An algorithm for the split-feasibility problems with application to the split-equality problem.

In this paper, we study the split-feasibility problem in Hilbert spaces by using the projected reflected gradient algorithm. As applications, we study the convex linear inverse problem and the split-equality problem in Hilbert spaces, and we give new algorithms for these problems. Finally, numerical results are given for our main results.

Introduction

The split-feasibility problem was first introduced by Censor et al. [1]: where C is a nonempty closed convex subset of a real Hilbert space , Q is a nonempty closed convex subset of a real Hilbert space , and is a linear and bounded operator. The split-feasibility problem was originally introduced by Censor and Elfving [2] for modeling phase retrieval problems, and it later was studied extensively as an extremely powerful tool for the treatment of a wide range of inverse problems, such as medical image reconstruction and intensity-modulated radiation therapy problems. For examples, one may refer to [2-4].

In 2002, Byrne [5] proposed the CQ algorithm to study the split-feasibility problem: where C is a nonempty closed convex subset of , Q is a nonempty closed convex subset of , is a sequence in the interval , is the metric projection from onto C, is the metric projection from onto Q, A is an matrix, and is the transpose of A.

In 2005, Qu and Xiu [6] presented modifications of the CQ algorithm in the setting of finite dimensional spaces by adopting the Armijo-like searches, which need not compute the matrix inverses and the largest eigenvalue of the matrix . In 2007, Censor, Motova, and Segal [4] studied the multiple-sets split-feasibility problem that requires one 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 by using a perturbed projection method.

In 2010, Xu [7] gave the following modified CQ algorithm and gave a weak convergence theorem for the split-feasibility problem in infinite dimensional Hilbert spaces: where is chosen in the interval , C is a nonempty closed convex subset of a real Hilbert space , Q is a nonempty closed convex subset of a real Hilbert space , and is a linear and bounded operator, and let be the adjoint of A.

Besides, Xu [7] also gave a regularized algorithm for the split-feasibility problem and proposed a strong convergence theorem under suitable conditions: where C is a nonempty closed convex subset of a real Hilbert space , Q is a nonempty closed convex subset of a real Hilbert space , and is a linear and bounded operator, and is the adjoint of A.

In 2015, Qu, Liu, and Zheng [8] gave the following modified CQ algorithm to study the split-feasibility problem: where , and . Indeed, Qu et al. [8] thought that the CQ-like algorithm not only need not compute the largest eigenvalue of the related matrix but also need not use any line search scheme.

For more details as regards various algorithms for the split-feasibility problems and related problems, one may refer to [5-20] and related references.

Motivated by the above work, in this paper, we study the split-feasibility problem in Hilbert spaces by using the projected reflected gradient algorithm. As applications, we study the convex linear inverse problem and the split-equality problem in Hilbert spaces, and give new algorithms for these problems. Final, numerical results are given for our main results.

Preliminaries

Let H be a real Hilbert space with inner product and norm . We denote the strong convergence and weak convergence to by and , respectively. From [21], for each and , we have

Definition 2.1

Let C be a nonempty closed convex subset of a real Hilbert space H, and be a mapping, and set . Thus,

T is a nonexpansive mapping if for every .

T is a firmly nonexpansive mapping if for every , that is, for every .

T is a quasi-nonexpansive mapping if and for every and .

Remark 2.1

If T is a firmly nonexpansive mapping, then T is a nonexpansive mapping.

Lemma 2.1

([22])

Let C be a nonempty closed convex subset of a real Hilbert space H. Let be a nonexpansive mapping, and be a sequence in C. If and , then .

Let C be a nonempty closed convex subset of a real Hilbert space H. For each , there is a unique element such that In this study, we set , and is called the metric projection from H onto C.

Lemma 2.2

([21])

Let C be a nonempty closed convex subset of a real Hilbert space H, and let be the metric projection from H onto C. Then the following are satisfied:

for all and ;

for all and ;

is a firmly nonexpansive mapping.

Lemma 2.3

([23])

Let and be two real Hilbert spaces, be a linear mapping, and be the adjoint of A. Let C be a nonempty closed convex subset of . Let . Then T is a monotone mapping. In fact, we have for all .

