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AdvancesinAppliedMathematicsA^êÆ?Ð,2021,10(11),3850-3864
PublishedOnlineNovemb er2021inHans.http://www.hanspub.org/journal/aam
https://doi.org/10.12677/aam.2021.1011409
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ResearchonSmoothingMethod
forSparsePhaseRetrieval
Problems
XiaojiaLiu,DingtaoPeng
∗
SchoolofMathematicsandStatistics,GuizhouUniversity,GuiyangGuizhou
Received:Oct.17
th
,2021;accepted:Nov.7
th
,2021;published:Nov.19
th
,2021
∗ÏÕŠö"
©ÙÚ^:4¡Z,$½7.DÕƒ u¢¯K1wz•{ïÄ[J].A^êÆ?Ð,2021,10(11):3850-3864.
DOI:10.12677/aam.2021.1011409
4¡Z§$½7
Abstract
Forsparsephaseretrievalproblems,thispaperconstructedaclassofsparseleast
absolutedeviationoptimizationmodel.First,weusedthecontinuousCapped-L1
regularizationfunctiontorelaxthediscontinuoussparsityfunction.Byvirtueof
directionalderivativesandthedirectionalstationarypoints,weprovidedthefirst-
orderoptimalityconditionfortherelaxedoptimizationproblem.Furthermore,we
derived thelowerboundpropertyofthedirectionalstationary points, basedonwhich
weprovedtheequivalencebetweentheoriginalproblemandtherelaxedproblemin
thesenseofglobalsolutions.Finally,weadvisedusingsmoothingmethodstosolve
therelaxedproblem.Weproposedaclassofsmoothfunctiontoapproximatethe
objectivefunction,andestablishedtheconsistencyofthesolutionsofthesmoothing
problemandtherelaxedproblem.Ourresultprovidesatheoreticalbasisforusing
smoothingmethodstosolvesparsephaseretrievalproblems.
Keywords
SparsePhaseRetrievalProblem,LeastAbsoluteEstimate,Directional
StationaryPoint,SmoothingMethod
Copyright
c
2021byauthor(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.2021.10114093855A^êÆ?Ð
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1

X
i∈I
1
sgn((a
>
i
bx)
2
−y
i
)·a
>
i
bxa
i

j
(x
j
−bx
j
)+
X
j∈Γ
1

X
i∈I
2
|a
>
i
bxa
i
|

j
|x
j
−bx
j
|

.
5¿,
X
j∈Γ
1

X
i∈I
1
sgn((a
>
i
bx)
2
−y
i
)·a
>
i
bxa
i

j
(x
j
−bx
j
) ≤
X
j∈Γ
1

X
i∈I
1
|a
>
i
bxa
i
|

j
|x
j
−bx
j
|.
u´,
F
0
(bx;x−bx)≤
2
N

X
j∈Γ
1

X
i∈I
1
|a
>
i
bxa
i
|

j
|x
j
−bx
j
|+
X
j∈Γ
1

X
i∈I
2
|a
>
i
bxa
i
|

j
|x
j
−bx
j
|

=
2
N

X
j∈Γ
1

N
X
i=1
|a
>
i
bxa
i
|

j
|x
j
−bx
j
|

.(15)
DOI:10.12677/aam.2021.10114093856A^êÆ?Ð
4¡Z§$½7
-x
j
=bx
j
−bx
j
,>0,j∈Γ
1
.(Ü(14)†(15),
X
j∈Γ
1
φ
0
(bx
j
)|bx
j
|≤
2
N

X
j∈Γ
1

N
X
i=1
|a
>
i
bxa
i
|

j
|bx
j
|

≤
2
N
L(x
∗
)
X
j∈Γ
1
|bx
j
|.
dué?¿j∈Γ
1
,φ
0
(bx
j
) = λ/v.u´,λ/v≤(2/N)L(x
∗
),†®•^‡λ/v>(2/N)L(x
∗
)gñ.T
(ؤá.
e¡,·‚?دK(4)†¯K(6))'X.
½n2.4λ/v≥(2/N)L(x
∗
),
(i)ebx∈R
n
´¯K(6)Û•`),…÷vF(bx) ≤Υ,Kbx´¯K(4)Û•`).
(ii)ebx∈R
n
´¯K(4)Û•`),…¯K(6)Û)x
0
•3¿÷vF(x
0
)≤Υ,Kbx´¯
K(6)Û•`).
y².(i)bx∈R
n
´¯K(6)Û•`),Kbx∈R
n
´¯K(6)••-½:.dF(bx)≤Υ
9e.½n2.3,Φ(bx) = λkbxk
0
.u´
F(bx)+λkbxk
0
= F(bx)+Φ(bx) <F(x)+Φ(x) ≤F(x)+λkxk
0
.
Ïd,bx´¯K(4)Û•`).
(ii)bx∈R
n
´¯K(4)Û•` ),bx∈R
n
Ø´¯K(6)Û• `).x
0
´¯K(6)
Û•`),KF(x
0
) ≤Υ.d½n®•^‡9e.½n2.3,Φ(x
0
) = λkx
0
k
0
.u´
F(x
0
)+λkx
0
k
0
= F(x
0
)+Φ(x
0
) <F(bx)+Φ(bx) ≤F(bx)+λkbxk
0
.
ù†bx∈R
n
´¯K(4)Û•`)gñ.Ïdbx´¯K(6)Û•`).
5¿,¯K(6)Û)x
0
´˜½•3,ù´Ï•UCÛ)©þÎÒ,جUC¯
K(6)8I¼êŠ,ù´dƒ u¢¯K5Ÿû½.
3.1wz•{
¯K(6)´˜šà!š1w`z¯K,†¦)´(J.!·‚ï¯K(6)˜a1w¼
ê.¿y²1w¯K••-½:?¿à:´¯K••-½:.ùnØ•¯K(6)¦)
JønØy.
£[22],ý銼ê|t|,t∈R˜a1w¼êŒ±½Â•
e
θ(t,µ) :=



