# Strong Convergence of Modified Algorithms Based on the Regularization for the Constrained Convex Minimization Problem

Authors: Ming Tian; Jun-Ying Gong
Journal: Abstract and Applied Analysis
 (2014)

---

## Abstract

As is known, the regularization method plays an important role in solving constrained convex minimization problems. Based on the idea of regularization, implicit and explicit iterative algorithms are proposed in this paper and the sequences generated by the algorithms can converge strongly to a solution of the constrained convex minimization problem, which also solves a certain variational inequality. As an application, we also apply the algorithm to solve the split feasibility problem.


---

## Body

## 1. Introduction

Assume thatH is a Hilbert space with inner product 〈·〉 and norm ∥·∥ induced by its inner product. Let C be a nonempty, closed, and convex subset of H. Recall that the projection from H onto C, denoted by ProjC, assigns, to each x∈H, the unique point ProjCx∈C with the property
(1)∥x-ProjCx∥=inf⁡{∥x-y∥:y∈C}.Below we introduce some nonlinear operators. LetT,A: H→H be nonlinear operators.(i)T is L-Lipschitzian if ∥Tx-Ty∥≤L∥x-y∥, for all x,y∈H, where L≥0 is a constant.In particular, ifL∈[0,1), then T is called a contraction on H; if L=1, then T is called a nonexpansive mapping on H. We know that the projection is nonexpansive.(ii)A is monotone if, for all x,y∈H,
(2)〈x-y,Ax-Ay〉≥0.(iii) Given numberβ>0, A is said to be β-strongly monotone, if
(3)〈x-y,Ax-Ay〉≥β∥x-y∥2,x,y∈H.(iv) Given numberv>0, A is said to be v-inverse strongly monotone (v-ism) if
(4)〈x-y,Ax-Ay〉≥v∥Ax-Ay∥2,x,y∈H.IfT is nonexpansive, then I-T is monotone.We know that the gradient-projection algorithm can be used to solve the constrained convex minimization problem. Let us recall the concrete analysis below. Consider the following constrained convex minimization problem:(5)min⁡x∈Cf(x),
where f:C→R is a real-valued function. Assume that f is Fréchet differentiable; define a sequence {xn} by
(6)xn+1=ProjC(I-γ∇f)(xn),n≥0,
where the initial guess is taken from C and the parameter γ is a real number which satisfies certain conditions. The convergence of algorithm (6) depends on the property of ∇f. In fact, if ∇f is only inverse strongly monotone, then algorithm (6) can converge weakly to a solution of the minimization problem (5). In 2011, Xu [1] provided an alternative averaged mapping approach to the gradient-projection algorithm; he also constructed a counterexample to prove that algorithm (6) has weak convergence only, in infinite-dimensional space. He also provided two modifications to ensure that the gradient-projection algorithms can converge strongly to a solution of (5). More investigations about the gradient-projection algorithm and its important role in solving the constrained convex minimization problem can be seen in [2–11]. Recently, the method has also been applied to solve the split feasibility problems which find application in image reconstruction and the intensity modulated radiation therapy (see [12–17]). However, sometimes the minimization problem has more than one solution, so regularization is needed.Consider the regularized minimization problem(7)min⁡x∈Cfα(x)∶=f(x)+α2∥x∥2;
here α>0 is the regularization parameter, and again f is Fréchet differentiable and the gradient ∇f is (1/L)-ism.On the gradient-projection method based on the regularization, we have weak convergence result as follows: define a sequence by(8)xn+1∶=ProjC(I-γ∇fαn)(xn),n≥0,
where αn>0, 0<λ<2/L, and ∑n=1∞αn<∞. Then the sequence generated by (8) converges weakly to a minimizer of (5) in the setting of infinite-dimensional space (see [17]).On the other hand, Tian [18] proposed the following iterative method:
(9)xn+1=snλh(xn)+(I-snμF)Txn,
where T is a nonexpansive mapping on H with a fixed point, h is a contraction on H with coefficient 0<α<1, and F:H→H is a k-Lipschitzian and η-strongly monotone operator with k, η>0. Letting 0<μ<2η/k2, 0<λ<μ(η-μk2/2)/ρ=τ/ρ, he proved that the sequence {xn} generated by (9) can converge strongly to a fixed point x~∈Fix⁡(T), which solves the variational inequality 〈(λh-μF)x~,x-x~〉≤0,  x∈Fix⁡(T).Combing the idea of regularization with Tian’s iterative scheme, in this paper, we will construct a new algorithm. The algorithm can not only find the minimum-solution of the constrained convex minimization problem by a single step but also ensure the strong convergence. In fact, the sequence generated by the constructed algorithm can converge strongly to a minimizer of the constrained convex minimization problem. The obtained point is also a solution of a certain variational inequality. Then we also apply the constructed algorithm to solve a split feasibility problem.


