Research Article Modified Halfspace-Relaxation Projection Methods for Solving the Split Feasibility Problem

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1 Advances in Operations Research Volume 01, Article ID , 17 pages doi: /01/ Research Article Modified Halfspace-Relaxation Projection Methods for Solving the Split Feasibility Problem Min Li School of Economics and Management, Southeast University, Nanjing 10096, China Correspondence should be addressed to Min Li, Received 4 March 01; Accepted 11 May 01 Academic Editor: Abdellah Bnouhachem Copyright q 01 Min Li. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. This paper presents modified halfspace-relaxation projection HRP methods for solving the split feasibility problem SFP. Incorporating with the techniques of identifying the optimal step length with positive lower bounds, the new methods improve the efficiencies of the HRP method Qu and Xiu 008. Some numerical results are reported to verify the computational preference. 1. Introduction Let C and Q be nonempty closed convex sets in R n and R m, respectively, and A an m n real matrix. The problem, to find x C with Ax Q if such x exists, was called the split feasibility problem SFP by Censor and Elfving 1. In this paper, we consider an equivalent reformulation of the SFP: minimize f z x subject to z Ω, y 1.1 where f z 1 Bz 1 y Ax, B A I, Ω C Q. 1. For convenience, we only consider the Euclidean norm. It is obvious that f z is convex. If z x T,y T T Ω and f z 0, then x solves the SFP. Throughout we assume that the solution

2 Advances in Operations Research set of the SFP is nonempty. And thus the solution set of 1.1, denoted by Ω, is nonempty. In addition, in this paper, we always assume that the set Ω is given by Ω {z R n m c z 0}, 1.3 where c : R n m R is a convex not necessarily differentiable function. This representation of Ω is general enough, because any system of inequalities {c j z 0, j J}, where c j z are convex and J is an arbitrary index set, can be reformulated as the single inequality c z 0 with c z sup{c j z j J}. For any z R n m, at least one subgradient ξ c z can be calculated, where c z is a subgradient of c z at z and is defined as follows: c z { ξ R n m c u c z u z T ξ, u R n m}. 1.4 Qu and Xiu proposed a halfspace-relaxation projection method to solve the convex optimization problem 1.1. Starting from any z 0 R n R m, the HRP method iteratively updates z k according to the formulae: z k P Ωk [z k α k f z k], [ z k 1 z k γ k z k z k α k f z k f z k], where Ω k { z R n m c z k z z k } T ξ k 0, 1.7 ξ k is an element in c z k, α k γl m k and m k is the smallest nonnegative integer m such that α k z k z k T f z k f z k 1 ρ z k z k, ρ 0, 1, 1.8 γ k θρz k z k z k z k α k f z k f z k, θ 0,. 1.9 The notation P Ωk v denotes the projection of v onto Ω k under the Euclidean norm, that is, P Ωk v arg min{ u v u Ω k } Here the halfspace Ω k contains the given closed convex set Ω and is related to the current iterative point z k. From the expressions of Ω k, the projection onto Ω k is simple to be computed for details, see Proposition 3.3. The idea to construct the halfspace Ω k and replace P Ω by P Ωk is from the halfspace-relaxation projection technique presented by Fukushima 3. This technique is often used to design algorithms see, e.g.,, 4, 5 to solve the SFP. The drawback

3 Advances in Operations Research 3 of the HRP method in is that the step length γ k defined in 1.9 may be very small since lim k z k z k 0. Note that the reformulation 1.1 is equivalent to a monotone variational inequality VI : z Ω, z z T f z 0, z Ω, 1.11 where f z B T Bz. 1.1 The forward-backward splitting method 6 and the extragradient method 7, 8 are considerably simple projection-type methods in the literature. They are applicable for solving monotone variational inequalities, especially for For given z k,let z k P Ω [z k α k f z k] Under the assumption f α k z k f z k z ν k z k, ν 0, 1, 1.14 the forward-backward FB splitting method generates the new iterate via [ z k 1 P Ω z k α k f z k f z k], 1.15 while the extra-gradient EG method generates the new iterate by z k 1 P Ω [z k α k f z k] The forward-backward splitting method 1.15 can be rewritten as [ z k 1 P Ω z k γ k z k z k α k f z k f z k], 1.17 where the descent direction z k z k α k f z k f z k is the same as 1.6 and the step length γ k along this direction always equals to 1. He et al. 9 proposed the modified versions of the FB method and EG method by incorporating the optimal step length γ k along

