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Iterative methods for constrained convex minimization problem in Hilbert spaces
Fixed Point Theory and Applications volume 2013, Article number: 105 (2013)
Abstract
In this paper, based on Yamada’s hybrid steepest descent method, a general iterative method is proposed for solving constrained convex minimization problem. It is proved that the sequences generated by proposed implicit and explicit schemes converge strongly to a solution of the constrained convex minimization problem, which also solves a certain variational inequality.
MSC:58E35, 47H09, 65J15.
1 Introduction
Let H be a real Hilbert space with inner product \u3008\cdot ,\cdot \u3009 and induced norm \parallel \cdot \parallel. Let C be a nonempty, closed and convex subset of H. We need some nonlinear operators which are introduced below.
Let T,A:H\to H be nonlinear operators.

T is nonexpansive if \parallel TxTy\parallel \le \parallel xy\parallel for all x,y\in H.

T is Lipschitz continuous if there exists a constant L>0 such that \parallel TxTy\parallel \le L\parallel xy\parallel, for all x,y\in H.

A:H\to H is monotone if \u3008xy,AxAy\u3009\ge 0, for all x,y\in H.

Given is a number \eta >0, A:H\to H is ηstrongly monotone if \u3008xy,AxAy\u3009\ge \eta {\parallel xy\parallel}^{2}, for all x,y\in H.

Given is a number \upsilon >0. A:H\to H is υinverse strongly monotone (υism) if \u3008xy,AxAy\u3009\ge \upsilon {\parallel AxAy\parallel}^{2}, for all x,y\in H.
It is known that inverse strongly monotone operators have been studied widely (see [1–3]), and applied to solve practical problems in various fields; for instance, in traffic assignment problems (see [4, 5]).

T:H\to H is said to be an averaged mapping if T=(1\alpha )I+\alpha S, where α is a number in (0,1) and S:H\to H is nonexpansive. In particular, projections are (1/2)averaged mappings.
Averaged mappings have received many investigations, see [6–10].
Consider the following constrained convex minimization problem:
where f:C\to R is a real valued convex function. Assume that the minimization problem (1.1) is consistent, and let S denote its solution set. It is known that the gradientprojection algorithm is one of the powerful methods for solving the minimization problem (1.1) (see [11–18]), and sometimes the minimization problem (1.1) has more than one solution. So, regularization is needed. We can use the idea of regularization to design an iterative algorithm for finding the minimumnorm solution of (1.1).
We consider the regularized minimization problem:
Here, \alpha >0 is the regularization parameter, f is convex function with LLipschitz continuous gradient ∇f. Let {x}_{\mathrm{min}} be minimumnorm solution of (1.1), namely, {x}_{\mathrm{min}} satisfies the property:
{x}_{\mathrm{min}} can be obtained by two steps. First, observing that the gradient \mathrm{\nabla}{f}_{\alpha}=\mathrm{\nabla}f+\alpha I is (L+\alpha )Lipschitzian and αstrongly monotone, the mapping {Proj}_{C}(I\gamma \mathrm{\nabla}{f}_{\alpha}) is a contraction with coefficient \sqrt{1\gamma (2\alpha \gamma {(L+\alpha )}^{2})}\le 1\frac{1}{2}\alpha \gamma, where 0<\gamma \le \frac{\alpha}{{(L+\alpha )}^{2}}. So, the regularized problem (1.2) has a unique solution, which is denoted as {x}_{\alpha}\in C and which can be obtained via the Banach contraction principle. Secondly, letting \alpha \to 0 yields {x}_{\alpha}\to {x}_{\mathrm{min}} in norm. The following result shows that for suitable choices of γ and α, the minimumnorm solution {x}_{\mathrm{min}} can be obtained by a single step.
Theorem 1.1 [9]
Assume that the minimization problem (1.1) is consistent and let S denote its solution set. Assume that the gradient ∇f is LLipschitz continuous. Let {\{{x}_{n}\}}_{n=0}^{\mathrm{\infty}} be generated by the following iterative algorithm:
Let \{{\gamma}_{n}\} and \{{\alpha}_{n}\} satisfy the following conditions:

(i)
0<{\gamma}_{n}\le {\alpha}_{n}/{(L+{\alpha}_{n})}^{2} for all n;

(ii)
{\alpha}_{n}\to 0 (and {\gamma}_{n}\to 0) as n\to \mathrm{\infty};

(iii)
{\sum}_{n=1}^{\mathrm{\infty}}{\alpha}_{n}{\gamma}_{n}=\mathrm{\infty};

(iv)
({\gamma}_{n}{\gamma}_{n1}+{\alpha}_{n}{\gamma}_{n}{\alpha}_{n1}{\gamma}_{n1})/{({\alpha}_{n}{\gamma}_{n})}^{2}\to 0 as n\to \mathrm{\infty}.
Then {x}_{n}\to {x}_{\mathrm{min}} as n\to \mathrm{\infty}.
In the assumptions of Theorem 1.1, the sequence \{{\gamma}_{n}\} is forced to tend to zero. If we keep it as a constant, then we have weak convergence as follows.
Theorem 1.2 [19]
Assume that the minimization problem (1.1) is consistent and let S denote its solution set. Assume that the gradient ∇f is LLipschitz continuous. Let {\{{x}_{n}\}}_{n=0}^{\mathrm{\infty}} be generated by the following iterative algorithm:
Assume that 0<\gamma <2/L and {\sum}_{n=1}^{\mathrm{\infty}}{\alpha}_{n}<\mathrm{\infty}. Then {\{{x}_{n}\}}_{n=0}^{\mathrm{\infty}} converges weakly to a solution of the minimization problem (1.1).
In 2001, Yamada [10] introduced the following hybrid steepest descent method:
where F:H\to H is kLipschitzian and ηstrongly monotone, and 0<\mu <2\eta /{k}^{2}. It is proved that the sequence {\{{x}_{n}\}}_{n=0}^{\mathrm{\infty}} generated by (1.5) converges strongly to {x}^{\ast}\in Fix(T), which solves the variational inequality:
In this paper, we introduce a modification of algorithm (1.4) which is based on Yamada’s method. It is proved that the sequence generated by our proposed algorithm converges strongly to a minimizer of (1.1), which is also a solution of a certain variational inequality.
2 Preliminaries
In this section, we introduce some useful properties and lemmas which will be used in the proofs for the main results in the next section.
Let the operators S,T,V:H\to H be given:

(i)
If T=(1\alpha )S+\alpha V, for some \alpha \in (0,1) and if S is averaged and V is nonexpansive, then T is averaged.

(ii)
The composition of finitely many averaged mappings is averaged. That is, if each of the mappings {\{{T}_{i}\}}_{i=1}^{N} is averaged, then so is the composite {T}_{1}\cdots {T}_{N}. In particular, if {T}_{1} is {\alpha}_{1}averaged and {T}_{2} is {\alpha}_{2}averaged, where {\alpha}_{1},{\alpha}_{2}\in (0,1), then the composite {T}_{1}{T}_{2} is αaveraged, where \alpha ={\alpha}_{1}+{\alpha}_{2}{\alpha}_{1}{\alpha}_{2}.

(iii)
If the mappings {\{{T}_{i}\}}_{i=1}^{N} are averaged and have a common fixed point, then
\bigcap _{i=1}^{N}Fix({T}_{i})=Fix({T}_{1}\cdots {T}_{N}).
Here, the notations Fix(T) denotes the set of fixed point of the mapping T; that is, Fix(T):=\{x\in H:Tx=x\}.
Let T:H\to H be given. We have:

(i)
T is nonexpansive, if and only if the complement IT is (1/2)ism;

(ii)
If T is υism, then for \gamma >0, γT is (\upsilon /\gamma)ism;