## 2. Preliminaries

Lemma 1 (see [18]).
LetC be a nonempty, closed, convex subset of a real Hilbert space H. Let h:C→C be a contraction with coefficient 0<ρ<1 and let F:C→C be k-Lipschitzian and η-strongly monotone with k, η>0. Then, for 0<λ<μη/ρ,
(10)〈x-y,(μF-λh)x-(μF-λh)y〉≥(μη-λρ)∥x-y∥2,∀x,y∈C.Lemma 2 (see [19]).
LetC be a closed and convex subset of a Hilbert space H and let T:C→C be a nonexpansive mapping with Fix⁡(T)≠∅. If {xn}n=1∞ is a sequence in C weakly converging to x and if {(I-T)}n=1∞ converges strongly to y, then (I-T)x=y.Lemma 3 (see [20]).
LetC be a closed and convex subset of a Hilbert space H. Given x∈H and y∈C, then y=ProjCx if and only if there holds the inequality
(11)〈x-y,y-z〉≥0,∀z∈C.Lemma 4 (see [1]).
Assume that{an}n=0∞ is a sequence of nonnegative real numbers such that
(12)an+1≤(1-γn)an+γnδn+βn,n≥0,
where {γn}n=0∞ and {βn}n=0∞ are sequences in (0,1) and {δn}n=0∞ is a sequence in R such that(i)
∑n=0∞γn=∞;(ii)
eitherlim⁡⁡sup⁡n→∞⁡δn≤0 or ∑n=0∞γn|δn|<∞;(iii)
∑n=0∞βn<∞.Then lim⁡n→∞⁡an=0.We will use the following notation:(i)
⇀ for weak convergence and → for strong convergence.