4 4 Advances in Operations Research the descent directions z k z k α k f z k f z k and α k f z k, respectively. Here γ k is defined by z k z k Tz k z k α k f z k f z k γ k θγ k, θ 0,, γ k z k z k α k f z k f z k Under the assumption 1.14, γ 1/ is lower bounded. k This paper is to develop two kinds of modified halfspace-relaxation projection methods for solving the SFP by improving the HRP method in. One is an FB type HRP method, the other is an EG type HRP method. The numerical results reported in 9 show that efforts of identifying the optimal step length usually lead to attractive numerical improvements. This fact triggers us to investigate the selection of optimal step length with positive lower bounds in the new methods to accelerate convergence. The preferences to the HRP method are verified by numerical experiments for the test problems arising in. The rest of this paper is organized as follows. In Section, we summarize some preliminaries of variational inequalities. In Section 3, we present the new methods and provide some remarks. The selection of optimal step length of the new methods is investigated in Section 4. Then, the global convergence of the new methods is proved in Section 5. Some preliminary numerical results are reported in Section 6 to show the efficiency of the new methods, and the numerical superiority to the HRP method in. Finally, some conclusions are made in Section 7.. Preliminaries In the following, we state some basic concepts for the variational inequality VI S,F : s S, s s T F s 0, s S,.1 where F is a mapping from R N into R N,andS R N is a nonempty closed convex set. The mapping F is said to be monotone on R N if s t T F s F t 0, s, t R N.. Notice that the variational inequality VI S,F is invariant when we multiply F by some positive scalar α. ThusVI S,F is equivalent to the following projection equation see 10 : s P S s αf s,.3 that is, to solve VI S,F is equivalent to finding a zero point of the residue function e s, α : s P S s αf s..4 Note that e s, α is a continuous function of s because the projection mapping is nonexpansive. The following lemma states a useful property of e s, α.

5 Advances in Operations Research 5 Lemma.1 4, Lemma.. Let F be a mapping from R N into R N. For any s R N and α>0, we have min{1,α} e s, 1 e s, α max{1,α} e s, 1..5 Remark.. Let S Sbe a nonempty closed convex set and let e S s, α be defined as follows: e S s, α s P S s αf s..6 Inequalities.5 still hold for e S s, α. Some fundamental inequalities are listed below without proof, see, for example, 10. Lemma.3. Let S be a nonempty closed convex set. Then the following inequalities always hold t P S t T P S t s 0, t R N, s S,.7 P S t s t s t P S t, t R N,s S..8 The next lemma lists some inequalities which will be useful for the following analysis. Lemma.4. Let S Sbe a nonempty closed convex set, s a solution of the monotone VI S,F.1 and especially F s 0. For any s R N and α>0, one has α s s T F P S s αf s αe S s, α T F P S s αf s, s s T{ e S s, α α[ F s F P S s αf s ]} e S s, α T{ e S s, α α[ F s F P S s αf s ]} Proof. Under the assumption that F is monotone we have { αf P S s αf s αf s } T{ P S s αf s s } 0, s R N..11 Using F s 0and the notation of e S s, α, from.11 the assertion.9 is proved. Setting t s αf s and s s in the inequality.7 and using the notation of e S s, α, weobtain { e S s, α αf s } T{ P S s αf s s } 0, s R N..1 Adding.11 and.1, andusingf s 0, we have.10. The proof is complete. Note that the assumption F s 0 in Lemma.4 is reasonable. The following proposition and remark will explain this.