Projected reflected gradient algorithm

Theorem 3.1

Let and be real Hilbert spaces, C and Q be nonempty closed convex subsets of and , respectively, and be a linear and bounded operator with adjoint operator . Let Ω be the solution set of the split-feasibility problem and assume that . For , suppose ρ satisfies Let be defined by Then there exists such that converges weakly to x̄.

Proof

Let , , and be fixed. Then, by Lemma 2.2, we have By Lemma 2.3, we know that Then, by (3.2) and (3.3), By Lemma 2.2, we know that and this implies that Therefore, by (3.6), That is, This implies that Also, we have By (3.4), (3.9), (3.10), and set , we have By (3.11), we have Hence, exists, and then Further, this implies that So, is a bounded sequence, and then there exist and a subsequence of such that . By (3.13), we determine that and . By Lemma 2.1, we know that and . So, . Final, by Opial’s condition, we know that . Therefore, the proof is completed. □

Remark 3.1

The algorithm in Theorem 3.1 are different from those in the references. For examples, one may refer to [6], Theorem 3.1, [16], Theorem 4.3, [8], Theorem 3.1, [24], Theorem 3.1, Theorem 4.1, and [7], Theorem 3.3.

Applications

Convex linear inverse problem

In this section, we consider the following convex linear inverse problem: where C is a nonempty closed convex subset of a real Hilbert space , b is given in a real Hilbert space , and is a linear and bounded operator.

Theorem 4.1

Let and be real Hilbert spaces, C be a nonempty closed convex subset of , , and be a linear and bounded operator with adjoint operator . Let Ω be the solution set of the convex linear inverse problem and assume that . For , suppose ρ satisfies Let be defined by Then there exists such that converges weakly to x̄.

Proof

Let . Then for all . Hence, we get the conclusion of Theorem 4.1 by using Theorem 3.1. □

Split equality problem

Let , , and be real Hilbert spaces. Let C and Q be nonempty closed convex subsets of and , respectively. Let and be linear and bounded operators with adjoint operators and , respectively. The following problem is the split-equality problem, which was studied by Moudafi [25, 26]: Let be the solution set of problem (SEP). Further, we observed that is a solution of the split-equality problem if and only if for all and , where is the metric projection from onto C, and is the metric projection from onto Q, [27].

As mentioned in Moudafi [25], the interest of the split-equality problem covers many situations, for instance in decomposition methods for PDEs, game theory, and modulated radiation therapy (IMRT). For details, see [3, 25, 28]. Besides, we also observed that problem are extended to many generalized problems, like the split-equality fixed point problem [29, 30].

To solve the split-equality problem, Moudafi [26] proposed the alternating CQ algorithm: where , , is the metric projection from onto C, and is the metric projection from onto Q, , A is a matrix, B is a matrix, and are the spectral radius of and , respectively, and is a sequence in .

In 2013, Byrne and Moudafi [31] presented a simultaneous algorithm, which was called the projected Landweber algorithm, to study the split-equality problem: where , , is the metric projection from onto C, and is the metric projection from onto Q, , A is a matrix, B is a matrix, and are the spectral radius of and , respectively, and is a sequence in .

Next, we need the following results to establish our results in the sequel. Let and be two real Hilbert spaces, with inner product for all , . Hence, W is a real Hilbert space with norm (For simple, and are written by .) Further, we know that converges weakly to if and only if converges weakly to u and converges weakly to v. Next, suppose that C and Q are nonempty closed convex subsets of and , respectively, and set . Then the metric projection for all .

Next, we give a reflected projected Landweber algorithm for the split-equality problem.

Theorem 4.2

Let , , and be real Hilbert spaces. Let C and Q be nonempty closed convex subsets of and , respectively. Let and be linear and bounded operators with adjoint operators and , respectively. Let Ω be the solution set of the split-equality problem and assume that . For , suppose ρ satisfies Let and be defined by Then there exists such that converges weakly to x̄ and converges weakly to ȳ.

Let , , , . Then Thus, and Therefore, we get the conclusion of Theorem 4.2 by using Theorem 4.1. □

In Theorem 4.2, if we set and B is the identity mapping on , then we can obtain a new algorithm and related convergence theorem for the split-feasibility problem.

Corollary 4.1

Let and be real Hilbert spaces. Let C and Q be nonempty closed convex subsets of and , respectively. Let be a linear and bounded operator with adjoint operator . Let Ω be the solution set of the split-feasibility problem (SFP) and assume that . For , suppose ρ satisfies Let and be defined by Then there exists such that converges weakly to x̄. Further, converges weakly to Ax̄.