(iii)
T is averaged, if and only if the complement IT is υism for some \upsilon >1/2; indeed, for \alpha \in (0,1), T is αaveraged, if and only if IT is (1/2\alpha)ism.
The socalled demiclosed principle for nonexpansive mappings will often be used.
Lemma 2.3 (Demiclosed Principle [21])
Let C be a closed and convex subset of a Hilbert space H and let T:C\to C be a nonexpansive mapping with Fix(T)\ne \mathrm{\varnothing}. If {\{{x}_{n}\}}_{n=1}^{\mathrm{\infty}} is a sequence in C weakly converging to x and if {\{(IT){x}_{n}\}}_{n=1}^{\mathrm{\infty}} converges strongly to y, then (IT)x=y. In particular, if y=0, then x\in Fix(T).
Recall the metric (nearest point) projection {Proj}_{C} from a real Hilbert space H to a closed convex subset C of H is defined as follows: given x\in H, {Proj}_{C}x is the unique point in C with the property
{Proj}_{C} is characterized as follows.
Lemma 2.4 Let C be a closed and convex subset of a real Hilbert space H. Given x\in H and y\in C, then y={Proj}_{C}x if and only if there holds the inequality
Lemma 2.5 Assume that {\{{a}_{n}\}}_{n=0}^{\mathrm{\infty}} is a sequence of nonnegative real numbers such that
where {\{{\gamma}_{n}\}}_{n=0}^{\mathrm{\infty}} and {\{{\beta}_{n}\}}_{n=0}^{\mathrm{\infty}} are sequences in (0,1) and {\{{\delta}_{n}\}}_{n=0}^{\mathrm{\infty}} is a sequence in ℝ such that

(i)
{\sum}_{n=0}^{\mathrm{\infty}}{\gamma}_{n}=\mathrm{\infty};

(ii)
either {lim\hspace{0.17em}sup}_{n\to \mathrm{\infty}}{\delta}_{n}\le 0 or {\sum}_{n=0}^{\mathrm{\infty}}{\gamma}_{n}{\delta}_{n}<\mathrm{\infty};

(iii)
{\sum}_{n=0}^{\mathrm{\infty}}{\beta}_{n}<\mathrm{\infty}.
Then {lim}_{n\to \mathrm{\infty}}{a}_{n}=0.
We adopt the following notation:

{x}_{n}\to x means that {x}_{n}\to x strongly;

{x}_{n}\rightharpoonup x means that {x}_{n}\to x weakly.
3 Main results
Recall that throughout this paper, we use S to denote the solution set of constrained convex minimization problem (1.1).
Let H be a real Hilbert space and C be a nonempty closed convex subset of Hilbert space H. Let F:C\to H be a kLipschitzian and ηstrongly monotone operator with constant k>0, \eta >0 such that 0<\mu <2\eta /{k}^{2}. Suppose that ∇f is LLipschitz continuous. We now consider a mapping {Q}_{s} on C defined by:
where s\in (0,1), and {T}_{{\lambda}_{s}} is nonexpansive. Let {T}_{{\lambda}_{s}} and {\lambda}_{s} satisfy the following conditions:

(i)
{Proj}_{C}(I\gamma \mathrm{\nabla}{f}_{{\lambda}_{s}})=(1{\theta}_{s})I+{\theta}_{s}{T}_{{\lambda}_{s}} and \gamma \in (0,2/L);

(ii)
{\theta}_{s}=\frac{2+\gamma (L+{\lambda}_{s})}{4};

(iii)
{\lambda}_{s} is continuous with respect to s and {\lambda}_{s}=o(s).
It is easy to see that {Q}_{s} is a contraction. Indeed, we have for each x,y\in C,
where \tau =\frac{1}{2}\mu (2\eta \mu {k}^{2}). Hence, {Q}_{s} has a unique fixed point in C, denoted by {x}_{s} which uniquely solves the fixedpoint equation
The following proposition summarizes the properties of the net \{{x}_{s}\}.
Proposition 3.1 Let {x}_{s} be defined by (3.1). Then the following properties for the net \{{x}_{s}\} hold:

(a)
\{{x}_{s}\} is bounded for s\in (0,1);

(b)
{lim}_{s\to 0}\parallel {x}_{s}{T}_{{\lambda}_{s}}{x}_{s}\parallel =0;