6 6 Advances in Operations Research Proposition.5, Proposition.. For the optimization problem 1.1, the following two statements are equivalent: i z Ω and f z 0, ii z Ω and f z 0. Remark.6. Under the assumption that the solution set of the SFP is nonempty, if z x T, y T T is a solution of 1.1, then we have f z B T Bz B T y Ax This point z is also the solution point of the VI The next lemma provides an important boundedness property of the subdifferential, see, for example, 11. Lemma.7. Suppose h : R N R is a convex function, then it is subdifferentiable everywhere and its subdifferentials are uniformly bounded on any bounded subset of R N. 3. Modified Halfspace-Relaxation Projection Methods In this section, we will propose two kinds of modified halfspace-relaxation projection methods Algorithms 1 and. Algorithm 1 is an FB type HRP method and Algorithm is an EG type HRP method. The relationship of these two methods is that they use the same optimal step length along different descent directions. The detailed procedures are presented as below. The Modified Halfspace-Relaxation Projection Methods Step 1. Let α 0 > 0, 0 < μ < ν < 1, z 0 R n m, θ 0,, ε > 0andk 0. In practical computation, we suggest to take μ 0.3, ν 0.9 andθ 1.8. Step. Set z k P Ωk [z k α k f z k], 3.1 where Ω k is defined in 1.7. If z k z k ε, terminate the iteration with the iterate z k x k T, y k T T, and then x k is the approximate solution of the SFP. Otherwise, go to Step 3. Step 3. If r k : f α k z k f z k z k z k ν, 3.

7 Advances in Operations Research 7 then set e k z k,α k e Ωk z k,α k z k z k, g k z k,α k α k f z k, 3.3 d k z k,α k e k z k,α k α k f z k g k, 3.4 γ k e k z k Tdk,α k z k,α k, γ dk zk,α k k θγ k, 3.5 z k 1 P Ωk [ z k γ k d k z k,α k ], Algorithm 1 : FB type HRP method 3.6 or z k 1 P Ωk [ z k γ k g k z k,α k ], Algorithm : EG type HRP method α k : α k if r k μ, α k otherwise, 3.8 α k 1 α k, k k 1, go to Step. Step 4. Reduce the value of α k by α k : /3 α k min{1, 1/r k }, set z k P Ωk [z k α k f z k] and go to Step Remark 3.1. In Step 3, if the selected α k satisfies 0 <α k ν/l L is the largest eigenvalue of the matrix B T B, then from 1.1, we have f α k z k f z k αk Lz k z k z ν k z k, 3.10 and thus Condition 3. is satisfied. Without loss of generality, we can assume that inf k {α k } α min > 0. Remark 3.. By the definition of subgradient, it is clear that the halfspace Ω k contains Ω.From the expressions of Ω k, the orthogonal projections onto Ω k may be directly calculated and then we have the following proposition see 3, 1. Proposition 3.3. For any z R n m, z c z k z z k T ξ k ξ P Ωk z k, if c z k z z k T ξ k > 0; ξ k z, otherwise, 3.11 where Ω k is defined in 1.7.

8 8 Advances in Operations Research Remark 3.4. For the FB type HRP method, taking z k 1 z k γ k d k z k,α k 3.1 as the new iterate instead of Formula 3.6 seems more applicable in practice. Since from Proposition 3.3 the projection onto Ω k is easy to be computed, Formula 3.6 is still preferable to generate the new iterate z k 1. Remark 3.5. The proposed methods and the HRP method in can be used to solve more general convex optimization problem minimize f z subject to z Ω, 3.13 where f z is a general convex function only with the property that f z 0 for any solution point z of 3.13, andω is defined in 1.3. The corresponding theoretical analysis is similar as these methods to solve The Optimal Step Length This section concentrates on investigating the optimal step length with positive lower bounds in order to accelerate convergence of the new methods. To justify the reason of choosing the optimal step length γ k in the FB type HRP method 3.6, we start from the following general form of the FB type HRP method: z k 1 FB γ PΩk [z k 1 ] PC γ, 4.1 where z k 1 PC γ z k γd k. 4. Let z Θ k FBγ : k z z k 1 FB γ z, 4.3 which measures the progress made by the FB type HRP method. Note that Θ k FB γ is a function of the step length γ. It is natural to consider maximizing this function by choosing an optimal parameter γ. Thesolutionz is not known, so we cannot maximize Θ k FB γ directly. The following theorem gives an estimate of Θ k FB γ which does not include the unknown solution z. Theorem 4.1. Let z be an arbitrary point in Ω. If the step length in the general FB type HRP method is taken γ>0, then we have z Θ k FBγ : k z z k 1 FB γ z Υk γ, 4.4