|t|if|t|≥µ,
t
2
2µ
+
µ
2
if|t|<µ.
DOI:10.12677/aam.2021.10114093857A^êÆ?Ð
4¡Z§$½7
Ù¥µ>0´1wëê.dd·‚¯K(6)¥›”¼ê1w¼ê
e
F(x,µ) :=
1
N
N
X
i=1
e
f
i
(x,µ),(16)
Ù¥,
e
f
i
(x,µ) =
e
θ((a
T
i
x)
2
−y
i
,µ).P{1,···,N}= C
µ
1
(x)∪C
µ
2
(x)∪C
µ
3
(x),Ù¥C
µ
1
(x) = {1,···,N:
|(a
T
i
x)
2
−y
i
|<µ},C
µ
2
(x) = {1,···,N: |(a
T
i
x)
2
−y
i
|>µ},C
µ
3
(x) = {1,···,N: |(a
T
i
x)
2
−y
i
|= µ}.
e¡,·‚©Û1w¼ê
e
F(x,µ)˜5Ÿ.
Ún3.1éu¼ê(16),eã(ؤá:
(1)lim
z→x,µ↓0
e
F(z,µ) = F(x).
(2)('uxLipschitzëY5)•3˜‡~êL>0¦é?¿µ∈(0,µ],¦∇
e
F(x,µ) ´Lipschitz
ëY,…Lipschitz~ê´Lµ
−1
.
(3)('uµLipschitzëY5)|
e
F(x,µ
1
)−
e
F(x,µ
2
)|≤|µ
1
−µ
2
|.
(4)(˜—5†f˜—5)?¿i∈I
1
(x),klim
z→x,µ↓0
h∇
e
f
i
(z,µ),wi=f
0
i
(x;w),∀w∈R
n
¤á;?
¿i∈I
2
(x),klimsup
z→0,µ↓0
h∇
e
f
i
(z,µ),wi= f
0
i
(x;w),∀w∈R
n
¤á.
y².(1)dC
µ
1
(x),C
µ
2
(x),C
µ
3
(x)½Â,
|
e
F(z,µ)−F(x)|=
N
X
i=1

e
f
i
(z,µ)−f
i
(x,µ)

=
N
X
i=1

e
f
i
(z,µ)−
e
f
i
(x,µ)+
e
f
i
(x,µ)−f
i
(x,µ)

=
N
X
i=1

e
f
i
(z,µ)−
e
f
i
(x,µ))

+
X
i∈C
µ
1
(x)

e
f
i
(x,µ)−f
i
(x,µ)

=Π
1
+Π
2
.
d¼ê
e
fëY5Œ•
lim
z→x,µ↓0
Π
1
= 0.(17)
éuΠ
2
,N´
X
i∈C
µ
1
(x)

e
f
i
(x,µ)−f
i
(x,µ)

≤|C
µ
1
(x)|
µ
2
→0, (µ→0).(18)
(Ü(17)†(18),u´(Ø(1)¤á.
(2)d1w¼ê½Â,∇
e
F(x,µ) =
1
N
P
N
i=1
∇
e
f
i
(x,µ).
DOI:10.12677/aam.2021.10114093858A^êÆ?Ð
4¡Z§$½7
i∈C
µ
1
(x)ž,
∇
e
f
i
(x,µ) = 2

