Numerical and Analytical Methods for Variational Inequalities and Related Problems with ApplicationsView this Special Issue
A Note on Approximating Curve with 1-Norm Regularization Method for the Split Feasibility Problem
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.
Let and be nonempty closed convex subsets of real Hilbert spaces and , respectively. The problem under consideration in this paper is formulated as finding a point satisfying the property: where is a bounded linear operator. Problem (1.1), referred to by Censor and Elfving  as the split feasibility problem (SFP), attracts many authors’ attention due to its application in signal processing . Various algorithms have been invented to solve it (see [2–13] and references therein).
Using the idea of Tikhonov's regularization, Wang and Xu  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  so that strong convergence is guaranteed and its limit is the minimum-norm solution of SFP (1.1).
In the practical application, and are often and , respectively. Moreover, scientists and engineers are more willing to use 1-norm regularization method in the calculation process (see, e.g., [15–18]). Inspired by the above results of Wang and Xu , 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.
Let be a normed linear space with norm , and let be the dual space of . We use the notation to denote the value of at . In particular, if is a Hilbert space, we will denote it by , and and are the inner product and its induced norm, respectively.
We recall some definitions and facts that are needed in our study.
Let denote the projection from onto a nonempty closed convex subset of ; that is, It is well known that is characterized by the inequality
Definition 2.1. Let be a convex functional, . Set If , is said to be subdifferentiable at and is called the subdifferential of at . For any , we say is a subgradient of at .
Lemma 2.2. There holds the following property: where denotes the subdifferential of the functional at .
Proof. The process of the proof will be divided into two parts.
Case 1. In the case of , for any such that and any , there holds the inequality so we have , and thus, Conversely, for any , we have from the definition of subdifferential that Consequently, and this implies that . Thus, we have verified that Combining (2.6) and (2.9), we immediately obtain
Case 2. If , for any , we obviously have which means that , and thus, Conversely, if , we have hence, On the other hand, using (2.14), we get and consequently, that is, Equation (2.17) together with (2.15) implies that hence, . Note that (2.14) implies that ; we assert that Thus we have from (2.14) and (2.19) that The proof is finished by combining (2.12) and (2.20).
and will stand for -norm and -norm of any Euclidean space; respectively, that is, for any , we have
Corollary 2.3. In l-dimensional Euclidean space , there holds the following result: Let be a Hilbert space and a functional. Recall that (i)is convex if , for all , for all ;(ii) is strictly convex if , for all , for all with ;(iii) is coercive if whenever . See  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 ). Let be a Hilbert space and a nonempty closed convex subset of . Let be a convex and subdifferentiable functional. Then is a solution of the problem if and only if there exists some satisfying the following optimality condition:
3. Main Results
It is well known that SFP (1.1) is equivalent to the minimization problem Using the idea of Tikhonov's regularization method, Wang and Xu  studied the minimization problem in Hilbert spaces: where is the regularization parameter.
In what follows, and in SFP (1.1) are restricted to and , respectively, and will stand for the usual 2-norm of any Euclidean space ; that is, for any , 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: where is the regularization parameter. Let It is easy to see that 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 is not necessarily strictly convex. Denote by the solution set of problem (3.4); thus we can assert that is a nonempty closed convex set but may contain more than one element. The following simple example illustrates this fact.
Example 3.1. Let , and Then is a bounded linear operator. Obviously, and it contains more than one element.
Proposition 3.2. For any , if and only if there exists some satisfying the following inequality:
Theorem 3.3. Denote by an arbitrary element of , then the following assertions hold:(i) is decreasing for ;(ii) is increasing for .
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) in the sense of norm (induced by the inner product).
Definition 3.4 (see ). An element is said to be the minimum-norm solution of SFP (1.1) in the sense of norm if . In other words, 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 .
We can also give the concept of minimum-norm solution of SFP (1.1) in other senses.
Definition 3.5. An element is said to be a minimum-norm solution of SFP (1.1) in the sense 1-norm if . We use to stand for all minimum-norm solutions of SFP (1.1) in the sense of 1-norm and is called the minimum-norm solution set of SFP (1.1) in the sense of 1-norm.
Obviously, is a closed convex subset of . Moreover, it is easy to see that . Indeed, taking a sequence such that as , then is bounded. There exists a convergent subsequence of . Set , then since is closed. On the other hand, using lower semicontinuity of the norm, we have and this implies that .
However, may contain more than one elements, in general (see Example 3.1, .
Theorem 3.6. Let and . Then , where .
Proof. Taking arbitrarily, for any , we always have
Since is a solution of SFP (1.1), . This implies that
thus is bounded.
Take arbitrarily, then there exists a sequence such that and as . Put . By Proposition 3.2, we deduce that there exists some such that This implies that Since , the characterizing inequality (2.2) gives then, Combining (3.23) and (3.25), we have Consequently, we get Furthermore, noting the fact that and and are all continuous operators, we have ; that is, ; thus, . Since is a minimum-norm solution of SFP (1.1) in the sense of 1-norm, using (3.21) again, we get Thus we can assert that and this completes the proof.
Corollary 3.7. If contains only one element , then , ().
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, may not belong to ! The following simple example shows this fact.
Example 3.9. Let , , and It is not hard to see that is a bounded linear operator and , for all . Obviously, , , but . Hence, .
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).
H. K. Xu, “A variable Krasnosel'skii-Mann algorithm and the multiple-set split feasibility problem,” Inverse Problems, vol. 22, no. 6, pp. 2021–2034, 2006.View at: Google Scholar
X. Yu, N. Shahzad, and Y. Yao, “Implicit and explicit algorithms for solving the split feasibility problem,” Optimization Letters. In press.View at: Google Scholar
X. Nan, N. Wang, P. Gong, C. Zhang, Y. Chen, and D. Wilkins, “Gene selection using 1-norm regularization for multi-class microarray data,” in Proceedings of the IEEE International Conference on Bioinformatics and Biomedicine (BIBM '10), pp. 520–524, December 2010.View at: Google Scholar
J. P. Aubin, Optima and Equilibria: An Introduction to Nonliear Analysis, vol. 140 of Graduate Texts in Mathematics, Springer, Berlin, Germany, 1993.
H. W. Engl, M. Hanke, and A. Neubauer, Regularization of Inverse Problems, vol. 375 of Mathematics and Its Applications, Kluwer Academic, Dordrecht, The Netherlands, 1996.