9 Advances in Operations Research 9 where Υ k γ : γek z k,α k Tdk z k,α k γ dk. 4.5 Proof. Since z k 1 FB γ P Ω k z k 1 PC γ and z Ω Ω k, it follows from.8 that z k 1 FB γ z z k 1 PC γ z z k 1 PC γ z k 1 FB γ z k 1 PC γ z, 4.6 and consequently z Θ k FBγ k z z k 1 PC γ z. 4.7 Setting α α k, s z k, s z and S Ω k in the equality.10 and using the notation of e k z k,α k see 3.3 and d k z k,α k see 3.4, we have z k z T dk e k Tdk. 4.8 Using this and 4., weget z k z z k 1 γ z z k z z k γd k z PC γ z k z T dk γ dk z k,α k γe k z k,α k Tdk z k,α k γ dk, 4.9 and then from 4.7 the theorem is proved. Similarly, we start from the general form of the EG type HRP method z k 1 ] EG γ PΩk [z k γg k to analyze the optimal step length in the EG type HRP method 3.7. The following theorem estimates the progress in the sense of Euclidean distance made by the new iterate and thus motivates us to investigate the selection of the optimal length γ k in the EG type HRP method 3.7. Theorem 4.. Let z be an arbitrary point in Ω. If the step length in the general EG type HRP method is taken γ>0, then one has z Θ k EGγ : k z z k 1 EG γ z Υk γ, 4.11 where Υ k γ is defined in 4.5 and z k 1 PC γ is defined in 4..

10 10 Advances in Operations Research Proof. Since z k 1 EG γ P Ω k z k γg k z k,α k and z Ω Ω k, it follows from.8 that γ z z k γg k z z k γg k z k 1 γ, 4.1 z k 1 EG EG and consequently we get z Θ k EGγ k z z k z γg k z k z,α k k z k 1 γ γgk z k,α k z k z k 1 γ γ z k z T gk EG EG γ z k z k 1 EG T γ gk Setting α α k, s z k, s z and S Ω k in the equality.9 and using the notation of e k z k,α k and g k z k,α k see 3.3, we have z k z T gk e k Tgk From the above inequality, we obtain z Θ k EGγ k z k 1 Tgk EG γ γek γ z k z k 1 T EG γ gk Using g k z k,α k d k z k,α k e k z k,α k α k f z k see 3.4, it follows that z Θ k EGγ k z k 1 T EG γ γek {d k [e k α k f z k]} γ z k z k 1 EG γ T{d k [e k α k f z k]}, 4.16 which can be rewritten as z Θ k EGγ k z k 1 EG γ γdk z k,α k γ dk z k Tdk,α k γek γ z k z k 1 T EG γ ek e k α k f z k Υ k γ γ z k z k 1 T EG γ ek e k α k f z k Now we consider the last term in the right-hand side of Noticethat z k z k 1 [ EG γ ek P Ωk z k α k f z k] z k 1 EG γ. 4.18

