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. []: (SFP) Findx ∈ H  such thatx ∈ C and Ax ∈ Q, where C is a nonempty closed convex subset of a real Hilbert space H  , Q is a nonempty closed convex subset of a real Hilbert space H  , and A : H  → H  is a linear and bounded operator. The split-feasibility problem was originally introduced by Censor and Elfving [] 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 [-].
In , Byrne [] proposed the CQ algorithm to study the split-feasibility problem: x  is chosen arbitrarily in R n , x n+ = P C (x nρ n A (I -P Q )Ax n ), n ∈ N, where C is a nonempty closed convex subset of R , Q is a nonempty closed convex subset of R m , {ρ n } n∈N is a sequence in the interval (, / A  ), P C is the metric projection from R onto C, P Q is the metric projection from R m onto Q, A is an m × matrix, and A is the transpose of A.
In , Qu and Xiu [] 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 A A. In , Censor, Motova, and Segal [] 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 , Xu [] gave the following modified CQ algorithm and gave a weak convergence theorem for the split-feasibility problem in infinite dimensional Hilbert spaces: where {ρ n } n∈N is chosen in the interval (, / A  ), C is a nonempty closed convex subset of a real Hilbert space H  , Q is a nonempty closed convex subset of a real Hilbert space H  , and A : H  → H  is a linear and bounded operator, and let A * be the adjoint of A.
Besides, Xu [] 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 H  , Q is a nonempty closed convex subset of a real Hilbert space H  , and A : H  → H  is a linear and bounded operator, and A * is the adjoint of A.
In , Qu, Liu, and Zheng [] gave the following modified CQ algorithm to study the split-feasibility problem: where  < w ≤ w n ≤ w < , and r n = (P Q -I)Ax n  A (P Q -I)Ax n  . Indeed, Qu et al. [] 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 [-] 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 {x n } n∈N to x ∈ H by x n → x and x n x, respectively.
From [], for each x, y, u, v ∈ H and λ ∈ R, we have Definition . Let C be a nonempty closed convex subset of a real Hilbert space H, and T : C → H be a mapping, and set Fix(T) := {x ∈ C : Tx = x}. Thus, x ∈ C and y ∈ Fix(T).
Remark . If T is a firmly nonexpansive mapping, then T is a nonexpansive mapping. Let C be a nonempty closed convex subset of a real Hilbert space H. For each x ∈ H, there is a unique elementx ∈ C such that x -x = min y∈C xy .
In this study, we set P C x =x, and P C is called the metric projection from H onto C.
Lemma . ([]) Let C be a nonempty closed convex subset of a real Hilbert space H, and let P C be the metric projection from H onto C. Then the following are satisfied: for all x, y ∈ H  .

Projected reflected gradient algorithm
Let {x n } n∈N be defined by Then there existsx ∈ such that {x n } n∈N converges weakly tox. Proof Let v ∈ C, w ∈ , and n ∈ N be fixed. Then, by Lemma ., we have By Lemma ., we know that Then, by (.) and (.), By Lemma ., we know that and this implies that Therefore, by (.), That is, By (.), (.), (.), and set v = w, we have By (.), we have Further, this implies that So, {x n } n∈N is a bounded sequence, and then there existx ∈ C and a subsequence {x n k } k∈N of {x n } n∈N such that x n k x. By (.), we determine that y n k x and Ay n k Ax. By Lemma ., we know that Ax = P Q Ax and Ax ∈ Q. So,x ∈ . Final, by Opial's condition, we know that x n x. Therefore, the proof is completed.

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 H  , b is given in a real Hilbert space H  , and A : H  → H  is a linear and bounded operator.

Theorem . Let H  and H  be real Hilbert spaces, C be a nonempty closed convex subset of H  , b ∈ H  , and A : H  → H  be a linear and bounded operator with adjoint operator A * .
Let be the solution set of the convex linear inverse problem and assume that = ∅. For k > , suppose ρ satisfies Let {x n } n∈N be defined by x n+ := P C (x n -ρA * (Ay nb)), y n+ := x n+x n , n ∈ N.
Proof Let Q = {b}. Then P Q (y) = b for all y ∈ H  . Hence, we get the conclusion of Theorem . by using Theorem .. (SEP) Findx ∈ C andȳ ∈ Q such that Ax = Bȳ.