(c)
{x}_{s} defines a continuous curve from (0,1) into C.
Proof It is well known that: \tilde{x}\in C solves the minimization problem (1.1) if and only if \tilde{x} solves the fixedpoint equation
where 0<\gamma <2/L is a constant. It is clear that \tilde{x}=T\tilde{x}, i.e., \tilde{x}\in S=Fix(T).
(a) Take a fixed p\in S, we obtain that
It follows that
For x\in C, note that
and
where {\theta}_{s}=\frac{2+\gamma (L+{\lambda}_{s})}{4} and \theta =\frac{2+\gamma L}{4}.
Then we get
Since {\theta}_{s}=\frac{2+\gamma (L+{\lambda}_{s})}{4} and \theta =\frac{2+\gamma L}{4}, there exists a real positive number M>0 such that
It follows from (3.2) and (3.3) that
Since {\lambda}_{s}=o(s), there exists a real positive number {M}^{\mathrm{\prime}}>0 such that \frac{{\lambda}_{s}}{s}\le {M}^{\mathrm{\prime}}, and
Hence, \{{x}_{s}\} is bounded.
(b) Note that the boundedness of \{{x}_{s}\} implies that \{F{T}_{{\lambda}_{s}}({x}_{s})\} is also bounded. Hence, by the definition of \{{x}_{s}\}, we have
(c) For \gamma \in (0,2/L), there exists
and
where {\theta}_{s}=\frac{2+\gamma (L+{\lambda}_{s})}{4} and {\theta}_{{s}_{0}}=\frac{2+\gamma (L+{\lambda}_{{s}_{0}})}{4}.
So for {x}_{s}\in C, we get
for some appropriate constant N>0 such that
Now take s,{s}_{0}\in (0,1) and calculate
It follows that
Since \{F{T}_{{\lambda}_{s}}({x}_{s})\} is bounded, and {\lambda}_{s} is continuous with respect to s, {x}_{s} defines a continuous curve from (0,1) into C. □
The following theorem shows that the net \{{x}_{s}\} converges strongly as s\to 0 to a minimizer of (1.1), which solves some variational inequality.
Theorem 3.2 Let H be a real Hilbert space and C be a nonempty, closed and convex subset of Hilbert space H. Let F:C\to H be a kLipschitzian and ηstrongly monotone operator with constant k>0, \eta >0 such that 0<\mu <2\eta /{k}^{2}. Suppose that the minimization problem (1.1) is consistent and let S denote its solution set. Assume that the gradient ∇f is Lipschitzian with constant L>0. Let {x}_{s} be defined by (3.1), where the parameter s\in (0,1) and {T}_{{\lambda}_{s}} is nonexpansive. Let {T}_{{\lambda}_{s}} and {\lambda}_{s} satisfy the following conditions:

(i)
{Proj}_{C}(I\gamma \mathrm{\nabla}{f}_{{\lambda}_{s}})=(1{\theta}_{s})I+{\theta}_{s}{T}_{{\lambda}_{s}} and \gamma \in (0,2/L);

(ii)
{\theta}_{s}=\frac{2+\gamma (L+{\lambda}_{s})}{4};

