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Journal of Applied Mathematics
Volume 2012 (2012), Article ID 683890, 10 pages
http://dx.doi.org/10.1155/2012/683890
Research Article

A Note on Approximating Curve with 1-Norm Regularization Method for the Split Feasibility Problem

1College of Science, Civil Aviation University of China, Tianjin 300300, China
2Tianjin Key Laboratory for Advanced Signal Processing, Civil Aviation University of China, Tianjin 300300, China

Received 21 March 2012; Accepted 6 June 2012

Academic Editor: Hong-Kun Xu

Copyright © 2012 Songnian He and Wenlong Zhu. 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.

Abstract

Inspired by the very recent results of Wang and Xu (2010), we study properties of the approximating curve with 1-norm regularization method for the split feasibility problem (SFP). The concept of the minimum-norm solution set of SFP in the sense of 1-norm is proposed, and the relationship between the approximating curve and the minimum-norm solution set is obtained.

1. Introduction

Let 𝐶 and 𝑄 be nonempty closed convex subsets of real Hilbert spaces 𝐻1 and 𝐻2, respectively. The problem under consideration in this paper is formulated as finding a point 𝑥 satisfying the property: 𝑥𝐶,𝐴𝑥𝑄,(1.1) where 𝐴𝐻1𝐻2 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 [213] and references therein).

Using the idea of Tikhonov's regularization, Wang and Xu [14] studied the properties of the approximating curve for the SFP. They gave the concept of the minimum-norm solution of the SFP (1.1) and proved that the approximating curve converges strongly to the minimum-norm solution of the SFP (1.1). Together with some properties of this approximating curve, they introduced a modification of Byrne’s CQ algorithm [2] so that strong convergence is guaranteed and its limit is the minimum-norm solution of SFP (1.1).

In the practical application, 𝐻1 and 𝐻2 are often 𝑁 and 𝑀, respectively. Moreover, scientists and engineers are more willing to use 1-norm regularization method in the calculation process (see, e.g., [1518]). Inspired by the above results of Wang and Xu [14], we study properties of the approximating curve with 1-norm regularization method. We also define the concept of the minimum-norm solution set of SFP (1.1) in the sense of 1-norm. The relationship between the approximating curve and the minimum-norm solution set is obtained.

2. Preliminaries

Let 𝑋 be a normed linear space with norm , and let 𝑋 be the dual space of 𝑋. We use the notation 𝑥,𝑓 to denote the value of 𝑓𝑋 at 𝑥𝑋. In particular, if 𝑋 is a Hilbert space, we will denote it by 𝐻, and , and are the inner product and its induced norm, respectively.

We recall some definitions and facts that are needed in our study.

Let 𝑃𝐶 denote the projection from 𝐻 onto a nonempty closed convex subset 𝐶 of 𝐻; that is, 𝑃𝐶𝑥=argmin𝑦𝐶𝑥𝑦,𝑥𝐻.(2.1) It is well known that 𝑃𝐶𝑥 is characterized by the inequality 𝑥𝑃𝐶𝑥,𝑦𝑃𝐶𝑥0,𝑦𝐶.(2.2)

Definition 2.1. Let 𝜑𝑋{+} be a convex functional, 𝑥0dom(𝜑)={𝑥𝑋𝜑(𝑥)<+}. Set 𝑥𝜕𝜑0=𝜉𝑋𝑥𝜑(𝑥)𝜑0+𝑥𝑥0,𝜉,𝑥𝑋.(2.3) If 𝜕𝜑(𝑥0), 𝜑 is said to be subdifferentiable at 𝑥0 and 𝜕𝜑(𝑥0) is called the subdifferential of 𝜑 at 𝑥0. For any 𝜉𝜕𝜑(𝑥0), we say 𝜉 is a subgradient of 𝜑 at 𝑥0.

Lemma 2.2. There holds the following property: 𝑥𝜕(𝑥)=𝑋𝑥=1,𝑥,𝑥𝑥=𝑥,𝑥0,𝑋𝑥1,𝑥=0,(2.4) where 𝜕(𝑥) denotes the subdifferential of the functional 𝑥 at 𝑥𝑋.

