1
)
≥(τ
k+1
−σ)


x
k+1
−y
k+1


2
+2(τ
k+1
−σ)

x
k
−y
k+1
,y
k+1
−x
k+1

(2.27)
3(2.26)ü>Ó¦±σ
−σ
2
L
k+1
α
k+1
1
≥σ


x
k+1
−y
k+1


2
+2σ

x
∗
−y
k+1
,y
k+1
−x
k+1

(2.28)
(2.27)\(2.28)
2
L
k+1
((τ
k+1
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k
1
−τ
k+1
α
k+1
1
)≥τ
k+1


x
k+1
−y
k+1


2
+2

τ
k+1
y
k+1
−(τ
k+1
−σ)x
k
−σx
∗
,x
k+1
−y
k+1

(2.29)
3(2.29)ü>Ó¦±τ
k+1
,…A^(2.9),Œ
2
L
k+1
(%τ
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k
1
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k+1
α
k+1
1
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k+1
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k+1
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
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τ
k+1
y
k+1
−(τ
k+1
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k
−σx
∗
,x
k+1
−y
k+1

(2.30)
-a:= τ
k+1
y
k+1
,b:= τ
k+1
x
k+1
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k+1
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k
+σx
∗
.dPythagoras'X
kb−ak
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−ka−ck
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,
Ú(2.30),Œ
2
L
k+1
(%τ
2
k
α
k
1
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2
k+1
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k+1
1
)≥


τ
k+1
x
k+1
−(τ
k+1
−σ)x
k
−σx
∗


2
−


τ
k+1
y
k+1
−(τ
k+1
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k
−σx
∗


2
(2.31)
DOI:10.12677/aam.2023.1241801738A^êÆ?Ð
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d(2.10),w,Œτ
k+1
y
k+1
= τ
k+1
x
k
+(τ
k
−σ)(x
k
−x
k−1
).Šâ(2.31)Úα
k
2
½Â,Œ
2
L
k+1
(%τ
2
k
α
k
1
−τ
2
k+1
α
k+1
1
) ≥


α
k+1
2


2
−


α
k
2


2
.(2.32)
2(Ü(2.14),Œ
%ηkA
>
Ak
β
2
L
k
τ
2
k
α
k
1
−
2
L
k+1
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k+1
α
k+1
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α
k+1
2


2
−


α
k
2


2
.(2.33)
Ún2.4d(2.9)ê{τ
k
}÷vτ
k
≥
k
2
(k≥1),Ù¥τ
1
= 1.
y².|^êÆ8B{y².k= 1ž,dτ
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1
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1
2
.
bkž·K¤á,=τ
k
≥
k
2
.d(2.9),¿A^σ≥1,%≥1,Œ
τ
k+1
=
σ+
p
σ
2
+%τ
2
k
2
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σ+
√
%k
2
≥
1+k
2
.
ÏdUýÏ(J.
y3,y²Ž{2.1Âñ(J.
½n2.1bx
∗
´(1.1)?¿˜),{x
k
}Ú{y
k
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F(x
k+1
)−F(x
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c
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2
.(2.34)
Ù¥c´~ê.
y².-a
k
=
2
L
k
τ
2
k
α
k
1
,b
k
=


α
k
2


2
,A^(2.21),
a
k+1
+b
k+1
≤
%ηkA
>
Ak
β
a
k
+b
k
.(2.35)
(Ü0 <
%ηkA
>
Ak
β
<1,ÏLOŽŒ
a
k+1
≤a
k+1
+b
k+1
≤a
k
+b
k
≤a
k−1
+b
k−1
≤···≤a
1
+b
1
,(2.36)
…a
1
=
2
L
1
(F(x
1
)−F(x
∗
)),b
1
=k(x
1
−x
0
)+σ(x
0
−x
∗
)k
2
.2(Ü(2.14),¿A^Ún2.4,Œ

F(x
k+1
)−F(x
∗
) ≤
2ηkA
>
Ak[
2
L
1
(F(x
1
)−F(x
∗
))+k(x
1
−x
0
)+σ(x
0
−x
∗
)k
2
]
(k+1)
2
.
-c= 2ηkA
>
Ak[
2
L
1
(F(x
1
)−F(x
∗
))+k(x
1
−x
0
)+σ(x
0
−x
∗
)k
2
],Kk(J¤á.
DOI:10.12677/aam.2023.1241801739A^êÆ?Ð
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3.ꊢ
!Jø˜'u¯K(1.1)êŠÿÁ,±y²Ž{2.1k5.¤k“è^MATLAB
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13
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ã1.äk\5pdxD(DÕ&Ò¡E(n= 2
13
)
DOI:10.12677/aam.2023.1241801740A^êÆ?Ð
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,,éuØÓ‘ê!ØÓ&Ò¯K(1.1),A^Ž{2.1?1\5pdxD(‚¸[D
Õ&Ò¡E.CPUžm(CPUTime)!S“gê(Iter)ÚƒéØ(RelErr)3L1.lL¥Œ±w
Ñ,Ž{2.1UÐ¡EØÓ‘êDÕ&Ò,…CPUžm,•O(,é‘êO\دa.
L²Ž{2.1kÐ-½5.
Table1.SignalrecoveryonadditiveGaussianwhitenoisewiththedifferentdimension
L1.äkØÓ‘ê\5pdxD(DÕ&Ò¡E
‘êDÕ&ÒCPUTimeIterRelErr
10241×2
5
0.48441294.9128
20482×2
5
1.15621384.8728
30723×2
5
2.73431314.6985
40964×2
5
4.04681284.4007
51205×2
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5.65121364.6486
61446×2
5
7.60931354.5438
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9.68751414.4214
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DOI:10.12677/aam.2023.1241801741A^êÆ?Ð
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ë•©z
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