## 3. Main Results

Assume thatH is a Hilbert space with inner product 〈·〉 and norm ∥·∥ induced by its inner product. Let C be a nonempty, closed, and convex subset of H. Assume that the minimization problem (5) is consistent and its solution set is denoted by S.Rewrite the regularized constrained convex minimization problem:(13)min⁡x∈Cfα(x)∶=f(x)+α2∥x∥2.Recall thath is a ρ-contraction on C with 0<ρ<1, and F is k-Lipschitzian and η-strongly monotone on C with k, η>0. Given t∈(0,1), assume that αt is continuous with respect to t and that αt=o(t)(t→0); then there exists a constant D>0 such that |αt/t|<D for t≤t0. Let the gradient ∇f be (1/L)-ism. For each t∈(0,min⁡(1,1/τ)), we consider the mapping Xt on C defined by
(14)Yt(x)∶=tλh(x)+(I-tμF)ProjC(I-γ∇fαt)(x),Xt(x)∶=ProjCYt(x),
where 0<μ<2η/k2, 0<λ<μ(η-μk2/2)/ρ=τ/ρ, and 0<γ<2/L.We will use the following notation in Lemma5, Proposition 6, and Theorem 7:
(15)Tαt∶=ProjC(I-γ∇fαt);T∶=ProjC(I-γ∇f).Lemma 5.
There exists an implicit algorithm{xt}; here xt is the fixed point of Xt; that is,
(16)xt=Xt(xt)=ProjCYt(xt),
where Xt, Yt are defined by (14).Proof.
Below we will show thatXt is a contraction. Indeed, we have
(17)∥(I-tμF)Tαt(x)-(I-tμF)Tαt(y)∥2WW=∥Tαt(x)-Tαt(y)∥2+t2μ2∥FTαt(x)-FTαt(y)∥2WWW-2tμ〈Tαt(x)-Tαt(y),FTαt(x)-FTαt(y)〉WW≤∥Tαt(x)-Tαt(y)∥2+t2μ2k2∥Tαt(x)-Tαt(y)∥2WWW-2tμη∥Tαt(x)-Tαt(y)∥2WW≤(1-tμ(2η-tμk2)2)2∥Tαt(x)-Tαt(y)∥2WW≤(1-tτ)2∥x-y∥2.
It follows easily that
(18)∥Xt(x)-Xt(y)∥=∥ProjCYt(x)-ProjCYt(y)∥≤∥Yt(x)-Yt(y)∥=∥tλh(x)+(I-tμF)Tαt(x)-tλh(y)-(I-tμF)Tαt(y)∥≤∥tλh(x)-tλh(y)∥W+∥(I-tμF)Tαt(x)-(I-tμF)Tαt(y)∥≤(1-t(τ-λρ))∥x-y∥.
Hence Xt has a unique fixed point; we denote it by xt, which uniquely solves the fixed point equation
(19)xt=ProjCYt(xt);
here Yt(xt)=tλh(xt)+(I-tμF)ProjC(I-γ∇fαt)(xt).Proposition 6.
Letxt be defined by (19):(i)
{xt} is bounded for t∈(0,1).(ii)
lim⁡t→0⁡∥xt-ProjC(I-γ∇fαt)(xt)∥=0.(iii)
xt defines a continuous curve from (0,1/τ) into C.Proof.
(i) Forx¯∈S, we have