Split equality problem
Let := {(x, y) ∈ C × Q : Ax = By} be the solution set of problem (SEP). Further, we observed that (x, y) is a solution of the split-equality problem if and only if for all ρ  >  and ρ  > , where P C is the metric projection from H  onto C, and P Q is the metric projection from H  onto Q, [].
As mentioned in Moudafi [], 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 [, , ]. Besides, we also observed that problem are extended to many generalized problems, like the split-equality fixed point problem [, ].
To solve the split-equality problem, Moudafi [] proposed the alternating CQ algorithm: x  and y  are chosen arbitrarily in H  and H  , respectively, x n+ := P C (x nρ n A * (Ax n -By n )), y n+ := P Q (y n + ρ n B * (Ax n+ -By n )), n ∈ N, where H  = R N , H  = R M , P C is the metric projection from H  onto C, and P Q is the metric projection from H  onto Q, ε > , A is a J × N matrix, B is a J × M matrix, λ A and λ B are the spectral radius of A * A and B * B, respectively, and {ρ n } is a sequence in (ε, min{  . In , Byrne and Moudafi [] presented a simultaneous algorithm, which was called the projected Landweber algorithm, to study the split-equality problem: x  and y  are chosen arbitrarily in H  and H  , respectively, x n+ := P C (x nρ n A * (Ax n -By n )), y n+ := P Q (y n + ρ n B * (Ax n -By n )), n ∈ N, where H  = R N , H  = R M , P C is the metric projection from H  onto C, and P Q is the metric projection from H  onto Q, ε > , A is a J × N matrix, B is a J × M matrix, λ A and λ B are the spectral radius of A * A and B * B, respectively, and {ρ n } is a sequence in (ε,  λ A +λ B ). Next, we need the following results to establish our results in the sequel. Let H  and H  be two real Hilbert spaces, W := H  × H  with inner product (For simple, ·, · H  and ·, · H  are written by ·, · .) Further, we know that {w n = (u n , v n )} ⊆ W = H  × H  converges weakly to w = (u, v) if and only if {u n } converges weakly to u and {v n } converges weakly to v. Next, suppose that C and Q are nonempty closed convex subsets of H  and H  , respectively, and set D = C × Q ⊆ W . Then the metric projection P D (w) = (P C (u), P Q (v)) for all z = (u, v) ∈ W . Next, we give a reflected projected Landweber algorithm for the split-equality problem.
Let {x n } n∈N and {y n } n∈N be defined by x  and y  are chosen arbitrarily in H  and H  , respectively, x n+ := P C (x n -ρA * (Au n -Bv n )), y n+ := P Q (y n + ρB * (Au n -Bv n )), v n+ := y n+y n , n ∈ N.

(.)
Then there exists (x,ȳ) ∈ such that {x n } n∈N converges weakly tox and {y n } n∈N converges weakly toȳ. Then Thus, Therefore, we get the conclusion of Theorem . by using Theorem ..
In Theorem ., if we set H  = H  and B is the identity mapping on H  , then we can obtain a new algorithm and related convergence theorem for the split-feasibility problem.
Corollary . Let H  and H  be real Hilbert spaces. Let C and Q be nonempty closed convex subsets of H  and H  , respectively. Let A : H  → H  be a linear and bounded operator with adjoint operator A * . Let be the solution set of the split-feasibility problem (SFP) and assume that = ∅. For k > , suppose ρ satisfies x n+ := P C (x n -ρA * (Au nv n )), y n+ := P Q (y n + ρ(Au nv n )), u n+ := x n+x n , v n+ := y n+y n , n ∈ N.

(.)
Then there existsx ∈ such that {x n } n∈N converges weakly tox. Further, {y n } n∈N converges weakly to Ax.
Remark . The results in this section are different from those in the references. For example, one may refer to [], Theorem ..
Remark . 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 .. (--)). The R Foundation for Statistical Computing Platform: x--w-mingw/x (-bit).
We give numerical results for problem (SFP) by using algorithm (PRGA), CQ algorithm, and CQ-like algorithm. Let ε >  and the algorithm stop if x n -x < ε.
In Tables  and , we set x  = (, ) T , ρ n = ρ = . for all n ∈ N. From Table , we see that the proposed algorithm in Theorem . reaches the required errors faster than the CQ algorithm and CQ-like algorithms with w n =  (resp. w n = .). From Tables  and , we see that the proposed algorithm in Theorem . only need ,, iteration number and . seconds to reach the required error ε =  - , but the other algorithms could not reach the required error.
In Tables  and , we set x  = (, ) T , ρ n = ρ = . for all n ∈ N. From Table , we see that the proposed algorithm in Theorem . reaches the required errors faster than the   CQ algorithm. From Tables  and , we see that the proposed algorithm in Theorem . only needs ,, iterations and . seconds to reach the required error ε =  - , but the CQ algorithm could not reach the required error.

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.