R E S E A R C H
Open Access
An improved method for solving
multiple-sets split feasibility problem
Yunfei Du and Rudong Chen
**Correspondence:
Department of Mathematics, Tianjin Polytechnic University, Tianjin, 300387, China
Abstract
The multiple-sets split feasibility problem (MSSFP) has a variety of applications in the real world such as medical care, image reconstruction and signal processing. Censor
et al.proposed solving the MSSFP by a proximity function, and then developed a class of simultaneous methods for solving split feasibility. In our paper, we improve a simultaneous method for solving the MSSFP and prove its convergence.
Keywords: multiple-sets split feasibility problem; convergence
1 Introduction
Throughout this paper, let H be a Hilbert space,·,·denote the inner product and· de-note the corresponding norm. The multiple-sets split feasibility problem (MSSFP) is a gen-eralization of the split feasibility problem (SFP) and the convex feasibility problem (CFP); see []. LetCi andQj be closed convex sets in theN-dimensional andM-dimensional
Euclidean spaces, respectively. The MSSFP is to find a vectorx*satisfying
x*∈C:=
t
i=
Ci such thatAx*∈Q:= r
j=
Qj, ()
where A is a matrix ofM×N, andt,r> are integers. Whent=r= , the problem becomes to find a vectorx*∈C, such thatAx*∈Q, which is just the two-sets split feasibility problem
(SFP) introduced in []. The MSSFP has many applications in our real life such as image restoration, signal processing and medical care (e.g., [–]). In order to solve the MSSFP, Censoret al.considered the MSSFP in the following form:
x*∈C:=X∩
t
i=
Ci
and Ax*∈Q:=
r
j=
Qj, ()
X⊆Rnis a nonempty closed convex set. In fact, () is equivalent to (). Many methods
MSSFP () is equivalent to the minimization problem
min
x–PCx
+
x–PQAx
,
wherePCandPQdenote the orthogonal projections onto C and Q, respectively. The
pro-jections of a point onto C and Q are difficult to compute. In practical applications, however, the projections onto individual setsCiare more easily calculated than the projection onto
the intersection C. For this purpose, Censoret al.[] defined a proximity functionp(x) to measure the distance of a point to all sets
p(x) :=
t
i=
aiPCix–x +
r
j=
bjPQjAx–Ax
, ()
whereai> ,bj> for alliandj, respectively, and t
i=
ai+ r
j=
bj= .
With the proximity function (), they proposed an optimization model
minp(x)|x∈X ()
to approach the () and exerted the projection gradient method to solve it with
xk+=PC
xk–s∇pxk ,
where∇pdenotes the gradient ofp(x) and can be shown as follows (see []):
∇p(x) =
t
i=
ai
x–PCi(x)
+
r
j=
bj
Ax–PQj(Ax)
.
In this paper, we continue the algorithmic improvement on the constrained MSSFP. More specifically, the constrained MSSFP [] is to findx*such that
x*∈X∩
t
i=
Ci and Ax*∈Y∩ r
j=
Qj
. ()
By the same idea of approaching ()viathe model (), we definep:Rn→Randp:
Rm→Ras follows:
p(x) =
t
i=
aix–PCix
, ()
p(y) =
r
j=
bjy–PQjy
Then we get the following optimization model which can solve ():
minp(x) +p(Ax)|x∈X,y∈Y . ()
It is easy to see that model () is nonnegative and with the minimal value zero. So, we can further reformulate () into the following separable form:
minp(x) +p(y)|Ax–y= ,x∈X,y∈Y . ()
2 Preliminaries
In this section, we present some concepts and properties of the MSSFP. LetMbe a positive definite matrix. We denote theM-norm byvM =
√
v,Mv. In particular,v=√v,vis the Euclidean norm of the vectorv∈Rn.
Lemma Let S be a nonempty closed convex subset of Rn.We denote PS(·)as the projection
onto S,i.e.,
PS(z) =arg min
z–x|x∈S .
Then the following properties hold: () x∈S⇔PS(x) =x;
() x–PS(x),z–PS(x) ≤,∀x∈Rnand∀z∈S;
() x–y,PS(x) –PS(y) ≥ PS(x) –PS(y),∀x,y∈Rn;
() PS(x) –z≤ x–z–PS(x) –x,∀x∈Rnand∀z∈S;
() PS(x) –PS(y) ≤ x–y,∀x,y∈Rn.