(20)∥xt-x¯∥W=∥ProjCYt(xt)-ProjC(x¯)∥W≤∥Yt(xt)-x¯∥W=∥tλh(xt)+(I-tμF)Tαt(xt)-x¯∥W=∥tλh(xt)-tλh(x¯)+(I-tμF)Tαt(xt)WWW-(I-tμF)Tαt(x¯)+(I-tμF)Tαt(x¯)WWW-(I-tμF)T(x¯)+tλh(x¯)-tμF(x¯)tλh(xt)-tλh(x¯)+(I-tμF)Tαt(xt)∥W≤∥tλh(xt)-tλh(x¯)∥WW+∥(I-tμF)Tαt(xt)-(I-tμF)Tαt(x¯)∥WW+∥(I-tμF)Tαt(x¯)-(I-tμF)T(x¯)∥WW+t∥λh(x¯)-μF(x¯)∥W≤tλρ∥xt-x¯∥+(1-tτ)∥xt-x¯∥WW+(1+tμk)γαt∥x¯∥+t∥λh(x¯)-μF(x¯)∥W≤(1-t(τ-λρ))∥xt-x¯∥+(1+μk)γαt∥x¯∥WW+t∥λh(x¯)-μF(x¯)∥W≤(1-t(τ-λρ))∥xt-x¯∥+(1+μk)γtD∥x¯∥WW+t∥λh(x¯)-μF(x¯)∥.
It follows that ∥xt-x¯∥≤(∥λh(x¯)-μF(x¯)∥+(1+μk)γD∥x¯∥)/(τ-λρ). Hence, {xt} is bounded.
(ii) We can easily see that(21)∥xt-ProjC(I-γ∇fαt(xt))∥WW=∥ProjCYt(xt)-ProjCTαt(xt)∥WW≤∥Yt(xt)-Tαt(xt)∥WW=∥tλh(xt)+(I-tμF)Tαt(xt)-Tαt(xt)∥WW=t∥λh(xt)-μFProjC(I-γ∇fαt)(xt)∥WW≤t(λ∥h(xt)∥+μ∥FProjC(I-γ∇fαt)(xt)∥).
The boundedness of {xt} implies that {h(xt)} and {FProjC(I-γ∇fαt)(xt)} are also bounded. Hence (ii) follows.
(iii) Fort,t0∈(0,1/τ), we have
(22)∥xt-xt0∥W=∥ProjCYt(xt)-ProjCYt0(xt0)∥W≤∥Yt(xt)-Yt0(xt0)∥W=∥tλh(xt)+(I-tμF)Tαt(xt)-t0λh(xt0)Tαt0(xt0)WWW-(I-t0μF)Tαt0(xt0)∥W≤|t-t0|λ∥h(xt)∥+t0λ∥h(xt)-h(xt0)∥WW+∥(I-tμF)Tαt(xt)-(I-t0μF)Tαt0(xt0)∥W≤|t-t0|λ∥h(xt)∥+t0λρ∥xt-xt0∥WW+∥(I-tμF)Tαt(xt)-(I-t0μF)Tαt0(xt0)∥WW+∥(I-t0μF)Tαt0(xt)-(I-t0μF)Tαt0(xt)∥W≤|t-t0|λ∥h(xt)∥+t0λρ∥xt-xt0∥WW+(1-t0τ)∥xt-xt0∥WW+∥(I-tμF)Tαt(xt)-(I-t0μF)Tαt0(xt)∥W≤|t-t0|λ∥h(xt)∥+t0λρ∥xt-xt0∥WW+(1-t0τ)∥xt-xt0∥+∥Tαt(xt)-Tαt0(xt)∥WW+∥tμFTαt(xt)-t0μFTαt0(xt)∥W≤|t-t0|λ∥h(xt)∥+(1-t0(τ-λρ))∥xt-xt0∥WW+|αt-αt0|γ∥xt∥+|t-t0|μ∥FTαt(xt)∥WW+t0μkγ|αt-αt0|∥xt∥.
Hence, ∥xt-xt0∥≤((λ∥h(xt)∥+μ∥FProjC(I-γ∇fαt)(xt)∥)/t0(τ-λρ))|t-t0| + ((1+t0μk)γ∥xt∥/t0(τ-λρ))|αt-αt0|. Noting that αt is continuous with respect to t, we get xt→xt0 as t→t0, and therefore xt is continuous.Theorem 7.
Letxt be defined by (19). Then xt converges in norm, as t→0, to a minimizer x* of (5), which solves the following variational inequality:
(23)〈(μF-λh)x*,x~-x*〉≥0,∀x~∈S.Proof.
It follows from Lemma1 that the variational inequality (23) has only one solution x*∈S. To prove convergence we will use Lemma 3 in the following calculations. It holds that
