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

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 (2008)). Some numerical results are reported to verify the computational preference.


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 2 of the SFP: 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 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 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: Qu and Xiu 2 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: where 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 1.9 The notation P Ω k v denotes the projection of v onto Ω k under the Euclidean norm, that is, 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 Ω k is from the halfspace-relaxation projection technique presented by Fukushima 3 .This technique is often used to design algorithms see, e.g., 2, 4, 5 to solve the SFP.The drawback of the HRP method in 2 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 : where 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 1.11 .For given z k , let Under the assumption the forward-backward FB splitting method generates the new iterate via while the extra-gradient EG method generates the new iterate by The forward-backward splitting method 1.15 can be rewritten as where the descent direction 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 and −α k ∇f z k , respectively.Here γ k is defined by

1.18
Under the assumption 1.14 , γ * k ≥ 1/2 is lower bounded.This paper is to develop two kinds of modified halfspace-relaxation projection methods for solving the SFP by improving the HRP method in 2 .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 2 .
The rest of this paper is organized as follows.In Section 2, 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 2 .Finally, some conclusions are made in Section 7.

Preliminaries
In the following, we state some basic concepts for the variational inequality VI S, F : where F is a mapping from R N into R N , and S ⊆ R N is a nonempty closed convex set.The mapping F is said to be monotone on R N if Notice that the variational inequality VI S, F is invariant when we multiply F by some positive scalar α.Thus VI S, F is equivalent to the following projection equation see 10 :

2.8
The next lemma lists some inequalities which will be useful for the following analysis.

2.10
Proof.Under the assumption that F is monotone we have

2.11
Using F s * 0 and the notation of e S s, α , from 2.11 the assertion 2.9 is proved.Setting t s − αF s and s s * in the inequality 2.7 and using the notation of e S s, α , we obtain e S s, α − αF s T P S s − αF s − s * ≥ 0, ∀s ∈ R N .

2.12
Adding 2.11 and 2.12 , and using F s * 0, we have 2.10 .The proof is complete.
Note that the assumption F s * 0 in Lemma 2.4 is reasonable.The following proposition and remark will explain this.

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Proposition 2.5 2 , Proposition 2.2 .For the optimization problem 1.1 , the following two statements are equivalent: ii z * ∈ Ω and ∇f z * 0.
Remark 2.6.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 This point z * is also the solution point of the VI 1.11 .
The next lemma provides an important boundedness property of the subdifferential, see, for example, 11 .
Lemma 2.7.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 .

Modified Halfspace-Relaxation Projection Methods
In this section, we will propose two kinds of modified halfspace-relaxation projection methods-Algorithms 1 and 2. Algorithm 1 is an FB type HRP method and Algorithm 2 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.
Step 2. Set 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 or

3.8
Step 4. Reduce the value of α k by α k : 2/3 α k * min{1, 1/r k }, set z k P Ω k z k − α k ∇f z k and go to Step 3.

3.9
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 1.12 , we have and thus Condition 3.2 is satisfied.Without loss of generality, we can assume that inf k {α k } α min > 0.
Remark 3.2.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, 12 .

3.11
where Ω k is defined in 1.7 .
Remark 3.4.For the FB type HRP method, taking 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 2 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 , and Ω is defined in 1.3 .The corresponding theoretical analysis is similar as these methods to solve 1.1 .

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: where 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.The solution z * 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 where and consequently Setting α α k , s z k , s * z * and S Ω k in the equality 2.10 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 Using this and 4.2 , we get 4.9 and then from 4.7 the theorem is proved.
Similarly, we start from the general form of the EG type HRP method 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 4.2.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 where Υ k γ is defined in 4.5 and z k 1 PC γ is defined in 4.2 .