Proof. The process of the proof will be divided into two parts.
Case  1. In the case of 𝑥=0, for any 𝑥𝑋 such that 𝑥1 and any 𝑦𝑋, there holds the inequality 𝑦𝑦,𝑥=𝑥+𝑦𝑥,𝑥,(2.5) so we have 𝑥𝜕(𝑥), and thus, 𝑥𝑋𝑥1𝜕(𝑥).(2.6) Conversely, for any 𝑥𝜕(𝑥), we have from the definition of subdifferential that 𝑦𝑥+𝑦𝑥,𝑥=𝑦,𝑥,𝑦𝑋,𝑦=𝑦𝑦,𝑥=𝑦,𝑥.(2.7) Consequently, ||𝑦,𝑥||𝑦,𝑦𝑋,(2.8) and this implies that 𝑥1. Thus, we have verified that 𝜕𝑥(𝑥)𝑋𝑥1.(2.9) Combining (2.6) and (2.9), we immediately obtain 𝜕𝑥(𝑥)=𝑋𝑥1.(2.10)
Case  2. If 𝑥0, for any 𝑥{𝑥𝑋𝑥=1,𝑥,𝑥=𝑥}, we obviously have 𝑦𝑥,𝑥=𝑦,𝑥𝑥𝑦𝑥,𝑦𝑋,(2.11) which means that 𝑥𝜕(𝑥), and thus, 𝑥𝑋𝑥=1,𝑥,𝑥=𝑥𝜕(𝑥).(2.12) Conversely, if 𝑥𝜕(𝑥), we have 𝑥,𝑥0𝑥=𝑥,𝑥,𝑥2𝑥𝑥=𝑥;(2.13) hence, 𝑥,𝑥=𝑥.(2.14) On the other hand, using (2.14), we get 𝑦𝑥+𝑦𝑥,𝑥=𝑥+𝑦,𝑥𝑥,𝑥=𝑦,𝑥,𝑦𝑋,(2.15) and consequently, 𝑦=𝑦𝑥+𝑦𝑥,𝑥=𝑥𝑦,𝑥𝑥,𝑥=𝑦,𝑥;(2.16) that is, 𝑦𝑦,𝑥.(2.17) Equation (2.17) together with (2.15) implies that ||𝑦,𝑥||𝑦,𝑦𝑋;(2.18) hence, 𝑥1. Note that (2.14) implies that 𝑥𝑥,𝑥/𝑥=1; we assert that 𝑥=1.(2.19) Thus we have from (2.14) and (2.19) that 𝑥𝑋𝑥=1,𝑥,𝑥=𝑥𝜕(𝑥).(2.20) The proof is finished by combining (2.12) and (2.20).

and 1 will stand for -norm and 1-norm of any Euclidean space; respectively, that is, for any 𝑥=(𝑥1,𝑥2,,𝑥𝑙)𝑙, we have 𝑥=max1𝑗𝑙||𝑥𝑗||,𝑥1=𝑙𝑗=1||𝑥𝑗||.(2.21)

Corollary 2.3. In l-dimensional Euclidean space 𝑙, there holds the following result: 𝜕𝑥1=𝜉𝑙𝜉=1,𝑥,𝜉=𝑥1,𝑥0,𝜉𝑙𝜉=1,𝑥=0,𝜉𝑙𝜉𝑖=𝑥𝑖||𝑥𝑖||,if𝑥𝑖0;𝜉𝑖[]1,1,if𝑥𝑖=0,𝑥0,𝜉𝑙𝜉1,𝑥=0.(2.22) Let 𝐻 be a Hilbert space and 𝑓𝐻 a functional. Recall that (i)𝑓is convex if 𝑓(𝜆𝑥+(1𝜆)𝑦)𝜆𝑓(𝑥)+(1𝜆)𝑓(𝑦), for all 0<𝜆<1, for all 𝑥,𝑦𝐻;(ii)𝑓 is strictly convex if 𝑓(𝜆𝑥+(1𝜆)𝑦)<𝜆𝑓(𝑥)+(1𝜆)𝑓(𝑦), for all 0<𝜆<1, for all 𝑥,𝑦𝐻 with 𝑥𝑦;(iii)𝑓 is coercive if 𝑓(𝑥) whenever 𝑥. See [19] 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 2.4 (see [20]). Let 𝐻 be a Hilbert space and 𝐶 a nonempty closed convex subset of 𝐻. Let 𝑓𝐻 be a convex and subdifferentiable functional. Then 𝑥𝐶 is a solution of the problem min𝑥𝐶𝑓(𝑥)(2.23) if and only if there exists some 𝜉𝜕𝑓(𝑥) satisfying the following optimality condition: 𝜉,𝑣𝑥0,𝑣𝐶.(2.24)