(24)∥xt-x~∥2=〈xt-x~,xt-x~〉=〈ProjCyt-x~,xt-x~〉=〈ProjCyt-yt+yt-x~,xt-x~〉=〈ProjCyt-yt,ProjCyt-x~〉+〈yt-x~,xt-x~〉≤〈yt-x~,xt-x~〉=〈tλh(xt)+(I-tμF)Tαt(xt)-x~,xt-x~〉=〈tλh(xt)-tλh(x~),xt-x~〉W+〈(I-tμF)Tαt(xt)-(I-tμF)Tαt(x~),xt-x~〉W+〈(I-tμF)Tαt(x~)-(I-tμF)T(x~),xt-x~〉W+〈tλh(x~)-tμF(x~),xt-x~〉≤tλρ∥xt-x~∥2+(1-tτ)∥xt-x~∥2W+(1+tμk)γαt∥x~∥∥xt-x~∥+t〈λh(x~)-μF(x~),xt-x~〉≤(1-t(τ-λρ))∥xt-x~∥2W+(1+μk)γαt∥x~∥∥xt-x~∥W+t〈λh(x~)-μF(x~),xt-x~〉.
Hence,
(25)∥xt-x~∥2≤1τ-λρ〈λh(x~)-μF(x~),xt-x~〉W+αtt(1+μk)γτ-λρ∥xt-x~∥.
Since xt is bounded for t∈(0,1), and αt=o(t)(t→∞), we see that if tn is a sequence in (0,1) such that tn→0 and xtn⇀x¯, then, by (25), xtn→x¯. We may further assume that αtn→0. It turns out that
(26)∥xtn-ProjC(I-γ∇f)xtn∥W≤∥xtn-ProjC(I-γ∇fαtn)xtn∥+γαtn∥xtn∥.
From the boundedness of {xt} and limt→0∥xt-ProjC(I-γ∇fαt)xt-xt∥=0, we conclude that
(27)lim⁡n→∞⁡∥xtn-ProjC(I-γ∇f)xtn∥=0.
It then follows from Lemma 2 that x¯=ProjC(I-γ∇f)x¯. This shows that x¯∈S.
We next prove thatx¯ is a solution of the variational inequality (23). Since ProjC(I-γ∇fαt) is nonexpansive, we see that I-ProjC(I-γ∇fαt) is monotone. By (19), we have
(28)xt=ProjCYt(xt)-Yt(xt)+tλh(xt)+(I-tμF)Tαt(xt);
then it follows that
(29)λh(xt)=-1t[Yt(xt)-Yt(xt)+(I-tμF)Tαt(xt)-xt],(μF-λh)(xt)=μF(xt)W+1t[ProjCYt(xt)-Yt(xt)+(I-tμF)Tαt(xt)-xt],〈(μF-λh)(xt),xt-x~〉=〈μF(xt)+1t[ProjCYt(xt)-Yt(xt)WWWWWWW+(I-tμF)Tαt(xt)-xt],12xt-x~〉=1t〈ProjCYt(xt)-Yt(xt),xt-x~〉W+〈μF(xt)+1t[(I-tμF)Tαt(xt)-xt],xt-x~〉≤〈μF(xt)+1t[(I-tμF)Tαt(xt)-xt],xt-x~〉=μ〈F(xt)-FTαt(xt),xt-x~〉W-1t〈(I-Tαt)(xt),xt-x~〉=μ〈F(xt)-FTαt(xt),xt-x~〉W-1t〈(I-Tαt)(xt)-(I-Tαt)(x~)WWiW(I-Tαt)-Tαt(x~)+T(x~),xt-x~〉=μ〈F(xt)-FTαt(xt),xt-x~〉W-1t〈(I-Tαt)(xt)-(I-Tαt)(x~),xt-x~〉W+1t〈Tαt(x~)-T(x~),xt-x~〉≤μ〈F(xt)-FTαt(xt),xt-x~〉W+1tαtγ∥x~∥∥xt-x~∥.
Taking the limit through t=tn→0 ensures that x¯ is a solution of the variational inequality. This implies that 〈(μF-λh)(x¯),x~-x¯〉≥0. Therefore x¯=x*.Finally we consider the explicit version of our algorithm which is(30)yn=snλh(xn)+(I-snμF)ProjC(I-γ∇fαn)(xn),xn+1=ProjCyn,
