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Approximating Curve and Strong Convergence of the CQ Algorithm for the Split Feasibility Problem

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Abstract

Using the idea of Tikhonov's regularization, we present properties of the approximating curve for the split feasibility problem (SFP) and obtain the minimum-norm solution of SFP as the strong limit of the approximating curve. It is known that in the infinite-dimensional setting, Byrne's CQ algorithm (Byrne, 2002) has only weak convergence. We introduce a modification of Byrne's CQ algorithm in such a way that strong convergence is guaranteed and the limit is also the minimum-norm solution of SFP.

1. Introduction

Let and be nonempty closed convex subsets of real Hilbert spaces and , respectively. The problem under consideration in this article is formulated as finding a point satisfying the property:

(1.1)

where is a bounded linear operator. Problem (1.1), referred to by Censor and Elfving [1] as the split feasibility problem (SFP), attracts many authors' attention due to its application in signal processing [1]. Various algorithms have been invented to solve it (see [27] and reference therein).

In particular, Byrne [2] introduced the so-called algorithm. Take an initial guess arbitrarily, and define recursively as

(1.2)

where and where denotes the projector onto and is the spectral radius of the self-adjoint operator Then the sequence generated by (1.2) converges strongly to a solution of SFP whenever is finite-dimensional and whenever there exists a solution to SFP (1.1).

However, the algorithm need not necessarily converge strongly in the case when is infinite dimensional. Let us mention that the algorithm can be regarded as a special case of the well-known Krasnosel'skii-Mann algorithm for approximating fixed points of nonexpansive mappings [3]. This iterative method is introduced in [8] and defined as follows. Take an initial guess arbitrarily, and define recursively as

(1.3)

where satisfying If is nonexpansive with a nonempty fixed point set, then the sequence generated by (1.3) converges weakly to a fixed point of . It is known that Krasnosel'skii-Mann algorithm is in general not strongly convergent (see [9, 10] for counterexamples) and neither is the algorithm.

It is therefore the aim of this paper to construct a new algorithm so that strong convergence is guaranteed. The paper is organized as follows. In the next section, some useful lemmas are given. In Section 3, we define the concept of the minimal norm solution of SFP (1.1). Using Tikhonov's regularization, we obtain a continuous curve for approximating such minimal norm solution. Together with some properties of this approximating curve, we introduce, in Section 4, a modification of Byrne's algorithm so that strong convergence is guaranteed and its limit is the minimum-norm solution of SFP (1.1).

2. Preliminaries

Throughout the rest of this paper, denotes the identity operator on , the set of the fixed points of an operator and the gradient of the functional The notation "" denotes strong convergence and "" weak convergence.

Recall that an operator from into itself is called nonexpansive if

(2.1)

contractive if there exists such that

(2.2)

monotone if

(2.3)

Obviously, contractions are nonexpansive, and if is nonexpansive, then is monotone (see [11]).

Let denote the projection from onto a nonempty closed convex subset of ; that is,

(2.4)

It is well known that is characterized by the inequality

(2.5)

Consequently, is nonexpansive.

The lemma below is referred to as the demiclosedness principle for nonexpansive mappings (see [12]).

Lemma 2.1 (demiclosedness principle).

Let be a nonempty closed convex subset of and a nonexpansive mapping with If is a sequence in weakly converging to and if the sequence converges strongly to , then In particular, if , then .

Let be a a functional. Recall that

(i) is convex if

(2.6)

(ii) is strictly convex if the strict less than inequality in (2.6) holds for all distinct .

(iii) is strongly convex if there exists a constant such that

(2.7)

(iv) is coercive if whenever It is easily seen that if is strongly convex, then it is coercive. See [13] for more details about convex functions.

The following lemma gives the optimality condition for the minimizer of a convex functional over a closed convex subset.

Lemma 2.2 (see [14]).

