Abstract
Selfadaptive methods which permit stepsizes being selected selfadaptively are effective methods for solving some important problems, e.g., variational inequality problems. We devote this paper to developing and improving the selfadaptive methods for solving the split feasibility problem. A new improved selfadaptive method is introduced for solving the split feasibility problem. As a special case, the minimum norm solution of the split feasibility problem can be approached iteratively.
MSC:47J25, 47J20, 49N45, 65J15.
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1 Introduction
As we know, the original split feasibility problem (SFP) was introduced firstly by Censor and Elfving [1], and has received much attention since its inception in 1994. This is due to its applications in signal processing and image reconstruction, with particular progress in intensitymodulated radiation therapy; please, see [2–6].
Since the SFP is a special case of the convex feasibility problem (CFP), which is to find a point in the nonempty intersection of finitely many closed and convex sets, we next briefly review some historic approaches which relate to the CFP. The CFP is an important problem because many realworld inversion or estimation problems in engineering as well as in mathematics can be cast into this framework; see, e.g., Combettes [7], Bauschke and Borwein [8] and Kiwiel [9]. Traditionally, iterative projection methods for solving the CFP employ orthogonal projections onto convex sets (i.e., nearest point projections with respect to the Euclidean distance function); see, e.g., [10–14]. Much work has been done with generalized distance functions and the generalized projections associated with them suggested by Bregman [15].
In 1994, Censor and Elfving [1] investigated the use of different kinds of generalized projections in a single iterative process for solving the SFP. Their proposal is an iterative algorithm, which involves the computation of the inverse of a matrix, which is known to be a difficult task. That is why Byrne [16, 17] proposed the socalled CQ algorithm, which generates a sequence by a recursive procedure with suitable stepsize. The CQ algorithm only involves the computations of the projections onto the sets C and Q, respectively, and is therefore implementable in the case where these projections have closedform expressions (e.g., C and Q are the closed balls or halfspaces). There is a large number of references on the CQ method in the literature; see, for instance, [18–34]. However, we have to remark that the determination of the stepsize depends on the operator (matrix) norm (or the dominant eigenvalue of a matrix product). This means that in order to implement the CQ algorithm, one has first to compute (or, at least, to estimate) the matrix norm of an operator, which is in general not an easy work in practice.
To overcome the above difficulty, the socalled selfadaptive method which permits stepsize being selected selfadaptively was developed. Note that this method is the application of the projection method of Goldstein [35], Levitin and Polyak [36] to a suitable variational inequality problem, which is among the simplest numerical methods for solving variational inequality problems. Nevertheless, the efficiency of this projection method depends strongly on the choice of the stepsize parameter. If one chooses a small parameter such that it guarantees the convergence of the iterative sequence, the recursion leads to slow speed of convergence. On the other hand, if one chooses a large stepsize to improve the speed of convergence, the generated sequence may not converge. In real applications to solving variational inequality problems, the Lipschitz constant may be difficult to estimate, even if the underlying mapping is linear, the case such as the SFP. Some selfadaptive methods for solving variational inequality problems have been developed according to the original GoldsteinLevitinPolyak method [35, 36]. See, e.g., [37–45].
Motivated by the selfadaptive strategy, Zhang et al. [45] proposed their method by using variable stepsizes instead of the fixed stepsizes as in Censor et al. [46]. Also, a selfadaptive projection method was introduced by Zhao and Yang [29], and it was adopted by using the Armijolike searches. The advantage of these algorithms lies in the fact that neither prior information about the matrix norm A nor any other conditions on Q and A are required, and still convergence is guaranteed.
In this paper, we further develop and improve the selfadaptive methods for solving the SFP. An improved selfadaptive method is introduced for solving the SFP. As a special case, the minimum norm solution of the SFP can be approached iteratively.
2 Framework and preliminary results
Let {H}_{1} and {H}_{2} be two Hilbert spaces, and let C and Q be two closed and convex subsets of {H}_{1} and {H}_{2}, respectively. Let A:{H}_{1}\to {H}_{2} be a bounded linear operator. The split feasibility problem (SFP) is to find a point {x}^{\ast} such that