3. Main Results

It is well known that SFP (1.1) is equivalent to the minimization problem min𝑥𝐶𝐼𝑃𝑄𝐴𝑥2.(3.1) Using the idea of Tikhonov's regularization method, Wang and Xu [14] studied the minimization problem in Hilbert spaces: min𝑥𝐶𝐼𝑃𝑄𝐴𝑥2+𝛼𝑥2,(3.2) where 𝛼>0 is the regularization parameter.

In what follows, 𝐻1 and 𝐻2 in SFP (1.1) are restricted to 𝑁 and 𝑀, respectively, and will stand for the usual 2-norm of any Euclidean space 𝑙; that is, for any 𝑥=(𝑥1,𝑥2,,𝑥𝑙)𝑙, 𝑥=𝑥21++𝑥2𝑙.(3.3) Inspired by the above work of Wang and Xu, we study properties of the approximating curve with 1-norm regularization scheme for the SFP, that is, the following minimization problem: min𝑥𝐶12𝐼𝑃𝑄𝐴𝑥2+𝛼𝑥1,(3.4) where 𝛼>0 is the regularization parameter. Let 𝑓𝛼1(𝑥)=2𝐼𝑃𝑄𝐴𝑥2+𝛼𝑥1.(3.5) It is easy to see that 𝑓𝛼 is convex and coercive, so problem (3.4) has at least one solution. However, the solution of problem (3.4) may not be unique since 𝑓𝛼 is not necessarily strictly convex. Denote by 𝑆𝛼 the solution set of problem (3.4); thus we can assert that 𝑆𝛼 is a nonempty closed convex set but may contain more than one element. The following simple example illustrates this fact.

Example 3.1. Let 𝐶={(𝑥,𝑦)𝑥+𝑦=1}, 𝑄={(𝑥,𝑦)𝑥+𝑦=1/2} and 1𝐴=20012.(3.6) Then 𝐴22 is a bounded linear operator. Obviously, 𝑆𝛼={(𝑥,𝑦)𝑥+𝑦=1,𝑥0,𝑦0} and it contains more than one element.

Proposition 3.2. For any 𝛼>0, 𝑥𝛼𝑆𝛼 if and only if there exists some 𝜉𝜕(𝑥1) satisfying the following inequality: 𝐴𝐼𝑃𝑄𝐴𝑥𝛼+𝛼𝜉,𝑣𝑥𝛼0,𝑣𝐶.(3.7)

Proof. Let 1𝑓(𝑥)=2𝐼𝑃𝑄𝐴𝑥2,(3.8) then 𝑓𝛼(𝑥)=𝑓(𝑥)+𝛼𝑥1.(3.9) Since 𝑓 is convex and differentiable with gradient 𝑓(𝑥)=𝐴𝐼𝑃𝑄𝐴𝑥,(3.10)𝑓𝛼 is convex, coercive, and subdifferentiable with the subdifferential 𝜕𝑓𝛼(𝑥)=𝜕𝑓(𝑥)+𝛼𝜕𝑥1;(3.11) that is, 𝜕𝑓𝛼(𝑥)=𝐴𝐼𝑃𝑄𝐴𝑥+𝛼𝜕𝑥1.(3.12) By Corollary 2.3 and Lemma 2.4, the proof is finished.