(iii)
{\lambda}_{s} is continuous with respect to s and {\lambda}_{s}=o(s).
Then the net \{{x}_{s}\} converges strongly as s\to 0 to a minimizer {x}^{\ast} of (1.1), which solves the variational inequality
Equivalently, we have {Proj}_{S}(I\mu F){x}^{\ast}={x}^{\ast}.
Proof It is easy to see the uniqueness of a solution of the variational inequality (3.4). Indeed, suppose both \tilde{x}\in S and \stackrel{\u02c6}{x}\in S are solutions to (3.4), then
and
Adding up (3.5) and (3.6) gets
The strong monotonicity of F implies that \tilde{x}=\stackrel{\u02c6}{x} and the uniqueness is proved. Below we use {x}^{\ast}\in S to denote the unique solution of the variational inequality (3.4).
Let us show that {x}_{s}\to {x}^{\ast} as s\to 0. Set
Then we have {x}_{s}={Proj}_{C}{y}_{s}. For any given z\in S, we get
Since {Proj}_{C} is the metric projection from H onto C, we have
Note that {Proj}_{C}(I\gamma \mathrm{\nabla}f)z=z and {Proj}_{C}(I\gamma \mathrm{\nabla}f)=\frac{2\gamma L}{4}I+\frac{2+\gamma L}{4}T, so we get z=Tz, i.e., z\in S=Fix(T).
It follows from (3.7) that
By (3.3), we obtain that
Since \{{x}_{s}\} is bounded, it is obvious that if \{{s}_{n}\} is a sequence in (0,1) such that {s}_{n}\to 0, and {x}_{{s}_{n}}\rightharpoonup \overline{x}.
By Proposition 3.1(b) and (3.3), we have
So, by Lemma 2.3, we get \overline{x}\in Fix(T)=S.
Since {\lambda}_{s}=o(s), we obtain from (3.8) that {x}_{{s}_{n}}\to \overline{x}\in S.
Next, we show that \overline{x} solves the variational inequality (3.4). Observe that
Hence, we conclude that
Since {T}_{{\lambda}_{s}} is nonexpansive, I{T}_{{\lambda}_{s}} is monotone. Note that, for any given z\in S, z=Tz and \u3008{Proj}_{C}{y}_{s}{y}_{s},{Proj}_{C}{y}_{s}z\u3009\le 0.
By (3.3), it follows that
Since {\lambda}_{s}=o(s), by Proposition 3.1(b), we obtain from (3.9) that
So \overline{x}\in S is a solution of the variational inequality (3.4). We get \overline{x}={x}^{\ast} by uniqueness. Therefore, {x}_{s}\to {x}^{\ast} as s\to 0.
The variational inequality (3.4) can be rewritten as
So in terms of Lemma 2.4, it is equivalent to the following fixed point equation:
Next, we study the following iterative method. For a given arbitrary initial guess {x}_{0}\in C, we propose the following explicit scheme that generates a sequence {\{{x}_{n}\}}_{n=0}^{\mathrm{\infty}} in an explicit way:
where the parameters \{{s}_{n}\}\subset (0,1). Let {T}_{{\lambda}_{n}} and {\lambda}_{n} satisfy the following conditions:

(i)
{Proj}_{C}(I\gamma \mathrm{\nabla}{f}_{{\lambda}_{n}})=(1{\theta}_{n})I+{\theta}_{n}{T}_{{\lambda}_{n}} and 0<\gamma <2/L;

(ii)
{\theta}_{n}=\frac{2+\gamma (L+{\lambda}_{n})}{4};

(iii)
{\lambda}_{n}=o({s}_{n}).
It is proved that the sequence {\{{x}_{n}\}}_{n=0}^{\mathrm{\infty}} converges strongly to a minimizer {x}^{\ast}\in S of (1.1), which also solves the variational inequality (3.4). □
Theorem 3.3 Let H be a real Hilbert space and C be a nonempty, closed and convex subset of Hilbert space H. Let F:C\to H be a kLipschitzian and ηstrongly monotone operator with constant k>0, \eta >0 such that 0<\mu <2\eta /{k}^{2}. Suppose that the minimization problem (1.1) is consistent and let S denote its solution set. Assume that the gradient ∇f is Lipschitzian with constant L>0. Let {\{{x}_{n}\}}_{n=0}^{\mathrm{\infty}} be generated by algorithm (3.10) and the parameters \{{s}_{n}\}\subset (0,1). Let {T}_{{\lambda}_{n}}, {\lambda}_{n} and {s}_{n} satisfy the following conditions:

(C1)
{Proj}_{C}(I\gamma \mathrm{\nabla}{f}_{{\lambda}_{n}})=(1{\theta}_{n})I+{\theta}_{n}{T}_{{\lambda}_{n}} and \gamma \in (0,2/L);

(C2)
{\theta}_{n}=\frac{2+\gamma (L+{\lambda}_{n})}{4} for all n;

(C3)
{lim}_{n\to \mathrm{\infty}}{s}_{n}=0 and {\sum}_{n=0}^{\mathrm{\infty}}{s}_{n}=\mathrm{\infty};

(C4)
{\sum}_{n=0}^{\mathrm{\infty}}{s}_{n+1}{s}_{n}<\mathrm{\infty};

(C5)
{\lambda}_{n}=o({s}_{n}) and {\sum}_{n=0}^{\mathrm{\infty}}{\lambda}_{n+1}{\lambda}_{n}<\mathrm{\infty}.
Then the sequence \{{x}_{n}\} generated by the explicit scheme (3.10) converges strongly to a minimizer {x}^{\ast} of (1.1), which is also a solution of the variational inequality (3.4).
Proof It is well known that:

(a)
\tilde{x}\in C solves the minimization problem (1.1) if and only if \tilde{x} solves the fixedpoint equation
\tilde{x}={Proj}_{C}(I\gamma \mathrm{\nabla}f)\tilde{x}=\frac{2\gamma L}{4}\tilde{x}+\frac{2+\gamma L}{4}T\tilde{x},
where 0<\gamma <2/L is a constant. It is clear that \tilde{x}=T\tilde{x}, i.e., \tilde{x}\in S=Fix(T).

(b)
the gradient ∇f is 1/Lism.

(c)
{Proj}_{C}(I\gamma \mathrm{\nabla}{f}_{{\lambda}_{n}}) is \frac{2+\gamma (L+{\lambda}_{n})}{4} averaged for \gamma \in (0,2/L), in particular, the following relation holds:
{Proj}_{C}(I\gamma \mathrm{\nabla}{f}_{{\lambda}_{n}})=\frac{2\gamma (L+{\lambda}_{n})}{4}I+\frac{2+\gamma (L+{\lambda}_{n})}{4}{T}_{{\lambda}_{n}}=(1{\theta}_{n})I+{\theta}_{n}{T}_{{\lambda}_{n}}.
We observe that \{{x}_{n}\} is bounded. Indeed, take a fixed p\in S, we get
It follows that
Note that, by using the same argument as in the proof of (3.3), there exists a real positive number M>0 such that
Since {\lambda}_{n}=o({s}_{n}), there exists a real positive number {M}^{\mathrm{\prime}}>0 such that \frac{{\lambda}_{n}}{{s}_{n}}\le {M}^{\mathrm{\prime}} and by (3.11) we get
It follows from induction that
Consequently, \{{x}_{n}\} is bounded. It implies that \{{T}_{{\lambda}_{n}}({x}_{n})\} is also bounded.
We claim that
Indeed, since
we obtain that
By using the same argument as in the proof of Proposition 3.1(c), we obtain that
for some appropriate constant K>0 such that
Thus, we get
for some appropriate constant E>0 such that
Consequently, we get
By Lemma 2.5, we obtain \parallel {x}_{n+1}{x}_{n}\parallel \to 0.
Next, we show that
Indeed, it follows from (3.13) that
Now we show that
where {x}^{\ast}\in S is a solution of the variational inequality (3.4).
Indeed, take a subsequence \{{x}_{{n}_{k}}\} of \{{x}_{n}\} such that
Without loss of generality, we may assume that {x}_{{n}_{k}}\rightharpoonup \tilde{x}.
We observe that
It follows from (3.11) that
By (3.14), we get \parallel {x}_{n}T{x}_{n}\parallel \to 0.
In terms of Lemma 2.3, we get \tilde{x}\in Fix(T)=S.
Consequently, from (3.16) and the variational inequality (3.4), it follows that
Finally, we show that {x}_{n}\to {x}^{\ast}.
As a matter of fact, set
Then, {x}_{n+1}={Proj}_{C}{y}_{n}{y}_{n}+{y}_{n}.
In terms of Lemma 2.4 and (3.11), we obtain
It follows that
since \{{x}_{n}\} is bounded, we can take a constant {L}^{\mathrm{\prime}}>0 such that
It then follows that
where {\delta}_{n}=\frac{2}{1+{s}_{n}\tau}\u3008\mu F({x}^{\ast}),{x}_{n+1}{x}^{\ast}\u3009+\frac{2{\lambda}_{n}}{{s}_{n}}{L}^{\mathrm{\prime}}.
By (3.15) and {\lambda}_{n}=o({s}_{n}), we get {lim\hspace{0.17em}sup}_{n\to \mathrm{\infty}}{\delta}_{n}\le 0. Now applying Lemma 2.5 to (3.17) concludes that {x}_{n}\to {x}^{\ast} as n\to \mathrm{\infty}. □
4 Application
In this section, we give an application of Theorem 3.3 to the split feasibility problem (say SFP, for short), which was introduced by Censor and Elfving [22]. Since its inception in 1994, the split feasibility problem (SFP) has received much attention (see [7, 23, 24]) due to its applications in signal processing and image reconstruction, with particular progress in intensitymodulated radiation therapy.
The SFP can mathematically be formulated as the problem of finding a point x with the property
where C and Q are nonempty, closed and convex subset of Hilbert space {H}_{1} and {H}_{2}, respectively. B:{H}_{1}\to {H}_{2} is a bounded linear operator.
It is clear that {x}^{\ast} is a solution to the split feasibility problem (4.1) if and only if {x}^{\ast}\in C and B{x}^{\ast}{Proj}_{Q}B{x}^{\ast}=0. We define the proximity function f by
and consider the constrained convex minimization problem
Then {x}^{\ast} solves the split feasibility problem (4.1) if and only if {x}^{\ast} solves the minimization problem (4.2) with the minimize equal to 0. Byrne [7] introduced the socalled CQ algorithm to solve the (SFP).
where 0<\gamma <2/{\parallel B\parallel}^{2}. He obtained that the sequence \{{x}_{n}\} generated by (4.3) converges weakly to a solution of the (SFP).
In order to obtain strong convergence iterative sequence to solve the (SFP), we propose the following algorithm:
where the parameters \{{s}_{n}\}\subset (0,1) and \{{T}_{{\lambda}_{n}}\} satisfy the following conditions:

(C1)
{Proj}_{C}(I\gamma ({B}^{\ast}(I{Proj}_{Q})B+{\lambda}_{n}I))=(1{\theta}_{n})I+{\theta}_{n}{T}_{{\lambda}_{n}} and \gamma \in (0,2/L);

(C2)
{\theta}_{n}=\frac{2+\gamma (L+{\lambda}_{n})}{4} for all n,
where F:C\to H is kLipschitzian and ηstrongly monotone operator with constant k>0, \eta >0 such that 0<\mu <2\eta /{k}^{2}. We can show that the sequence \{{x}_{n}\} generated by (4.4) converges strongly to a solution of the (SFP) (4.1) if the sequence \{{s}_{n}\}\subset (0,1) and the sequence \{{\lambda}_{n}\} of parameters satisfy appropriate conditions.
Applying Theorem 3.3, we obtain the following result.
Theorem 4.1 Assume that the split feasibility problem (4.1) is consistent. Let the sequence \{{x}_{n}\} be generated by (4.4). Where the sequence \{{s}_{n}\}\subset (0,1) and the sequence \{{\lambda}_{n}\} satisfy the conditions (C3)(C5). Then the sequence \{{x}_{n}\} converges strongly to a solution of the split feasibility problem (4.1).
Proof By the definition of the proximity function f, we have
and ∇f is Lipschitz continuous, i.e.,
where L={\parallel B\parallel}^{2}.
Set {f}_{{\lambda}_{n}}(x)=f(x)+\frac{{\lambda}_{n}}{2}{\parallel x\parallel}^{2}, consequently
Then the iterative scheme (4.4) is equivalent to
where the parameters \{{s}_{n}\}\subset (0,1). \{{T}_{{\lambda}_{n}}\} satisfy the following conditions:

(C1)
{Proj}_{C}(I\gamma \mathrm{\nabla}{f}_{{\lambda}_{n}})=(1{\theta}_{n})I+{\theta}_{n}{T}_{{\lambda}_{n}} and \gamma \in (0,2/L);

(C2)
{\theta}_{n}=\frac{2+\gamma (L+{\lambda}_{n})}{4} for all n.
Due to Theorem 3.3, we have the conclusion immediately. □
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Acknowledgements
The authors wish to thank the referees for their helpful comments, which notably improved the presentation of this manuscript. This work was supported in part by The Fundamental Research Funds for the Central Universities (the Special Fund of Science in Civil Aviation University of China: No. ZXH2012K001), and by the Science Research Foundation of Civil Aviation University of China (No. 2012KYM03).
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Tian, M., Huang, LH. Iterative methods for constrained convex minimization problem in Hilbert spaces. Fixed Point Theory Appl 2013, 105 (2013). https://doi.org/10.1186/168718122013105
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DOI: https://doi.org/10.1186/168718122013105