(a
T
i
x)
2
−y
i
µ

a
i
a
T
i
x,(19)
i∈C
µ
2
(x)∪C
µ
3
(x)ž,
∇
e
f
i
(x,µ) = 2sgn((a
T
i
x)
2
−y
i
)a
i
a
T
i
x.(20)
d©z[23]¥Š½n2.3.7,é?¿x,z∈R
n
,
∇
e
F(x,µ)−∇
e
F(z,µ) ∈(∂(∇
e
F(u,µ)))
T
(x−z),(21)
Ù¥u´‚ã[z,x]þ,˜:,∂(∇
e
F(u,µ))L«•þ¼ê∇
e
F(·,µ)3:u?Clarkeg‡©.i∈
C
µ
1
(u)ž,∂(∇
e
f
i
(u,µ)) = ∇
2
e
f
i
(u,µ) = [
2(a
T
i
u)
2
µ
+2]a
i
a
T
i
.i∈C
µ
2
(u)ž,∂(∇
e
f
i
(u,µ)) = ∇
2
e
f
i
(u,µ) =
2sgn((a
T
i
x)
2
−y
i
)a
i
a
T
i
.i∈C
µ
3
(u)ž,dClarkeg‡©½ÂŒ•,
∂(∇
e
f
i
(u,µ)) = con{2sgn((a
T
i
x)
2
−y
i
),[
2(a
T
i
u)
2
µ
+2]}a
i
a
T
i
.
¯¢þ,é?¿a∈con{2sgn((a
T
i
x)
2
−y
i
),[2(a
T
i
u)
2
µ
−1
+2]},a≤2y
i
µ
−1
+4 ≤(2y
i
+µ)µ
−1
.u´

∂(∇
e
F(u,µ))) ⊆
1
N

X
i∈C
µ
1
(u)∪C
µ
1
(u)
∇
2
e
f
i
(u,µ)+
X
i∈C
µ
3
(u)
∂(∇
e
f
i
(u,µ))

.(22)
(Ü(21),(22)Œ,•3V∈∂(∇
e
F(u,µ))¦eã'X¤á:
∇
e
F(x,µ)−∇
e
F(z,µ)=V
T
(x−z) ≤(2y+µ)µ
−1
λ
max
(
N
X
i=1
a
i
a
T
i
)kx−zk,
Ù¥y=max{y
i
,i=1,···,N}.5¿,(2y+µ)µ
−1
λ
max
(
P
N
i=1
a
i
a
T
i
)´˜~ê,…†CþxÃ',
u´(Ø(2)¤á.
(3)Ø”˜„5,bµ
2
≥µ
1
>0.é?¿i∈C
µ
2
2
(x)∪C
µ
2
3
(x),
e
θ((a
T
i
x)
2
−y
i
,µ
1
) =
e
θ((a
T
i
x)
2
−y
i
,µ
2
) = |(a
T
i
x)
2
−y
i
|.
Ïd,
e
f
i
(x,µ
1
)−
e
f
i
(x,µ
2
) = 0,i∈C
µ
2
2
(x)∪C
µ
2
3
(x).(23)
é?¿i∈C
µ
1
2
(x)∪C
µ
1
3
(x)∩C
µ
2
1
(x),
e
θ((a
T
i
x)
2
−y
i
,µ
2
) >
e
θ((a
T
i
x)
2
−y
i
,µ
1
) = |(a
T
i
x)
2
−y
i
|.(24)
DOI:10.12677/aam.2021.10114093859A^êÆ?Ð
4¡Z§$½7
dž,((a
T
i
x)
2
−y
i
)
2
/2µ
2
≤µ
2
2
/2µ
2
= µ
2
/2,…|(a
T
i
x)
2
−y
i
|≥µ
1
,u´d1w¼ê½Â9(24)
e
θ((a
T
i
x)
2
−y
i
,µ
2
)−
e
θ((a
T
i
x)
2
−y
i
,µ
1
) =
((a
T
i
x)
2
−y
i
)
2
2µ
2
+
µ
2
2
−|(a
T
i
x)
2
−y
i
|≤µ
2
−µ
1
.
Ïd,
e
f
i
(x,µ
2
)−
e
f
i
(x,µ
1
) ≤µ
2
−µ
1
,i∈C
µ
1
2
(x)∪C
µ
1
3
(x)∩C
µ
2
1
(x).(25)
é?¿i∈C
µ
1
1
(x),(a
T
i
x)
2
−y
i
<µ
1
≤µ
2
,
e
θ((a
T
i
x)
2
−y
i
,µ
1
) =
((a
T
i
x)
2
−y
i
)
2
2µ
1
+
µ
1
2
,
e
θ((a
T
i
x)
2
−y
i
,µ
2
) =
((a
T
i
x)
2
−y
i
)
2
2µ
2
+
µ
2
2
.
u´,
e
θ((a
T
i
x)
2
−y
i
,µ
2
)−
e
θ((a
T
i
x)
2
−y
i
,µ
1
) = (
1
2µ
2
−
1
2µ
1
)((a
T
i
x)
2
−y
i
)
2
+
µ
2
−µ
1
2
≤(
1
2µ
2
−
1
2µ
1
)µ
2
1
+
µ
2
−µ
1
2
≤
µ
2
2
2µ
2
−
µ
2
1
2µ
1
+
µ
2
−µ
1
2
= µ
2
−µ
1
.
Ïd,
e
f
i
(x,µ
2
)−
e
f
i
(x,µ
1
) ≤µ
2
−µ
1
,i∈C
µ
1
1
(x).(26)
,˜•¡, (C
µ
2
2
(x)∪C
µ
2
3
(x))∪(C
µ
1
2
(x)∪C
µ
1
3
(x)∩C
µ
2
1
(x))∪C
µ
1
1
(x) = {1,···,N},(Ü(23),(25),
(26)Œ
e
F(x,µ
2
)−
e
F(x,µ
1
) ≤
1
N
N(µ
2
−µ
1
) = µ
2
−µ
1
.
(Ø(3)¤á.
(4)é?¿i∈I
1
(x),=|(a
T
i
x)
2
−y
i
|6=0.•3µ>0¦|(a
T
i
x)
2
−y
i
|>µ.dž,i∈C
µ
2
(x).
k
e
f
i
=f
i
¤á,…§‚3:z
k
?Œ‡.u´lim
µ↓0
h∇
e
f
i
(x,µ),wi=h∇f
i
(x,µ),wi=f
0
i
(x;w).d
,d∇
e
f
i
ëY5Œ•,z→xž,∇
e
f
i
(z,µ) →∇
e
f
i
(x,µ).(Üþã'X,
lim
z→x,µ↓0
h∇
e
f(z,µ),wi=lim
z→x,µ↓0