11 Advances in Operations Research 11 Setting t : z k α k f z k, s : z k 1 EG γ and S Ω k in the basic inequality.7 of the projection mapping and using the notation of e k z k,α k,weget [ {P Ωk z k α k f z k] z k 1 EG γ } T{e k α k f z k} 0, 4.19 and therefore { z k z k 1 EG γ ek } T {e k α k f z k} Substituting 4.0 in 4.17, it follows that Θ k EGγ Υk γ 4.1 and the theorem is proved. Theorems 4.1 and 4. provide the basis of the selection of the optimal step length of the new methods. Note that Υ k γ is the profit-function since it is a lower-bound of the progress obtained by the new methods both the FB type HRP method and EG type HRP method. This motivates us to maximize the profit-function Υ k γ to accelerate convergence of the new methods. Since Υ k γ a quadratic function of γ, it reaches its maximum at γ k : e k z k Tdk,α k z k,α k. 4. d k zk,α k Note that under Condition 3., using the notation of d k z k,α k we have Tdk e k z k,α k z k,α k e k z k T,α k αk e k z k,α k f z k f z k 1 ν e k z k,α k. 4.3 In addition, since Tdk e k z k,α k z k,α k e k z k T,α k αk e k z k,α k f z k f z k 1 e k z k T,α k αk e k z k,α k f z k f z k 1 α k f z k f z k 1 d k, 4.4

12 1 Advances in Operations Research we have γ k : e k z k Tdk,α k z k,α k 1 dk zk,α k. 4.5 From numerical point of view, it is necessary to attach a relax factor to the optimal step length γ obtained theoretically to achieve faster convergence. The following theorem k concerns how to choose the relax factor. Theorem 4.3. Let z be an arbitrary point in Ω, θ a positive constant and γ defined in 4.. For k given z k Ω k, α k is chosen such that Condition 3. is satisfied. Whenever the new iterate z k 1 θγ k is generated by z k 1 θγ [ ] k PΩk z k θγ k d k or z k 1 θγ [ ] k PΩk z k θγ k g k, 4.6 we have z k 1 θγ k z z k z θ θ 1 ν e k z k,α k. 4.7 Proof. From Theorems 4.1 and 4. we have z k z z k 1 θγ k z Υk θγ k. 4.8 Using 4.5, 4.3, and 4.5, weobtain Tdk Υ k θγ k θγ k e k θγ dk k z k,α k Tdk Tdk θγ k e k θ γ k e k γ k θ θ e k θ θ 1 ν Tdk e k z k,α k, 4.9 and the assertion is proved. Theorem 4.3 shows theoretically that any θ 0, guarantees that the new iterate makes progress to a solution. Therefore, in practical computation, we choose γ k θγ k with θ 0, as the step length in the new methods. We need to point out that from numerical experiments, θ 1, is much preferable since it leads to better numerical performance.

13 Advances in Operations Research Convergence It follows from 4.7 that for both the FB type HRP method 3.6 and the EG type HRP method 3.7, there exists a constant τ>0, such that z k 1 z z k z τ ek, z Ω. 5.1 The convergence result of the proposed methods in this paper is based on the following theorem. Theorem 5.1. Let {z k } be a sequence generated by the proposed method 3.6 or 3.7. Then{z k } converges to a point z, which belongs to Ω. Proof. First, from 5.1 we get lim e k z k,α k k Note that e k z k,α k z k z k, see We have lim z k z k 0. k 5.4 Again, it follows from 5.1 that the sequence {z k } is bounded. Let z be a cluster point of {z k } and the subsequence {z k j } converges to z. We are ready to show that z is a solution point of 1.1. First, we show that z Ω. Since z k j Ω kj, then by the definition of Ω kj, we have c z k j z k j Tξ z k j k j 0, j 1,, Passing onto the limit in this inequality and taking into account 5.4 and Lemma.7, we obtain that c z Hence, we conclude z Ω. Next, we need to show z z T f z 0, z Ω. To do so, we first prove lim e kj z k j, j