where the initial guess x0∈C and {αn}n=0∞ and {sn}n=0∞⊂(0,1) are the parameter sequences satisfying certain conditions.Theorem 8.
Assume that the minimization problem (5) is consistent and let S denote its solution set. Let the gradient ∇f be (1/L)-ism. Let F:C→C be η-strongly monotone and k-Lipschitzian and let h:C→C be a contraction with coefficient 0<ρ<1. Fix a constant μ satisfying 0<μ<2η/k2, a constant λ with the property 0<λ<μ(η-μk2/2)/ρ=τ/ρ, and a constant γ satisfying 0<γ<2/L. Let {xn}n=0∞ be generated by the iterative algorithm (30). Assume(C1)
sn→0;(C2)
αn=o(sn)(sn→0,n→∞);(C3)
∑n=0∞sn=∞;(C4)
∑n=0∞|sn+1-sn|<∞;(C5)
∑n=0∞|αn+1-αn|<∞.Then the sequence {xn} converges in norm to x* as defined in Theorem 7.Proof.
We set(31)Tαn∶=ProjC(I-γ∇fαn);T:=ProjC(I-γ∇f).
Since αn=o(sn)(sn→0,n→∞), there exists M>0 such that αn/sn<M for n>n0.
We observe that{xn} is bounded. Indeed, taking a fixed point p∈S, we get
(32)∥xn+1-p∥=∥ProjCyn-ProjCp∥≤∥yn-p∥=∥snλh(xn)+(I-snμF)Tαn(xn)-p∥≤∥snλh(xn)-snλh(p)∥W+∥(I-snμF)Tαn(xn)-(I-snμF)Tαn(p)∥W+∥(I-snμF)Tαn(p)-(I-snμF)T(p)∥W+∥snλh(p)-snμF(p)∥≤snλρ∥xn-p∥+(1-snτ)∥xn-p∥W+(1+snμk)γαn∥p∥+sn∥λh(p)-μF(p)∥≤(1-sn(τ-λρ))∥xn-p∥W+(1+μk)γMsn∥p∥+sn∥λh(p)-μF(p)∥≤max⁡{∥xn-p∥,(1+μk)γM∥p∥+∥λh(p)-μF(p)∥τ-λρ}.
By induction ∥xn-p∥≤max⁡{∥x0-p∥,((1+μk)γM∥p∥+∥λh(p)-μF(p)∥)/(τ-λρ)},  n≥0, then {xn} is bounded, so are {h(xn)} and {FProjC(I-γ∇fαn)(xn)}.
We claim that∥xn+1-xn∥→0.
Because of the boundedness of{xn}, it can be easily seen that {h(xn)} and {FTαn(xn)} are also bounded. So we can take two constants N, E such that N≥λ∥h(xn-1)∥+μ∥FTαn(xn-1)∥ and E≥(1+μk)γ∥xn-1∥. Consider
(33)∥xn+1-xn∥=∥ProjCyn-ProjCyn-1∥≤∥yn-yn-1∥=∥snλh(xn)+(I-snμF)Tαn(xn)-sn-1λh(xn-1)WW-(I-sn-1μF)Tαn-1(xn-1)∥≤∥snλh(xn)-snλh(xn-1)∥W+∥snλh(xn-1)-sn-1λh(xn-1)∥W+∥(I-snμF)Tαn(xn)-(I-sn-1μF)Tαn-1(xn-1)∥≤snλρ∥xn-xn-1∥+|sn-sn-1|λ∥h(xn-1)∥W+∥(I-snμF)Tαn(xn)-(I-sn-1μF)Tαn-1(xn-1)∥≤(1-sn(τ-λρ))∥xn-xn-1∥W+|sn-sn-1|λ∥h(xn-1)∥W+∥Tαn(xn-1)-Tαn-1(xn-1)∥W+∥snμFTαn(xn-1)-sn-1μFTαn-1(xn-1)∥≤(1-sn(τ-λρ))∥xn-xn-1∥W+|sn-sn-1|λ∥h(xn-1)∥W+γ|αn-αn-1|∥xn-1∥W+∥snμFTαn(xn-1)-sn-1μFTαn-1(xn-1)∥≤(1-sn(τ-λρ))∥xn-xn-1∥W+|sn-sn-1|λ∥h(xn-1)∥+γ|αn-αn-1|∥xn-1∥W+|sn-sn-1|μ∥FTαn(xn-1)∥W+sn-1μkγ|αn-αn-1|∥xn-1∥≤(1-sn(τ-λρ))∥xn-xn-1∥W+(λ∥h(xn-1)∥+μ∥FTαn(xn-1)∥)|sn-sn-1|W+(1+μk)γ∥xn-1∥|αn-αn-1|≤(1-sn(τ-λρ))∥xn-xn-1∥+N|sn-sn-1|W+E|αn-αn-1|.