Remark 4.1

The results in this section are different from those in the references. For example, one may refer to [25], Theorem 2.1.

Remark 4.2

From the results in this section, we know that the split-equality problem is a special case of the split-feasibility problem. This is an important contribution in this paper since many researchers thought that the split-feasibility problem is a special case of the split-equality problem.

Numerical results

All codes were written in R language (version 3.2.4 (2016-03-10)). The R Foundation for Statistical Computing Platform: x86-64-w64-mingw32/x64 (64-bit).

Example 5.1

Let , , , , where is identity matrix. Then (SFP) has the unique solution . Indeed, and .

We give numerical results for problem (SFP) by using algorithm (PRGA), CQ algorithm, and CQ-like algorithm. Let and the algorithm stop if .

In Tables 1 and 2, we set , for all . From Table 1, we see that the proposed algorithm in Theorem 3.1 reaches the required errors faster than the CQ algorithm and CQ-like algorithms with (resp. ). From Tables 2 and 1, we see that the proposed algorithm in Theorem 3.1 only need 6,402,868 iteration number and 150.65 seconds to reach the required error , but the other algorithms could not reach the required error.

Table 1

Numerical results for Example  ( , for all )

ε	CPU(s)	Iteration	Approximate solution	CPU(s)	Iteration	Approximate solution
	CQ algorithm			PRGA
10−3	-	2	(0.5994553, 0.8004082)	0.01	314	(0.6006783, 0.7994908)
10−4	375.25	1,630,698	(0.5999200, 0.8000600)	0.03	1334	(0.5999466, 0.8000400)
	CQ-like algorithm (w_n=1)			CQ-like algorithm (w_n=1.9)
10−3	6.47	249,918	(0.6007997, 0.7993996)	3.41	131,247	(0.6007997, 0.7993996)
10−4	652.44	24,999,920	(0.6000800, 0.7999400)	342.80	13,157,560	(0.6000800, 0.7999400)

Table 2

Numerical results for Example 

	PRGA[x_1=(10,10)^T,ρ=0.06]
ε	CPU(s)	Iteration	Approximate solution
10−3	0.01	314	(0.6006783, 0.7994908)
10−4	0.03	1334	(0.5999466, 0.8000400)
10−5	0.10	3741	(0.6000052, 0.7999961)
10−6	14.67	650,838	(0.5999999, 0.8000001)
10−7	150.65	6,402,868	(0.6000001, 0.8000000)

In Tables 3 and 4, we set , for all . From Table 3, we see that the proposed algorithm in Theorem 3.1 reaches the required errors faster than the CQ algorithm. From Tables 4 and 3, we see that the proposed algorithm in Theorem 3.1 only needs 1,058,254 iterations and 374.21 seconds to reach the required error , but the CQ algorithm could not reach the required error.

Table 3

Numerical results for Example  ( , for all )

	CQ algorithm			PRGA
ε	CPU(s)	Iteration	Approximate solution	CPU(s)	Iteration	Approximate solution
10−3	3.89	166,658	(0.6007997, 0.7993996)	-	375	(0.5993223, 0.8005078)
10−4	374.21	16,666,660	(0.6000800, 0.7999400)	0.15	7086	(0.5999597, 0.8000302)

Table 4

Numerical results for Example 

	PRGA[x_1=(1,1)^T,ρ=0.06]
ε	CPU(s)	Iteration	Approximate solution
10−3	−	375	(0.5993223, 0.8005078)
10−4	0.16	7086	(0.5999597, 0.8000302)
10−5	0.22	9493	(0.5999947, 0.8000040)
10−6	1.00	44,211	(0.6000002, 0.7999999)
10−7	24.63	1,058,254	(0.6, 0.8)

Conclusions

In this paper, we study the split-feasibility problem in Hilbert spaces by using the projected reflected gradient algorithm. From the proposed numerical results, we know the projected reflected gradient algorithm is useful and faster than the CQ algorithm and CQ-like algorithms under suitable conditions. As applications, we study the convex linear inverse problem and the split-equality problem in Hilbert spaces. Here, we give an important connection between the linear inverse problem and the split-equality problem. Hence, many modified projected Landweber algorithms for the split-equality problem will be presented by using the related algorithms for the linear inverse problem.