14 14 Advances in Operations Research It follows from Remark 3.1 in Section 3 that inf j {α kj } inf k {α k } α min > 0. Then from Lemma.1, we have z k j z e kj z k j k j, 1 }, 5.8 min {1,α kj which, together with 5.4, implies that lim e kj z k j, 1 lim j j z k j z k j } lim min {1,α j kj z k j z k j 0. min{1,α min } 5.9 Setting t z k j f z k j, S Ω kj in the inequality.7, for any z Ω Ω kj,weobtain T z k j f z k j P Ωkj z k j f z k j P Ωkj z k j f z k j z From the fact that e kj z k j, 1 z k j P Ωkj z k j f z k j, we have T e kj z k j, 1 f z k j z k j e kj z k j, 1 z 0, z Ω, 5.11 that is, T f T z z k j z k j e kj z k j, 1 z k j e kj z k j, 1 z f z k j 0, z Ω. 5.1 Letting j, taking into account 5.7, we deduce z z T f z 0, z Ω, 5.13 which implies that z Ω. Then from 5.1, it follows that z k 1 z z k z τe k z k,α k Together with the fact that the subsequence {z k j } converges to z, we can conclude that {z k } converges to z. The proof is complete. 6. Numerical Results In this section, we implement the proposed methods to solve some numerical examples arising in and then report the results. To show the superiority of the new methods, we also compare them with the HRP method in. The codes for implementing the

15 Advances in Operations Research 15 Table 1: Results for Example 6.1 using the HRP method in. Starting points Number of iterations CPU Sec. Approximate solution 1,, 3, 0, 0, 0 T , 0.815, T 1, 1, 1, 1, 1, 1 T , , T 1,, 3, 4, 5, 6 T , , T Table : Results for Example 6.1 using FB type HRP method. Starting points Number of iterations CPU Sec. Approximate solution 1,, 3, 0, 0, 0 T 15 < , , T 1, 1, 1, 1, 1, 1 T 0 < , , T 1,, 3, 4, 5, 6 T 36 < , 0.957, T Table 3: Results for Example 6.1 using EG type HRP method. Starting points Number of iterations CPU Sec. Approximate solution 1,, 3, 0, 0, 0 T 15 < , , T 1, 1, 1, 1, 1, 1 T 0 < , , T 1,, 3, 4, 5, 6 T 38 < , 1.304, T proposed methods were written by Matlab R009b and run on an HP Compaq 6910p Notebook.00 GHz of Intel Core Duo CPU and.00 GB of RAM. The stopping criterion is e k z k,α k ε. For the new methods, we take ε 10 10, α 0 1, μ 0.3, ν 0.9 andθ 1.8. To compare with the HRP method and the new methods, we list the numbers of iterations, the computation times CPU Sec. and the approximate solutions in Tables 1,, 3, 4, 5, 6, 7, 8, and 9. For the HRP method in, we list the original numerical results in. Example 6.1 a convex feasibility problem. LetC {x R 3 x x 3 4 0}, Q {x R3 x 3 1 x 1 0}. Find some point x in C Q. Obviously this example can be regarded as an SFP with A I. For Example 6.1, it is easy to verify that the point 1, 1, 1, 1, 1, 1 T is a solution of 1.1. Therefore, the FB type and EG type HRP method only use 0 iteration when we choose the starting point z 0 1, 1, 1, 1, 1, 1 T. While applying the HRP method in to solve Example 6.1 and choosing the same starting point, the number of iterations is 67. This is the original numerical result listed in Table 1 of. 1 3 Example 6. a split feasibility problem. LetA 4 5,C {x R 3 x 1 x 0 x 3 0}, Q {x R 3 x 1 x x 3 0}. Find some point x C with Ax Q. Example 6.3 a nonlinear programming problem. Consider the problem Minimize f z n z i i 1 subject to i / jz i z j j 0, j 1,...,n. 6.1