Let be a convex and differentiable functional and let be a closed convex subset of . Then is a solution of the problem

(2.8)

if and only if satisfies the following optimality condition:

(2.9)

Moreover, if is, in addition, strictly convex and coercive, then problem (2.8) has a unique solution.

The following is a sufficient condition for a real sequence to converge to zero.

Lemma 2.3 (see [15]).

Let be a nonnegative real sequence satisfying

(2.10)

where the sequences and satisfy the conditions:

(1)

(2)

(3)either or

Then .

3. Approximating Curves

The convexly constrained linear problem requires to solve the constrained linear system (cf. [16, 17])

(3.1)

where . A classical way to deal with such a possibly ill-posed problem is the well-known Tikhonov regularization, which approximates a solution of problem (3.1) by the unique minimizer of the regularized problem

(3.2)

where is known as the regularization parameter.

We now try to transfer this idea of Tikhonov's regularization method for solving the constrained linear inverse problem (3.1) to the case of SFP (1.1).

It is not hard to find that SFP (1.1) is equivalent to the minimization problem

(3.3)

Motivated by Tikhonov's regularization, we consider the minimization problem

(3.4)

where is the regularization parameter. Denote by the unique solution of (3.4); namely,

(3.5)

Proposition 3.1.

For any the minimizer given by (3.5) is uniquely defined. Moreover, is characterized by the inequality

(3.6)

Proof.

Let

(3.7)

Since is convex and differentiable with gradient (see [13])

(3.8)

is strictly convex, coercive, and differentiable with gradient

(3.9)

Thus, applying Lemma 2.2 gets the assertion (3.6), as desired.

The next result collects some useful properties of .

Proposition 3.2.

The following assertions hold.

(a) is decreasing for .

(b) is increasing for .

(c) defines a continuous curve from to

Proof.

Let be fixed. Since and are the (unique) minimizers of and , respectively, we get

(3.10)
(3.11)

Adding up (3.10) and (3.11) yields

(3.12)

which implies that . Hence (a) holds.

It follows from (3.11) that

(3.13)

which together with (a) implies

(3.14)

and therefore (b) holds.

By Proposition 3.1, we have that

(3.15)

and also that

(3.16)

Adding up (3.15) and (3.16), we get

(3.17)

Since is monotone, we obtain from the last relation

(3.18)

It turns out that

(3.19)

Thus (c) holds.

Let , where . In what follows, we assume that that is, the solution set of SFP (1.1) is nonempty. The fact that is nonempty closed convex set thus allows us to introduce the concept of minimum-norm solution of SFP (1.1).

Definition 3.3.

An element is said to be the minimal norm solution of SFP (1.1) if . In other words, is the projection of the origin onto the solution set of SFP (1.1). Thus the minimum-norm solution for SFP (1.1) exists and is unique.

Theorem 3.4.

Let be given as (3.5). Then converges strongly as to the minimum-norm solution of SFP (1.1).

Proof.

We first show that the inequality

(3.20)

holds for any . To this end, observe that

(3.21)

Since , . It follows from (3.21) that

(3.22)

and (3.20) is proven.

Let now be a sequence such that as and let be abbreviated as All we need to prove is that contains a subsequence converging strongly to Since is bounded and since is bounded convex, by passing to a subsequence if necessary, we may assume that converges weakly to a point By Proposition 3.1, we deduce that

(3.23)

It turns out that

(3.24)

Since the characterizing inequality (2.5) gives

(3.25)

and this implies that

(3.26)

Now by combining (3.26) and (3.24), we get

(3.27)

where the last inequality follows from (3.20). Consequently, we get

(3.28)

Note that is also weakly continuous and hence . Now due to (3.28), we can use the demiclosedness principle (Lemma 2.1) to conclude that . That is, or ; therefore, We next prove that and this finishes the proof. To see this, we have that the weak convergence to of together with (3.20) implies that

(3.29)

This shows that is also a point in which assumes minimum norm. Due to uniqueness of minimum-norm element, we must have .

Remark 3.5.