Proof See Facchinei and Pang [, ].
Definition LetFbe a mapping fromS⊆RnintoRn, then
(a) Fis called monotone on S if
F(x) –F(y),x–y≥, ∀x,y∈S.
(b) Fis called strongly monotone on S if there is aμ> such that
F(x) –F(y),x–y≥μx–y, ∀x,y∈S.
(c) Fis called co-coercive (orν-inverse strongly monotone) onSif there is aν> such that
F(x) –F(y),x–y≥νF(x) –F(y), ∀x,y∈S.
(d) Fis called pseudo-monotone onSif
(e) Fis called Lipschitz continuous onSif there exists a constantL> such that
F(x) –F(y)≤Lx–y, ∀x,y∈S
andFis called nonexpansive iffL= .
Remark From Lemma and the above definition, we can infer that a monotone
map-ping is a pseudo-monotone mapmap-ping. An inverse strongly monotone mapmap-ping is mono-tone and Lipschitz continuous. A Lipschitz continuous and strongly monomono-tone mapping is a strongly monotone mapping. The projection operator is -ism and nonexpansive.
Lemma A mapping F is-ism if and only if the mapping I–F is-ism,where I is the
identity operator.
Proof See [, Lemma .].
Remark IfFis an inverse strongly monotone mapping, thenFis a nonexpansive
map-ping.
Definition LetSbe a nonempty closed convex subset ofHandxnbe a sequence inH,
then the sequencexnis calledFejérmonotone with respect toSif
xn+–z ≤ xn–z, ∀n≥,∀z∈S.
Lemma Let p(x)and p(x)be defined in()-(),then∇p(x)and∇p(y)are both
Lips-chitz continuous and inverse strongly monotone on X and Y,respectively.
Proof From the definition (),p(x) is differentiable onXand
∇p(x) = t
i=
ai
x–PCi(x)
.
Since the projection operatorPCiis -ism (from Remark ), then from Lemma , the
op-eratorI–PCiis -ism and is also nonexpansive. So, we have ∇p(x) –∇p(x)=t
i=
ai(I–PCi)(x–x)
≤
t
i=
ai(I–PCi)(x–x)
≤
t
i=
ai
x–x,
therefore,∇pi(x) is Lipschitz continuous onX, and the Lipschitz constant isL= t
i=ai.
It also follows from [, Corollary ] that∇p(x) is L-ism. Similarly, we can prove that
∇p(y) is Lipschitz continuous onY, and the Lipschitz constant isL= r
j=bj,
further-more, it is L
For notational convenience, let
M=
⎛ ⎜ ⎝
τIn
σIm βIm
⎞ ⎟ ⎠,
whereτ,σ andβare given positive scalars.
F(ω) =
⎛ ⎜ ⎝
∇p(x)
∇p(y)
Ax–y
⎞ ⎟ ⎠,
ξ(ω,ω) =ˆ
⎛ ⎜ ⎝
ξx(ω,ω)ˆ
ξy(ω,ω)ˆ
⎞ ⎟ ⎠= ⎛ ⎜ ⎝
∇p(x) –∇p(xˆ)
∇p(y) –∇p(yˆ)
⎞ ⎟ ⎠, () η(ω) = ⎛ ⎜ ⎝
ηx(ω)
ηy(ω)
⎞ ⎟ ⎠= ⎛ ⎜ ⎝
βAT(Ax–y)
–β(Ax–y)
⎞ ⎟
⎠, ()
d(ω,ω) =ˆ M(ω–ω) –ˆ ξ(ω,ω),ˆ
q(ω,ω) =ˆ F(ω) +ˆ η(ω),
ϕ(ω,ω) =ˆ ω–ω,ˆ d(ω,ω)ˆ +z–ˆz,Ax–y.
Furthermore, we letω= (x,y,z). Suppose that (x*,y*) is an optimal solution of the
prob-lem (). Then the constrained MSSFP () is equivalent to findingω*= (x*,y*,z*)∈W=
X×Y×Rnsuch that for anyω´= (x´,y´,´z)∈W, we have
⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩
´x–x*,∇p(x*) ≥;
´y–y*,∇p
(y*) ≥;
´z–z*,Ax*–y* ≥.