16 16 Advances in Operations Research Table 4: Results for Example 6. using the HRP method in. Starting points Number of iterations CPU Sec. Approximate solution 1,, 3, 0, 0, 0 T , 0.085, T 1, 1, 1, 1, 1, 1 T , , T 1,, 3, 4, 5, 6 T , 0.156,.3719 T Table 5: Results for Example 6. using FB type HRP method. Starting points Number of iterations CPU Sec. Approximate solution 1,, 3, 0, 0, 0 T , 0.074, T 1, 1, 1, 1, 1, 1 T , 0.073, T 1,, 3, 4, 5, 6 T , , T Table 6: Results for Example 6. using EG type HRP method. Starting points Number of iterations CPU Sec. Approximate solution 1,, 3, 0, 0, 0 T , , T 1, 1, 1, 1, 1, 1 T , , T 1,, 3, 4, 5, 6 T , , T Table 7: Results for Example 6.3 using the HRP method in. n dimension Number of iterations CPU Sec Table 8: Results for Example 6.3 using FB type HRP method. n dimension Number of iterations CPU Sec < Table 9: Results for Example 6.3 using EG type HRP method. n dimension Number of iterations CPU Sec < This example is a general nonlinear programming problem not the reformulation 1.1 for the SFP. Notice that it has a unique solution z 0,...,0 T and f z z 0,...,0 T. Then from Remark 3.5 in Section 3, the proposed methods and the HRP method in can be used to find its solution.

17 Advances in Operations Research 17 The computational preferences to the HRP method are revealed clearly in Tables 1 9. The numerical results demonstrate that the selection of optimal step length in both the FB type HRP method and the EG type HRP method reduces considerable computational load of the HRP method in. 7. Conclusions In this paper we consider the split feasibility problem, which is a special case of the multiplesets split feasibility problem With some new strategies for determining the optimal step length, this paper improves the HRP method in and thus develops modified halfspace-relaxation projection methods for solving the split feasibility problem. Compared to the HRP method in, the new methods reduce the number of iterations moderately with little additional computation. Acknowledgments This work was supported by National Natural Science Foundation of China Grant no and the Research Award Fund for Outstanding Young Teachers in Southeast University. References 1 Y. Censor and T. Elfving, A multiprojection algorithm using Bregman projections in a product space, Numerical Algorithms, vol. 8, no. 4, pp. 1 39, B. Qu and N. Xiu, A new halfspace-relaxation projection method for the split feasibility problem, Linear Algebra and its Applications, vol. 48, no. 5-6, pp , M. Fukushima, A relaxed projection method for variational inequalities, Mathematical Programming, vol. 35, no. 1, pp , B. Qu and N. Xiu, A note on the CQ algorithm for the split feasibility problem, Inverse Problems, vol. 1, no. 5, pp , Q. Yang, The relaxed CQ algorithm solving the split feasibility problem, Inverse Problems, vol. 0, no. 4, pp , P. Tseng, A modified forward-backward splitting method for maximal monotone mappings, SIAM Journal on Control and Optimization, vol. 38, no., pp , E. N. Khobotov, A modification of the extragradient method for solving variational inequalities and some optimization problems, USSR Computational Mathematics and Mathematical Physics, vol. 7, no. 10, pp , G. M. Korpelevič, An extragradient method for finding saddle points and for other problems, Èkonomika i Matematicheskie Metody, vol. 1, no. 4, pp , B. He, X. Yuan, and J. J. Z. Zhang, Comparison of two kinds of prediction-correction methods for monotone variational inequalities, Computational Optimization and Applications, vol. 7, no. 3, pp , B. C. Eaves, On the basic theorem of complementarity, Mathematical Programming, vol. 1, no. 1, pp , R. T. Rockafellar, Convex Analysis, Princeton University Press, Princeton, NJ, USA, B. T. Polyak, Minimization of unsmooth functionals, USSR Computational Mathematics and Mathematical Physics, vol. 9, pp. 14 9, Y. Censor, T. Elfving, N. Kopf, and T. Bortfeld, The multiple-sets split feasibility problem and its applications for inverse problems, Inverse Problems, vol. 1, no. 6, pp , H.-K. Xu, A variable Krasnosel skii-mann algorithm and the multiple-set split feasibility problem, Inverse Problems, vol., no. 6, pp , W. Zhang, D. Han, and Z. Li, A self-adaptive projection method for solving the multiple-sets split feasibility problem, Inverse Problems, vol. 5, no. 11, pp , 009.

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