Next, we use Γ to denote the solution set of the SFP, i.e., \mathrm{\Gamma}=\{x\in C:Ax\in Q\}.
In 1994, Censor and Elfving [1] investigated the use of different kinds of generalized projections in a single iterative process for solving the SFP. They were the first to propose the following algorithm which involved the computation of the inverse {A}^{1}:
where C and Q are closed and convex sets in {\mathbb{R}}^{n}, while A is a full rank n\times n matrix and A(C)=\{y\in {\mathbb{R}}^{n}\mid y=Ax,x\in C\}. Note that {A}^{1} is not easily executed. Consequently, Byrne [16, 17] proposed the socalled CQ algorithm which generates a sequence \{{x}_{n}\} by the recursive procedure
where the stepsize {\tau}_{n} is chosen in the interval (0,2/{\parallel A\parallel}^{2}). It is remarkable that the CQ algorithm only involves the computations of the projections {P}_{C} and {P}_{Q} onto the sets C and Q, respectively, and is therefore implementable in the case where {P}_{C} and {P}_{Q} have closedform expressions (e.g., C and Q are the closed balls or halfspaces). However, we observe that the determination of the stepsize {\tau}_{n} depends on the operator (matrix) norm \parallel A\parallel (or the largest eigenvalue of {A}^{\ast}A). This means that for practical implementation of the CQ algorithm, one has first to compute (or, at least, to estimate) the matrix norm of A, which is in general not an easy task in practice.
To overcome the above difficulty, the socalled selfadaptive method which permits stepsize {\tau}_{n} being selected selfadaptively was developed. If we set
then the convex objective f is differentiable and has a Lipschitz gradient given by
Thus, the CQ algorithm (2) can be obtained by minimizing the following convex minimization problem
We know that a point {x}^{\ast}\in C is a stationary point of problem (3) if it satisfies
Thus, we can use a gradient projection algorithm below to solve the (SFP)
where {\tau}_{n}, the stepsize at iteration n, is chosen in the interval (0,2/L), where L is the Lipschitz constant of ∇f.
The above method (5) has to be thought of as the application of the projection method of Goldstein [35], Levitin and Polyak [36] to the variational inequality problem (4), which is among the simplest numerical methods for solving variational inequality problems. Nevertheless, the efficiency of this projection method depends greatly on the choice of the parameter {\tau}_{n}. A small {\tau}_{n} guarantees the convergence of the iterative sequence, but the recursion leads to slow speed of convergence. On the other hand, a large stepsize will improve the speed of convergence, but the generated sequence may not converge. In real applications for solving variational inequality problems, the Lipschitz constant may be difficult to estimate, even if the underlying mapping is linear, the case such as the SFP.
The methods in Zhang et al. [45] and Censor et al. [46] were proposed for solving the multiplesets split feasibility problem.
Algorithm 2.1 S1. Given a nonnegative sequence {\tau}_{n} such that {\sum}_{n=0}^{\mathrm{\infty}}{\tau}_{n}<\mathrm{\infty}, \delta \in (0,1), \mu \in (0,1), \rho \in (0,1), \u03f5>0, {\beta}_{0}>0, and arbitrary initial point {x}_{0}. Set {\gamma}_{0}={\beta}_{0} and n=0.
S2. Find the smallest nonnegative integer {l}_{n} such that {\beta}_{n+1}={\mu}^{{l}_{k}}{\gamma}_{k} and
which satisfies
S3. If
then set {\gamma}_{n+1}=(1+{\tau}_{n+1}){\beta}_{n+1}; otherwise, set {\gamma}_{n+1}={\beta}_{n+1}.
S4. If \parallel e({x}_{n},{\beta}_{n})\parallel \le \u03f5, stop; otherwise, set n:=n+1 and go to S2.
The following selfadaptive projection method was introduced by Zhao and Yang [29], and it was adopted by using the Armijolike searches.
Algorithm 2.2 Given constants \beta >0, \sigma \in (0,1) and \gamma \in (0,1). Let {x}_{0} be arbitrary. For n=0,1,\dots , calculate
where {\tau}_{n}=\beta {\gamma}^{{l}_{n}} and {l}_{n} is the smallest nonnegative integer l such that
The advantage of Algorithm 2.1 and Algorithm 2.2 lies in the fact that neither prior information about the matrix norm A nor any other conditions on Q and A are required, and still convergence is guaranteed.
We shall introduce our improved selfadaptive method for solving the SFP. In this respect, we need the ingredients introduced right now.
Let C be a nonempty closed convex subset of a real Hilbert space H. A mapping T:C\to C is called nonexpansive if
A mapping \psi :C\to C is said to be δcontractive if there exists a constant \delta \in [0,1) such that
Recall that the (nearest point or metric) projection from H onto C, denoted by {P}_{C}, assigns to each x\in H the unique point {P}_{C}(x)\in C with the property
It is well known that the metric projection {P}_{C} of H onto C has the following basic properties:

(a)
\parallel {P}_{C}(x){P}_{C}(y)\parallel \le \parallel xy\parallel for all x,y\in H;

(b)
\u3008xy,{P}_{C}(x){P}_{C}(y)\u3009\ge {\parallel {P}_{C}(x){P}_{C}(y)\parallel}^{2} for every x,y\in H;

(c)
\u3008x{P}_{C}(x),y{P}_{C}(x)\u3009\le 0 for all x\in H and y\in C.
Next we adopt the following notation:

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

{x}_{n}\rightharpoonup x means that {x}_{n} converges weakly to x;

{\omega}_{w}({x}_{n}):=\{x:\mathrm{\exists}{x}_{{n}_{j}}\rightharpoonup x\} is the weak ωlimit set of the sequence \{{x}_{n}\}.
Recall that a function f:H\to \mathbb{R} is called convex if
It is known that a differentiable function f is convex if and only if the following relation holds:
Recall that an element g\in H is said to be a subgradient of f:H\to \mathbb{R} at x if
If the function f:H\to \mathbb{R} has at least one subgradient at x, it is said to be subdifferentiable at x. The set of subgradients of f at the point x is called the subdifferential of f at x, and is denoted by \partial f(x). A function f is called subdifferentiable if it is subdifferentiable at all x\in H. If f is convex and differentiable, then its gradient and subgradient coincide. A function f:H\to \mathbb{R} is said to be weakly lower semicontinuous (wlsc) at x if {x}_{n}\rightharpoonup x implies
f is said to be wlsc on H if it is wlsc at every point x\in H.
The first lemma is easy to prove.
Lemma 2.1 [14]
Let f(x):=\frac{1}{2}{\parallel Ax{P}_{Q}Ax\parallel}^{2}. Then

(i)
f is convex and differentiable;

(ii)
f is wlsc on C.
Lemma 2.2 [47]
Given {x}^{\ast}\in {H}_{1}. Then {x}^{\ast} solves the SFP if and only if {x}^{\ast} solves the fixed point equation
where \gamma >0.
Lemma 2.3 [48]
Assume that \{{a}_{n}\} is a sequence of nonnegative real numbers such that
where \{{\gamma}_{n}\} is a sequence in (0,1) and \{{\delta}_{n}\} is a sequence such that

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

(2)
{lim\hspace{0.17em}sup}_{n\to \mathrm{\infty}}\frac{{\delta}_{n}}{{\gamma}_{n}}\le 0 or {\sum}_{n=1}^{\mathrm{\infty}}{\delta}_{n}<\mathrm{\infty}.
Then {lim}_{n\to \mathrm{\infty}}{a}_{n}=0.
Lemma 2.4 [49]
Let \{{s}_{n}\} be a sequence of real numbers that does not decrease at infinity, in the sense that there exists a subsequence \{{s}_{{n}_{i}}\} of \{{s}_{n}\} such that {s}_{{n}_{i}}\le {s}_{{n}_{i}+1} for all i\ge 0. For every n\ge {n}_{0}, define an integer sequence \{\tau (n)\} as
Then \tau (n)\to \mathrm{\infty} as n\to \mathrm{\infty} and for all n\ge {n}_{0}
3 Main results
In this section we state and prove our main results.
Let C and Q be nonempty closed convex subsets of real Hilbert spaces {H}_{1} and {H}_{2}, respectively. Let \psi :C\to {H}_{1} be a δcontraction with \delta \in [0,\frac{\sqrt{2}}{2}). Let A:{H}_{1}\to {H}_{2} be a bounded linear operator.
Algorithm 3.1 For given {x}_{0}\in C, assume that \{{x}_{n}\} has been constructed. If \mathrm{\nabla}f({x}_{n})=0, then stop and {x}_{n} is a solution of SFP (1). Otherwise, continue and compute {x}_{n+1} by the recursion
where \{{\alpha}_{n}\}\subset (0,1) and \{{\rho}_{n}\}\subset (0,2).
Theorem 3.1 Suppose that the SFP is consistent, that is, \mathrm{\Gamma}\ne \mathrm{\varnothing}. Assume that the following conditions hold:

(a)
{lim}_{n\to \mathrm{\infty}}{\alpha}_{n}=0 and {\sum}_{n=1}^{\mathrm{\infty}}{\alpha}_{n}=\mathrm{\infty};