By Lemma 4, we obtain ∥xn+1-xn∥→0.
Next we show that(34)∥xn-ProjC(I-γ∇f)xn∥⟶0,(35)∥xn-ProjC(I-γ∇f)xn∥W≤∥xn-xn+1∥+∥xn+1-ProjC(I-γ∇f)xn∥W≤∥xn-xn+1∥WW+∥ProjCyn-ProjC[ProjC(I-γ∇f)xn]∥W≤∥xn-xn+1∥+∥yn-T(xn)∥W≤∥xn-xn+1∥WW+∥snλh(xn)+(I-snμF)Tαn(xn)-T(xn)∥W≤∥xn-xn+1∥WW+sn∥λh(xn)-μFProjC(I-γ∇fαn)(xn)∥WW+γαn∥xn∥⟶0.
We next show that(36)lim⁡n→∞⁡sup⁡〈xn-x*,(γh-μF)x*〉≤0,
where x* is obtained in Theorem 7. Indeed, take a subsequence {xnk} of {xn} such that
(37)lim⁡n→∞⁡sup⁡〈xn-x*,(γh-μF)x*〉WW=lim⁡k→∞⁡〈xnk-x*,(γh-μF)x*〉.
Since the sequence {xn} is bounded, we may assume that xnk⇀z; it follows from Lemma 2 and (34) that z∈S. Then we obtain limk→∞〈xnk-x*,(γh-μF)x*〉=〈z-x*,(γh-μF)x*〉≤0. By (37) we know that limn→∞sup⁡〈xn-x*,(γh-μF)x*〉≤0.
We finally show thatxn→x*. We have, using Lemma 3,
(38)∥xn+1-x*∥2=〈xn+1-x*,xn+1-x*〉=〈ProjCyn-yn+yn-x*,xn+1-x*〉=〈ProjCyn-yn,xn+1-x*〉+〈yn-x*,xn+1-x*〉≤〈yn-x*,xn+1-x*〉=〈snλh(xn)+(I-snμF)Tαn(xn)-x*,xn+1-x*〉=〈snλh(xn)-snλh(x*),xn+1-x*〉W+〈snλh(x*)-snμF(x*),xn+1-x*〉W+〈(I-snμF)Tαn(xn)-(I-snμF)Tαn(x*),xn+1-x*〉W+〈(I-snμF)Tαn(x*)-(I-snμF)(x*),xn+1-x*〉≤snλρ∥xn-x*∥∥xn+1-x*∥W+sn〈(λh-μF)x*,xn+1-x*〉W+(1-snτ)∥xn-x*∥∥xn+1-x*∥W+(1+snμk)γαn∥x*∥∥xn+1-x*∥≤12(1-sn(τ-λρ))(∥xn-x*∥2+∥xn+1-x*∥2)W+sn〈(λh-μF)x*,xn+1-x*〉W+(1+μk)γαn∥x*∥∥xn+1-x*∥.
Since {xn} is bounded, we can take a constant L′>0 such that
(39)L′≥(1+μk)γ∥x*∥∥xn+1-x*∥;
hence,
(40)∥xn+1-x*∥2W≤1-sn(τ-λρ)1+sn(τ-λρ)∥xn-x*∥2WW+2sn1+sn(τ-λρ)〈(λh-μF)x*,xn+1-x*〉WW+2(1+μk)γαn∥x*∥∥xn+1-x*∥1+sn(τ-λρ)W≤(1-sn(τ-λρ))∥xn-x*∥2WW+sn1+sn(τ-λρ)WW×[2〈(λh-μF)x*,xn+1-x*〉+2L′αnsn];
it then follows that
(41)∥xn+1-x*∥≤(1-sn(τ-λρ))∥xn-x*∥2+snδn,
where δn=(2〈(λh-μF)x*,xn+1-x*〉+2L′(αn/sn))/(1+sn(τ-λρ)).
Sincesn→0 and αn=o(sn)(sn→0,n→∞), by (37), we get limn→∞sup⁡  δn≤0. Applying Lemma 4 to (41) concludes that xn→x* as n→∞.