h∇
e
f(z,µ),wi−h∇
e
f(x,µ),wi

+lim
z→x,µ↓0
h∇
e
f(x,µ),wi
=0+lim
µ↓0
h∇
e
f(x,µ),wi= f
0
i
(x;w).
u´(4)¥1˜^(ؤá.
é?¿i∈I
2
(x),=|(a
T
i
x)
2
−y
i
|=0.é?¿µ>0þk|(a
T
i
x)
2
−y
i
|<µ.-{z
k
}´Âñx
DOI:10.12677/aam.2021.10114093860A^êÆ?Ð
4¡Z§$½7
?¿:.e|(a
T
i
z
k
)
2
−y
i
|<µ,=i∈C
µ
1
,(ܪf(19),K
limsup
z→x,µ↓0,
|(a
T
i
z)
2
−y
i
|<µ
h∇
e
f(z,µ),wi=limsup
z→x,µ↓0,
|(a
T
i
z)
2
−y
i
|<µ
2(
(a
T
i
z)
2
−y
i
µ
)a
T
i
za
T
i
w= 2|a
T
i
za
T
i
w|= f
0
i
(x;w),
Ù¥1‡ª¤á´Ï•k¿©Œž, •3÷v|(a
T
i
z
k
)
2
−y
i
|<µ, |(a
T
i
z
k
)
2
−y
i
|→µ…(a
T
i
z)
2
−
y
i
†a
T
i
za
T
i
wÓÒ:{z
k
}.e|(a
T
i
z
k
)
2
−y
i
|≥µ,Ki∈C
µ
2
.dž
e
f
i
=f
i
,…§‚3:z
k
?Œ‡.
(ܪf(20),u´
limsup
z→x,µ↓0,
|(a
T
i
z)
2
−y
i
|≥µ
h∇
e
f(z,µ),wi=limsup
z→x,µ↓0,
|(a
T
i
z)
2
−y
i
|≥µ
2sgn((a
T
i
z)
2
−y
i
)a
T
i
za
T
i
w= 2|a
T
i
za
T
i
w|= f
0
i
(x;w),
Ù¥1‡ª¤á´Ï•k¿©Œž,• 3÷v|(a
T
i
z
k
)
2
−y
i
|≥µ,…(a
T
i
z)
2
−y
i
†a
T
i
za
T
i
wÓ
Ò:{z
k
}.nþ¤ã(4)¥1^(ؤá.(Øy.
(Üþã1wzEâ,·‚˜‡›”¼êëYŒ‡`z¯K:
min
x∈R
n
e
F(x,µ)+Φ(x).(27)
ebx
µ
´þã¯K••-½:,K
h∇
e
F(bx
µ
),x−bx
µ
i+Φ
0
(bx
µ
;x−bx
µ
) ≥0,∀x∈R
n
,
=
e
F
0
(bx
µ
,x−bx
µ
)+Φ
0
(bx
µ
;x−bx
µ
) ≥0,∀x∈R
n
.
e¡,·‚òïÄ:{bx
µ
k
}à:5Ÿ,Ù¥{bx
µ
k
}•••-½:,µ
k
>0,k=1,2,···,…
k→∞ž,µ
k
→0.
½n3.1({bx
µ
k
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