(b)
{inf}_{n}{\rho}_{n}(2{\rho}_{n})>0.
Then \{{x}_{n}\} defined by (6) converges strongly to z, which solves the following variational inequality:
Proof First, it is obvious that the solution of the variational inequality (7) is unique (by the strong monotonicity of I\psi according to the related results in variational inequality), denoted by z. Then z={P}_{\mathrm{\Gamma}}(\psi (z)). We may assume that the sequence \{{x}_{n}\} is infinite, that is, Algorithm 3.1 does not terminate in a finite number of iterations. Thus, \mathrm{\nabla}f({x}_{n})\ne 0 for all n. From (6), we have
By the convexity of f (Lemma 2.1) and the fact that \mathrm{\nabla}f(z)=0 for z\in \mathrm{\Gamma}, we deduce that
Using the inequality {(a+b)}^{2}\le 2({a}^{2}+{b}^{2}) for all a,b\in \mathbb{R}, we have
From (8)(10), we get
By induction, we deduce
Hence, \{{x}_{n}\} is bounded.
By using the firm nonexpansivity of {P}_{C}, we derive that
It follows that
Next, we will prove that {x}_{n}\to z following the ideas in [49]. Set {s}_{n}={\parallel {x}_{n}z\parallel}^{2} for all n\ge 0. Since {\alpha}_{n}\to 0 and {inf}_{n}{\rho}_{n}(2{\rho}_{n})>0, we may assume, without loss of generality, that (1{\alpha}_{n}){\rho}_{n}(2{\rho}_{n})\ge \sigma for some \sigma >0. Thus, we can rewrite (11) as
Now, we consider two possible cases.
Case 1. Assume that \{{s}_{n}\} is eventually decreasing, i.e., there exists N>0 such that \{{s}_{n}\} is decreasing for n\ge N. In this case, \{{s}_{n}\} must be convergent, and from (12) it follows that
where M>0 is a constant such that {sup}_{n}\{2\parallel \psi (z)z\parallel \parallel {x}_{n+1}z\parallel \}\le M. Letting n\to \mathrm{\infty} in (13), we get
Since \{{x}_{n}\} is bounded, there exists a subsequence \{{x}_{{n}_{k}}\} of \{{x}_{n}\} converging weakly to \tilde{x}\in C.
From the weak lower semicontinuity of f, we have
Hence, f(\tilde{x})=0, i.e., A\tilde{x}\in Q. This indicates that
Furthermore, due to the property of the projection (c),
From (12), we obtain
Applying Lemma 2.3 to (14), we get {s}_{n}\to 0.
Case 2. Assume \{{s}_{n}\} is not eventually decreasing. That is, there exists an integer {n}_{0} such that {s}_{{n}_{0}}\le {s}_{{n}_{0}+1}. Thus, we can define an integer sequence \{{\tau}_{n}\} for all n\ge {n}_{0} as follows:
Clearly, \tau (n) is a nondecreasing sequence such that \tau (n)\to +\mathrm{\infty} as n\to \mathrm{\infty} and
for all n\ge {n}_{0}. In this case, we derive from (13) that
It follows that
This implies that every weak cluster point of \{{x}_{\tau (n)}\} is in the solution set Γ; i.e., {\omega}_{w}({x}_{\tau (n)})\subset \mathrm{\Gamma}.
On the other hand, we note that
from which we can deduce that
Since {s}_{\tau (n)}\le {s}_{\tau (n)+1}, we have from (12) that
Combining (15) and (16) yields
and hence
From (14), we have
Thus,
From Lemma 2.4, we have
Therefore, {s}_{n}\to 0. That is, {x}_{n}\to z. This completes the proof. □
From Theorem 3.1, we can deduce easily the following algorithm and corollary.
Algorithm 3.2 For given {x}_{0}\in C, assume that \{{x}_{n}\} has been constructed. If \mathrm{\nabla}f({x}_{n})=0, then stop and {x}_{n} is a solution of SFP (1). Otherwise, continue and compute {x}_{n+1} by the recursion
where \{{\alpha}_{n}\}\subset (0,1) and \{{\rho}_{n}\}\subset (0,2).
Theorem 3.2 Suppose that the SFP is consistent, that is, \mathrm{\Gamma}\ne \mathrm{\varnothing}. Assume that the following conditions hold:

(a)
{lim}_{n\to \mathrm{\infty}}{\alpha}_{n}=0 and {\sum}_{n=1}^{\mathrm{\infty}}{\alpha}_{n}=\mathrm{\infty};

(b)
{inf}_{n}{\rho}_{n}(2{\rho}_{n})>0.
Then \{{x}_{n}\} defined by (17) converges strongly to the minimum norm solution of the SFP.
4 Concluding remarks
This work contains our study dedicated to developing and improving selfadaptive methods for solving the split feasibility problem. We have introduced our improved selfadaptive method for solving the split feasibility problem. As a special case, the minimum norm solution of the split feasibility problem can be approached iteratively. This study is motivated by relevant applications for solving many realworld problems, which give rise to mathematical models in the sphere of variational inequality problems.
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Acknowledgements
The first author was supported in part by NSFC 11071279 and NSFC 71161001G0105. The third author was partially supported by NSC 1012628E230001MY3.
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Yao, Y., Postolache, M. & Liou, YC. Strong convergence of a selfadaptive method for the split feasibility problem. Fixed Point Theory Appl 2013, 201 (2013). https://doi.org/10.1186/168718122013201
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DOI: https://doi.org/10.1186/168718122013201