Theorem 3.3. Denote by 𝑥𝛼 an arbitrary element of 𝑆𝛼, then the following assertions hold:(i)𝑥𝛼1 is decreasing for 𝛼(0,);(ii)(𝐼𝑃𝑄)𝐴𝑥𝛼 is increasing for 𝛼(0,).

Proof. Let 𝛼>𝛽>0, for any 𝑥𝛼𝑆𝛼, 𝑥𝛽𝑆𝛽. We immediately obtain 12𝐼𝑃𝑄𝐴𝑥𝛼2𝑥+𝛼𝛼112𝐼𝑃𝑄𝐴𝑥𝛽2𝑥+𝛼𝛽1,1(3.13)2𝐼𝑃𝑄𝐴𝑥𝛽2𝑥+𝛽𝛽112𝐼𝑃𝑄𝐴𝑥𝛼2𝑥+𝛽𝛼1.(3.14) Adding up (3.13) and (3.14) yields 𝛼𝑥𝛼1𝑥+𝛽𝛽1𝑥𝛼𝛽1𝑥+𝛽𝛼1,(3.15) which implies 𝑥𝛼1𝑥𝛽1. Hence (i) holds.
Using (3.14) again, we have 12𝐼𝑃𝑄𝐴𝑥𝛽212𝐼𝑃𝑄𝐴𝑥𝛼2𝑥+𝛽𝛼1𝑥𝛽1,(3.16) which together with (i) implies 𝐼𝑃𝑄𝐴𝑥𝛽2𝐼𝑃𝑄𝐴𝑥𝛼2,(3.17) and hence (ii) holds.

Let =𝐶𝐴1(𝑄), where 𝐴1(𝑄)={𝑥𝑁𝐴𝑥𝑄}. 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) in the sense of norm (induced by the inner product).

Definition 3.4 (see [14]). An element 𝑥 is said to be the minimum-norm solution of SFP (1.1) in the sense of norm if 𝑥=inf𝑥𝑥. In other words, 𝑥 is the projection of the origin onto the solution set of SFP (1.1). Thus there exists only one minimum-norm solution of SFP (1.1) in the sense of norm , which is always denoted by 𝑥.
We can also give the concept of minimum-norm solution of SFP (1.1) in other senses.

Definition 3.5. An element ̃𝑥 is said to be a minimum-norm solution of SFP (1.1) in the sense 𝑜𝑓 1-norm if ̃𝑥1=inf𝑥𝑥1. We use 1 to stand for all minimum-norm solutions of SFP (1.1) in the sense of 1-norm and 1 is called the minimum-norm solution set of SFP (1.1) in the sense of 1-norm.
Obviously, 1 is a closed convex subset of . Moreover, it is easy to see that 1. Indeed, taking a sequence {𝑥𝑛} such that 𝑥𝑛1inf𝑥𝑥1 as 𝑛, then {𝑥𝑛} is bounded. There exists a convergent subsequence {𝑥𝑛𝑘} of {𝑥𝑛}. Set 𝑥=lim𝑘𝑥𝑛𝑘, then 𝑥 since is closed. On the other hand, using lower semicontinuity of the norm, we have 𝑥lim𝑘𝑥𝑛𝑘=inf𝑥𝑥1,(3.18) and this implies that 𝑥1.
However, 1 may contain more than one elements, in general (see Example 3.1, 1={(𝑥,𝑦)𝑥+𝑦=1,𝑥,𝑦0}).

Theorem 3.6. Let 𝛼>0 and 𝑥𝛼𝑆𝛼. Then 𝜔(𝑥𝛼)1, where 𝜔(𝑥𝛼)={𝑥{𝑥𝛼𝑘}{𝑥𝛼},𝑥𝛼𝑘𝑥𝑤𝑒𝑎𝑘𝑙𝑦}.