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Proof.Since 12 and consequently we get

4.13
Setting α α k , s z k , s * z * and S Ω k in the equality 2.9 and using the notation of e k z k , α k and g k z k , α k see 3.3 , we have From the above inequality, we obtain

4.15
Using which can be rewritten as

4.17
Now we consider the last term in the right-hand side of 4.17 .Notice that

4.18
Setting t : z k −α k ∇f z k , s : z k 1 EG γ and S Ω k in the basic inequality 2.7 of the projection mapping and using the notation of e k z k , α k , we get and therefore Substituting 4.20 in 4.17 , it follows that and the theorem is proved.
Theorems 4.1 and 4.2 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 :

4.22
Note that under Condition 3.2 , using the notation of d k z k , α k we have

4.23
In addition, since

4.24
Advances in Operations Research we have From numerical point of view, it is necessary to attach a relax factor to the optimal step length γ * k obtained theoretically to achieve faster convergence.The following theorem concerns how to choose the relax factor.
we have

4.27
Proof.From Theorems 4.1 and 4.2 we have

4.28
Using 4.5 , 4.23 , and 4.25 , we obtain and the assertion is proved.
Theorem 4.3 shows theoretically that any θ ∈ 0, 2 guarantees that the new iterate makes progress to a solution.Therefore, in practical computation, we choose γ k θγ * k with θ ∈ 0, 2 as the step length in the new methods.We need to point out that from numerical experiments, θ ∈ 1, 2 is much preferable since it leads to better numerical performance.
Advances in Operations Research 13

Convergence
It follows from 4.27 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 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 Note that We have 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 k j , then by the definition of Ω k j , we have Passing onto the limit in this inequality and taking into account 5.4 and Lemma 2.7, we obtain that c z ≤ 0.
Next, we need to show z − z T ∇f z ≥ 0, ∀z ∈ Ω.To do so, we first prove lim j → ∞ e k j z k j , 1 0.

5.7
It follows from Remark 3.1 in Section 3 that inf j {α k j } ≥ inf k {α k } α min > 0. Then from Lemma 2.1, we have which, together with 5.4 , implies that lim 5.9 Setting t z k j − ∇f z k j , S Ω k j in the inequality 2.7 , for any z ∈ Ω ⊆ Ω k j , we obtain From the fact that e k j z k j , 1 z k j − P Ω k j z k j − ∇f z k j , we have 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

5.14
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.

Numerical Results
In this section, we implement the proposed methods to solve some numerical examples arising in 2 and then report the results.To show the superiority of the new methods, we also compare them with the HRP method in 2 .The codes for implementing the  For the new methods, we take ε 10 −10 , α 0 1, μ 0.3, ν 0.9 and θ 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, 2, 3, 4, 5, 6, 7, 8, and 9.For the HRP method in 2 , we list the original numerical results in 2 .

Example 6.1 a convex feasibility problem . Let
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 2 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 i − z j − j ≤ 0, j 1, . . ., n.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 * 2z * 0, . . ., 0 T .Then from Remark 3.5 in Section 3, the proposed methods and the HRP method in 2 can be used to find its solution.

6.1
The computational preferences to the HRP method 1.5 -1.6 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 2 .

Conclusions
In this paper we consider the split feasibility problem, which is a special case of the multiplesets split feasibility problem 13-15 .With some new strategies for determining the optimal step length, this paper improves the HRP method in 2 and thus develops modified halfspace-relaxation projection methods for solving the split feasibility problem.Compared to the HRP method in 2 , the new methods reduce the number of iterations moderately with little additional computation.

Lemma 2 . 4 . 9 s
Let S ⊇ S be a nonempty closed convex set, s * a solution of the monotone VI S, F 2.1 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 , 2.− 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 .

Theorem 4 . 3 .
Let z * be an arbitrary point in Ω * , θ a positive constant and γ * k defined in 4.22 .For given z

Table 1 :
Results for Example 6.1 using the HRP method in 2 .

Table 2 :
Results for Example 6.1 using FB type HRP method.

Table 3 :
Results for Example 6.1 using EG type HRP method.
Tproposed methods were written by Matlab 7.9.0R2009b and run on an HP Compaq 6910p Notebook 2.00 GHz of Intel Core 2 Duo CPU and 2.00 GB of RAM .The stopping criterion is e k z k , α k ≤ ε.

Table 4 :
Results for Example 6.2 using the HRP method in 2 .

Table 5 :
Results for Example 6.2 using FB type HRP method.

Table 6 :
Results for Example 6.2 using EG type HRP method.

Table 7 :
Results for Example 6.3 using the HRP method in 2 .

Table 8 :
Results for Example 6.3 using FB type HRP method.

Table 9 :
Results for Example 6.3 using EG type HRP method.