()
3 Main results
In this section, we will present our method for solving the MSSFP and prove its conver-gence. Our algorithm is defined as follows:
Algorithm . Step . Give arbitraryν∈(, ),β> ,γ ∈(, ),μ> ,τ> ,σ> and
x,y,z. Letε> be the error tolerance for an approximate solution and setk= .
Step .
() Find the smallest nonnegative integerlksuch thatτk=μlkτk–and
ˆ
xk=PX
xk– τk
∇p
xk+βATAxk–yk, ()
which satisfies
xk–xˆk,∇p
xk–∇p
ˆ
xk+βAxk–Axˆk≤τkνxk–xˆk
() Find the smallest nonnegative integermksuch thatσk=μmkσk–and
ˆ
yk=PY
yk– σk
∇p
yk–βAxk–yk, ()
which satisfies
yk–yˆk,∇p
yk–∇p
ˆ
yk+βyk–ˆyk≤σkνyk–ˆyk
; ()
() Then we definezˆkby ˆ
zk=zk–βAxˆk–yˆk. ()
Step .
Letωk= (xk,yk,zk), then we get the new iterateωk+via
ωk+=ωk–γ α*kdωk,ωˆk, ()
where
α*k= ϕ(ω
k,ωˆk)
d(ωk,ωˆk)). ()
Step .
If pp(x(xk+))≤ε, stop. Otherwise, setk=k+ and go to Step .
Remark In fact, from Lemma , we know that∇p(x) is Lipschitz continuous with a
constantL, then the left-hand side of () satisfies
xk–xˆk,∇p
xk–∇p
ˆ
xk+βAxk–Axˆk≤L+βATAxk–xˆk
.
So, () holds as long asτk≥(L+βA TA)
υ . Since
(L+βATA)
υ > , it hasinf{τk}> denoted
byτ=inf{τk}. On the other hand, by a similar analysis as in [, Lemma .], it indicates
thatτk≤(L+βA TA)
υ , so we have
τ≤τk≤τmax=
(L+βATA)
υ , ∀k> . ()
Similarly, we can also have
σ≤σk≤σmax=
(L+β)
υ , ∀k> . ()
Next, we analyze the convergence of Algorithm .:
Lemma Supposeωkandωˆkare generated by Algorithm.,andω*= (x*,y*,z*)is a
so-lution of().Then there exits m> for any k≥such that
Proof First, we proveωk–ω*,d(ωk,ωˆk) ≥ϕ(ωk,ωˆk).
From the property of the projection operator in Lemma ,
x–PS(x),z–PS(x)
≤, ∀x∈Rnand∀z∈S.
Combining it with (), we have
x–xˆk,xˆk–xk+ τk
∇p
xk+βATAxk–yk≥, ∀x∈X.
Multiplying byτk, we get
x–xˆk,τk
ˆ
xk–xk+∇p
xk–∇p
ˆ
xk+∇p
ˆ
xk+βATAxk–yk≥, ∀x∈X. And from the definitions ofξx(ωk,ωˆk),ηx(ωk) in () and (), it is equivalent to
x–xˆk,τk
ˆ
xk–xk+ξx
ωk,ωˆk+∇p
ˆ
xk+ηx
ωk≥, ∀x∈X. () Similarly, from () and (), we can also get
y–ˆyk,σk
ˆ
yk–yk+ξy
ωk,ωˆk+∇p
ˆ
yk+ηy
ωk≥, ∀y∈Y ()
and
z–zˆk,Axˆk–yˆk+ β
ˆ
zk–zk≥, ∀z∈Rn. ()
Using the notation defined above, from ()-(), we have
ω–ωˆk,Fωˆk+ηωk+ξωk,ωˆk–Mk
ωk–ωˆk≥, ∀ω∈W, namely
ω–ωˆk,qωk,ωˆk–dωk,ωˆk≥, ∀ω∈W. () Note thatF(ω) is monotone onWbecause of the monotonicity of∇p(x) and∇p(y).
From (), we have
ˆ
ωk–ω*,Fωˆk≥ωˆk–ω*,Fωˆ*≥. () Consequently,
ˆ
ωk–ω*,qωk,ωˆk=ωˆk–ω*,Fωˆk+ωˆk–ω*,ηωk ≥ωˆk–ω*,ηωk
=xˆk–x*,βATAxk–yk+ˆyk–yk, –βAxk–yk
=βAxˆk–Ax*–yˆk+y*,Axk–yk
=βAxˆk–yˆk,Axk–yk
=zk–ˆzk,Axk–yk
=ϕωk,ωˆk–ωk–ωˆk,dωk,ωˆk, () where the first inequality follows from (), the second equality follows from the definition ofωˆk,ω*andη(ωk).