Proof. Taking ̃𝑥1 arbitrarily, for any 𝛼(0,), we always have 12𝐼𝑃𝑄𝐴𝑥𝛼2𝑥+𝛼𝛼112𝐼𝑃𝑄𝐴̃𝑥2+𝛼̃𝑥1.(3.19) Since ̃𝑥 is a solution of SFP (1.1), (𝐼𝑃𝑄)𝐴̃𝑥=0. This implies that 12(𝐼𝑃𝑄)𝐴𝑥𝛼2𝑥+𝛼𝛼1𝛼̃𝑥1,(3.20) then, 𝑥𝛼1̃𝑥1;(3.21) thus {𝑥𝛼} is bounded.
Take 𝜔𝜔(𝑥𝛼) arbitrarily, then there exists a sequence {𝛼𝑛} such that 𝛼𝑛0 and 𝑥𝛼𝑛𝜔 as 𝑛. Put 𝑥𝛼𝑛=𝑥𝑛. By Proposition 3.2, we deduce that there exists some 𝜉𝑛𝜕(𝑥𝑛1) such that 𝐴𝐼𝑃𝑄𝐴𝑥𝑛+𝛼𝑛𝜉𝑛,̃𝑥𝑥𝑛0.(3.22) This implies that 𝐼𝑃𝑄𝐴𝑥𝑛,𝐴̃𝑥𝑥𝑛𝛼𝑛𝜉𝑛,𝑥𝑛̃𝑥.(3.23) Since 𝐴̃𝑥𝑄, the characterizing inequality (2.2) gives 𝐼𝑃𝑄𝐴𝑥𝑛,𝐴̃𝑥𝑃𝑄𝐴𝑥𝑛0,(3.24) then, 𝐼𝑃𝑄𝐴𝑥𝑛2𝐼𝑃𝑄𝐴𝑥𝑛𝑥,𝐴𝑛.̃𝑥(3.25) Combining (3.23) and (3.25), we have 𝐼𝑃𝑄𝐴𝑥𝑛2𝛼𝑛𝜉𝑛,̃𝑥𝑥𝑛𝛼𝑛𝜉𝑛̃𝑥𝑥𝑛12𝛼𝑛̃𝑥1.(3.26) Consequently, we get lim𝑛𝐼𝑃𝑄𝐴𝑥𝑛=0.(3.27) Furthermore, noting the fact that 𝑥𝑛𝜔 and 𝐼𝑃𝑄 and 𝐴 are all continuous operators, we have (𝐼𝑃𝑄)𝐴𝜔=0; that is, 𝐴𝜔𝑄; thus, 𝜔. Since ̃𝑥 is a minimum-norm solution of SFP (1.1) in the sense of 1-norm, using (3.21) again, we get 𝜔1liminf𝑛𝑥𝑛1̃𝑥1=min𝑥1.𝑥(3.28) Thus we can assert that 𝜔1 and this completes the proof.

Corollary 3.7. If 1 contains only one element ̃𝑥, then 𝑥𝛼̃𝑥, (𝛼0).

Remark 3.8. It is worth noting that the minimum-norm solution of SFP (1.1) in the sense of norm is very different from the minimum-norm solution of SFP (1.1) in the sense of 1-norm. In fact, 𝑥 may not belong to 1! The following simple example shows this fact.

Example 3.9. Let 𝐶={(𝑥,𝑦)𝑥+2𝑦2,𝑥0,𝑦0},  𝑄={(𝑥,𝑦)𝑥+𝑦=1,𝑥0,𝑦0}, and 1𝐴=2001.(3.29) It is not hard to see that 𝐴22 is a bounded linear operator and 𝐴(𝑥,𝑦)𝑇=((1/2)𝑥,𝑦)𝑇, for  all (𝑥,𝑦)𝐶. Obviously, ={(𝑥,𝑦)𝑥+2𝑦=2,𝑥0,𝑦0}, 𝑥=(2/5,4/5), but 1={(0,1)}. Hence, 𝑥1.

Acknowledgments

This work was supported by the Fundamental Research Funds for the Central Universities (ZXH2012K001) and in part by the Foundation of Tianjin Key Lab for Advanced Signal Processing. W. Zhu was also supported by the Postgraduate Science and Technology Innovation Funds (YJSCX12-22).

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