## 4. An Application

Since the split feasibility problem (say SFP, for short) was proposed by Censor and Elfving in 1994, it has been widely used in signal processing and image reconstruction, with particular progress in intensity-modulated radiation therapy. We know that the gradient-projection method plays an important role in solving the SFP. In this section, we provide an application of Theorem8 to the SFP (see [21, 22]).The SFP can mathematically be formulated as the problem of finding a pointx with the property
(42)x∈C,Bx∈Q,
where C and Q are nonempty, closed, and convex subset of Hilbert spaces H1 and H2, respectively. B:H1→H2 is a bounded linear operator.It is clear thatx* is a solution to the split feasibility problem if and only if x*∈C and Bx*-ProjCBx*=0. We define the proximity function f by
(43)f(x)=12∥Bx-ProjQBx∥2,
and we consider the constrained convex minimization problem
(44)min⁡x∈C12∥Bx-ProjQBx∥2.Thenx* solves the split feasibility problem (42) if and only if x* solves the minimization problem (44) with the minimal value equal to 0. Byrne introduced the so-called CQ algorithm to solve the SFP:
(45)xn+1=ProjC(I-γB*(I-ProjQ)B)xn,n≥0,
where 0<γ<2/∥B∥2. He obtained that the sequence xn generated by (45) converges weakly to a solution of the SFP.Now we consider the regularization technique; let(46)fα(x)=12∥Bx-PQBx∥2+α2∥x∥2;
then we establish the iterative scheme as follows:
(47)yn=snλh(xn)WW+(I-snμF)ProjC(I-γ(B*(I-PQ)B+αnI))xn,xn+1=ProjCyn,
where h:C→H is a contraction with the coefficient ρ∈(0,1). Let F:C→H be η-strongly monotone and k-Lipschitzian.Theorem 9.
Assume that the split feasibility problem (42) is consistent. Let the sequence {xn} be generated by (47). The μ and λ are constants with the same property as in Theorem 8.   Fix⁡0<γ<2/∥B∥2. The sequences {αn}n=1∞ and {sn}n=1∞⊂(0,1) are the parameter sequences satisfying conditions (C1)–(C5) in Theorem 8. Then the sequence {xn} generated by (47) converges strongly to the solution of split feasibility problem (42).Proof.
By the definition of the proximity functionf, we have
(48)∇f=B*(I-ProjQ)Bx,
and we show that ∇f is 1/∥B∥2-ism.
SinceProjQ is (1/2)-averaged mapping, then I-ProjQ is 1-ism:
(49)〈∇f(x)-∇f(y),x-y〉-1∥B∥2·∥∇f(x)-∇f(y)∥2W=〈B*(I-ProjQ)Bx-B*(I-ProjQ)By,x-y〉WW-1∥B∥2·∥B*(I-ProjQ)Bx-B*(I-ProjQ)By∥2W=〈B*[(I-ProjQ)Bx-(I-ProjQ)By],x-y〉WW-1∥B∥2·∥B*[(I-ProjQ)Bx-(I-ProjQ)By]∥2W=〈(I-ProjQ)Bx-(I-ProjQ)By,Bx-By〉WW-1∥B∥2·∥B∥2∥(I-ProjQ)Bx-(I-ProjQ)By∥2W≥∥(I-ProjQ)Bx-(I-ProjQ)By∥2WW-∥(I-ProjQ)Bx-(I-ProjQ)By∥2W=0.
Hence, 〈∇f(x)-∇f(y),x-y〉≥1/∥B∥2·∥∇f(x)-∇f(y)∥2. Let L=∥B∥2; then the iterative scheme (47) is equivalent to
(50)xn+1=ProjC[snλh(xn)+(I-snμF)ProjC(I-γ∇fαn)xn],WWWWWWWWWWWWWWWWWWWWn≥0,
where 0<γ<2/L, and then, due to Theorem 8, we have the conclusion immediately.


## 5. Numerical Result

In this section, we present the following simple example to judge the numerical performance of our algorithm. We use the algorithm in Theorem9 to illustrate its realization in solving system of linear equations.Example 10.
In Theorem9, we assume that H1=H2=R3. Take h=(1/2)I, where I denotes the 3×3 identity matrix, and F=I with Lipschitz constant k=1 and strongly monotone constant η=1. Given the parameters sn=1/(n+2), αn=1/(n+2)2 for every n≥0. Fix μ=1, λ=1/2, and γ=1/10. Take
(51)B=(101-11012-3),b=(5-7-17).
The SFP can be formulated as the problem of finding a point x* with the property
(52)x∈C,Bx*∈Q,
where C=H1, Q={b}⊂H2. That is, x* is the solution of system of linear equations Bx=b, and
(53)x*=(2-53).
Then by Theorem 9, the sequence {xn} is generated by
(54)xn+1=14(n+2)xn+n+1n+2(xn-110B*Bxn+110B*b-110(n+2)2xn).
As n→∞, we have {xn}→x*=(2,-5,3)T (Table 1).Table 1
Numerical results as regards Example10.
n
x
n
1
x
n
2
x
n
3
E
n
0
0.0000
1.0000
1.0000
6.63
E
+
00
50
1.8793
−4.6691
3.1319
3.76
E
-
01
500
1.9875
−4.9675
3.0125
3.70
E
-
02
2000
1.9969
−4.9919
3.0031
9.21
E
-
03



---

*Source: 870102-2014-10-28.xml*