Settingω=ω*in (), sinceω*∈Wis a solution, we get
ˆ
ωk–ω*,dωk,ωˆk≥ωˆk–ω*,qωk,ωˆk. Then
ωk–ω*,dωk,ωˆk=ωk–ωˆk,dωk,ωˆk+ωˆk–ω*,dωk,ωˆk ≥ωk–ωˆk,dωk,ωˆk+ωˆk–ω*,qωk,ωˆk ≥ϕωk,ωˆk,
where the last inequality follows from (). Next, we prove
ϕωk,ωˆk≥mωk–ωˆk.
From the definition ofϕ(ωk,ωˆk) andd(ωk,ωˆk), we obtain
ϕωk,ωˆk=ωk–ωˆk,dωk,ωˆk+zk–zˆk,Axk–yk
=ωk–ωˆk,Mωk–ωˆk–ωk–ωˆk,ξωk–ωˆk+zk–ˆzk,Axk–yk
=ωk–ωˆkM
k–
xk–xˆk,ξx
ωk,ωˆk–yk–yˆk,ξy
ωk,ωˆk
+zk–zˆk,Axk–yk
=ωk–ωˆk
Mk–
xk–xˆk,∇p
xk–∇p
ˆ
xk–yk–yˆk,∇p
yk–p
ˆ
yk
+zk–zˆk,Axk–yk
≥ωk–ωˆkM
k–τkνx k–xˆk
+βAxk–Axˆk–σkνyk–yˆk
+βyk–yˆk+zk–zˆk,Axˆk–yˆk+zk–zˆk,Axk–Axˆk–yk+yˆk
≥τk( –ν)xk–xˆk
+σk( –ν)yk–yˆk
+ βz
k–ˆzk
+βAxk–Axˆk+zk–zˆk,Axk–Axˆk+βyk–yˆk
–zk–ˆzk,yk–ˆyk, ()
where the first inequality follows from () and (). Note that
βAxk–Axˆk+zk–zˆk,Axk–Axˆk=βAxk–Axˆk+ β
zk–zˆk
–
βz
and
βyk–yˆk–zk–zˆk,yk–yˆk=βyk–yˆk– β
zk–zˆk
–
βz
k–ˆzk.
Substituting them into (), we have
ϕωk,ωˆk≥τk( –ν)xk–xˆk
+σk( –ν)yk–yˆk
+
βz
k–zˆk
+βAxk–Axˆk+ β
zk–ˆzk
+βyk–yˆk– β
zk–zˆk
≥τk( –ν)xk–xˆk
+σk( –ν)yk–yˆk
+
βz
k–zˆk
.
For the sequences{τk}and{σk}are bounded from () and (), let
m=min
( –ν)τ, ( –ν)σ,
β
,
therefore,
ϕωk,ωˆk≥mωk–ωˆk.
This completes the proof.
Next, we prove the sequenceωkisFejérmonotone.
Theorem Supposeωk andωˆk are generated by Algorithm.,andω*= (x*,y*,z*)is a
solution of().Then there exits C> for any k≥such that
ωk+–ω*≤ωk–ω*–m
r( –r)
C ω k–ωˆk
.
Proof From (), we have
ωk+–ω* =ωk–γ α*kdωk,ωˆk–ω*
=ωk–ω*– γ αk*ωk–ω*,dωk,ωˆk+γ αk*dωk,ωˆk ≤ωk–ω*– γ α*kϕωk,ωˆk+γαk*ϕωk,ωˆk
=ωk–ω*–γ αk*( –γ)αk*ϕωk,ωˆk ≤ωk–ω*–γ αk*( –γ)mωk–ωˆk,
where the inequalities follow from Lemma and ().
Becauseξx(ωk,ωˆk) ≤Lxk–xˆkandξy(ωk,ωˆk) ≤Lyk–yˆk, and Mk
ωk–ωˆk=τkxk–xˆk+σkyk–yˆk+
βz
k–ˆzk
≤τmaxxk–xˆk+σmaxyk–yˆk+
βz
k–zˆk
we have
dωk,ωˆk=Mk
ωk–ωˆk–ξωk,ωˆk
≤Mk
ωk–ωˆk+ξωk,ωˆk
≤(L+τmax)xk–xˆk+ (L+σmax)yk–yˆk+
βz
k–ˆzk.