The above argument shows that if the solution set of SFP (1.1) is empty, then the net of norms, , diverges to as .

4. A Modified CQ Algorithm

It is a standard way to use contractions to approximate nonexpansive mappings. We follow this idea and use contractions to approximate the nonexpansive mapping in order to modify Byrne's algorithm. More precisely, we introduce the following algorithm which is viewed as a modification of Byrne's algorithm. The purpose for such a modification lies in the hope of strong convergence.

Algorithm 4.1.

For an arbitrary guess , the sequence is generated by the iterative algorithm

(4.1)

where is a sequence in such that

(1);

(2);

(3)either or

Note that a prototype of is for all .

To prove the convergence of algorithm (4.1) (see Theorem 4.3 below), we need a lemma below.

Lemma 4.2.

Set , where with being the spectral radius of the self-adjoint operator

(i) is an averaged mapping; namely, , where is a constant and is a nonexpansive mapping from into itself.

(ii); consequently, .

Proof.

  1. (i)

    That which is averaged is actually proved in [3].

To see (ii), we first observe that the inclusion holds trivially. It remains to prove the implication: . To see this, we notice that the relation is equivalent to the relation . It turns out that

(4.2)

Since the solution set , we can take . Now since , we have by (2.5),

(4.3)

It follows from (4.2) and (4.3) that

(4.4)

This shows that ; that is, .

Finally, since , and both and are averaged, we have .

Theorem 4.3.

The sequence generated by algorithm (4.1) converges strongly to the minimum-norm solution of SFP (1.1).

Proof.

Define operators and on by

(4.5)

where is averaged by Lemma 4.2.

It is readily seen that is a contraction with contractive constant . Namely,

(4.6)

Also we may rewrite algorithm (4.1) as

(4.7)

We first prove that is a bounded sequence. Indeed, since , we can take any (thus by Lemma 4.2) to deduce that

(4.8)

Note that

(4.9)

Substituting (4.9) into (4.8), we get

(4.10)

By induction, we can easily show that, for all ,

(4.11)

In particular, is bounded.

We now claim that

(4.12)

To see this, we compute

(4.13)

Letting be a constant such that for all , we find

(4.14)

Substituting (4.14) into (4.13), we arrive at

(4.15)

By virtue of the assumptions (a)–(c), we can apply Lemma 2.3 to (4.15) to obtain (4.12). Consequently we also have

(4.16)

This follows from the following computations:

(4.17)

Therefore, the demiclosedness principle (Lemma 2.1) ensures that each weak limit point of is a fixed point of the nonexpansive mapping , that is, a point of the solution set of SFP (1.1).

One of the key ingredients of the proof is the following conclusion:

(4.18)

where is the minimum-norm element of (i.e., the projection ). Since

(4.19)

to prove (4.18), it suffices to prove that

(4.20)
(4.21)

To prove (4.20), we use Lemma 4.2 to get and is averaged. Write for some and nonexpansive mapping . Then we derive, by taking a point , that

(4.22)

It turns out that (for some constant for all )

(4.23)

Now since , (4.23) implies (4.20).

To prove (4.21), we take a subsequence of so that

(4.24)

Since is bounded, we may further assume with no loss of generality that converges weakly to a point . Noticing that and that is the projection of the origin onto , and applying (2.5), we arrive at

(4.25)

This is (4.21).

Finally we prove in norm. To see this, we compute

(4.26)

where

(4.27)

satisfies the property (due to (4.18))

(4.28)

We therefore can apply Lemma 2.3 to (4.26) to conclude that . This completes the proof.

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Acknowledgment

H. K. Xu was supported in part by NSC 97-2628-M-110-003-MY3, and by DGES MTM2006-13997-C02-01.

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Correspondence to Hong-Kun Xu.

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Keywords

  • Minimization Problem
  • Weak Convergence
  • Nonexpansive Mapping
  • Strong Convergence
  • Bounded Linear Operator