LetC=max{(L+τmax), (L+σmax),β}, then we can getd(ω
k,ωˆk) ≤Cωk–ωˆk.
So, from Lemma , we can yield
α*k= ϕ(ω
k,ωˆk)
d(ωk,ωˆk) ≥
m C.
Therefore,
ωk+–ω*≤ωk–ω*–m
r( –r)
C ω k–ωˆk
.
Theorem The sequenceωkgenerated by Algorithm.converges to a solution of().
Proof Supposeω*is a solution of (). It follows from Theorem that
ωk+–ω*≤ · · · ≤ωk–ω*≤ω–ω*, ()
which means that the sequenceωkis bounded. Thus, it has at least a cluster point. Furthermore, Theorem also shows that
mr( –r)
C ω
k–ωˆk≤ωk–ω*–ωk+–ω*.
Summing both sides for allk, we obtain
mr( –r)
C
∞
k=
ωk–ωˆk≤ ∞
k=
ωk–ω*–ωk+–ω* ≤ω–ωˆ*,
which means that
lim
x→∞ω
k–ωˆk= .
So,{ωk}and{ ˆωk}have the same cluster points. Without loss of generality, letω¯ be a
cluster point of{ωk}and{ ˆωk},τ¯andσ¯ be the cluster points of{τk}and{σk}, respectively.
Let{ωkj},{ ˆωkj},{τ
kj},{σkj}be the subsequences converging to them. Then, by taking limits
over the subsequences in (), (), (), we have
¯
x=PX
¯
x– τ
∇p(x¯) +βAT(Ax¯–y¯)
,
¯
y=PY
¯
y– σ
∇p(y¯) –β(Ax¯–y¯))
¯
z=z¯–β(Ax¯–y¯).
It then follows from [] thatω¯ is a solution of ().
Because of the arbitraryω*, we can takeω*=ω¯ in () and obtain
ωk+–ω¯≤ωk–ω¯, ∀k≥.
Therefore, the whole sequence{ωk}converges toω. This completes the proof.¯
Remark Our iteration method is simpler in the form and is an improvement of the
corresponding result of [].
4 Applications
The multiple-sets split feasibility problem (MSSFP) requires to find a point closest to a family of closed convex sets in one space such that its image under a linear transformation will be closest to another family of closed convex sets in the image space. It serves as a model for real-word inverse problems where constraints are imposed on the solution in the domain of a linear operator as well as in the operator’s range.
In this paper, our algorithm converges to a solution of the multiple-sets split feasibility problem (MSSFP), for any starting vectorω= (x,y,z), whenever the MSSFP has a
so-lution. In the inconsistent case, it finds a point which is least violating the feasibility by being ‘closest’ to all sets as ‘measured’ by a proximity function.
In the general case, computing the projection in the MSSFP is difficult to implement. So, Yang [] solves this problem by the relaxed CQ-algorithm. Without loss of generality, take the two-sets split feasibility problem for instance. He assumes the sets C and Q are nonempty and given by
C=x∈RN|c(x)≤ , and Q=y∈RN|q(y)≤ ,
wherec:RN →Randq:RM →Rare convex functions, respectively. Here he uses the subgradient projections instead of the orthogonal projections. This is a huge achievement and it enables the split feasibility problem to achieve computer operation.
Lastly, we want to say that our work is related to significant real-world applications. The multiple-sets split feasibility problem was applied to the inverse problem of intensity-modulated radiation therapy (IMRT). In this field, beams of penetrating radiation are di-rected at the lesion (tumor) from external sources in order to eradicate the tumor without causing irreparable damage to surrounding healthy tissues; see,e.g., [].
Competing interests
The authors declare that they have no competing interests.
Authors’ contributions
YD proposed the algorithm, carried out the proof of this paper and drafted the manuscript. RC participated in the design of this paper and coordination. Both of the authors read and approved the final manuscript.
Acknowledgements
We wish to thank the referees for their helpful comments and suggestions. This research was supported by the National Natural Science Foundation of China, under the Grant No.11071279.
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