Journal of Applied Mathematics

Journal of Applied Mathematics / 2012 / Article
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Numerical and Analytical Methods for Variational Inequalities and Related Problems with Applications

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Research Article | Open Access

Volume 2012 |Article ID 683890 | 10 pages | https://doi.org/10.1155/2012/683890

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

Academic Editor: Hong-Kun Xu
Received21 Mar 2012
Accepted06 Jun 2012
Published17 Jul 2012

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: 𝑥∈𝐶,𝐴𝑥∈𝑄,(1.1) where 𝐴∶𝐻1→𝐻2 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 [2–13] 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 ℝ𝑁 and ℝ𝑀, 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 [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 𝑓∈𝑋∗ at 𝑥∈𝑋. 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=argmin𝑦∈𝐶‖𝑥−𝑦‖,𝑥∈𝐻.(2.1) It is well known that 𝑃𝐶𝑥 is characterized by the inequality ⟨𝑥−𝑃𝐶𝑥,𝑦−𝑃𝐶𝑥⟩≤0,∀𝑦∈𝐶.(2.2)

Definition 2.1. Let 𝜑∶𝑋→ℝ∪{+∞} be a convex functional, 𝑥0∈dom(𝜑)={𝑥∈𝑋∶𝜑(𝑥)<+∞}. Set 𝑥𝜕𝜑0=𝜉∈𝑋∗𝑥∶𝜑(𝑥)≥𝜑0+⟨𝑥−𝑥0,𝜉⟩,∀𝑥∈𝑋.(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‖)=∗∈𝑋∗∶‖𝑥∗‖=1,⟨𝑥,𝑥∗𝑥⟩=‖𝑥‖,𝑥≠0,∗∈𝑋∗∶‖𝑥∗‖≤1,𝑥=0,(2.4) where 𝜕(‖𝑥‖) denotes the subdifferential of the functional ‖𝑥‖ at 𝑥∈𝑋.

Proof. The process of the proof will be divided into two parts.
Case  1. In the case of 𝑥=0, for any 𝑥∗∈𝑋∗ such that ‖𝑥∗‖≤1 and any 𝑦∈𝑋, there holds the inequality ‖𝑦‖≥⟨𝑦,𝑥∗⟩=‖𝑥‖+⟨𝑦−𝑥,𝑥∗⟩,(2.5) so we have 𝑥∗∈𝜕(‖𝑥‖), and thus, 𝑥∗∈𝑋∗∶‖𝑥∗‖≤1⊂𝜕(‖𝑥‖).(2.6) Conversely, for any 𝑥∗∈𝜕(‖𝑥‖), we have from the definition of subdifferential that ‖𝑦‖≥‖𝑥‖+⟨𝑦−𝑥,𝑥∗⟩=⟨𝑦,𝑥∗⟩,∀𝑦∈𝑋,‖𝑦‖=‖−𝑦‖≥⟨−𝑦,𝑥∗⟩=−⟨𝑦,𝑥∗⟩.(2.7) Consequently, ||⟨𝑦,𝑥∗⟩||≤‖𝑦‖,∀𝑦∈𝑋,(2.8) and this implies that ‖𝑥∗‖≤1. Thus, we have verified that 𝜕𝑥(‖𝑥‖)⊂∗∈𝑋∗∶‖𝑥∗‖≤1.(2.9) Combining (2.6) and (2.9), we immediately obtain 𝜕𝑥(‖𝑥‖)=∗∈𝑋∗∶‖𝑥∗‖≤1.(2.10)
Case  2. If 𝑥≠0, for any 𝑥∗∈{𝑥∗∈𝑋∗∶‖𝑥∗‖=1,⟨𝑥,𝑥∗⟩=‖𝑥‖}, we obviously have ⟨𝑦−𝑥,𝑥∗⟩=⟨𝑦,𝑥∗⟩−‖𝑥‖≤‖𝑦‖−‖𝑥‖,∀𝑦∈𝑋,(2.11) which means that 𝑥∗∈𝜕(‖𝑥‖), and thus, 𝑥∗∈𝑋∗∶‖𝑥∗‖=1,⟨𝑥,𝑥∗⟩=‖𝑥‖⊂𝜕(‖𝑥‖).(2.12) Conversely, if 𝑥∗∈𝜕(‖𝑥‖), we have ⟨−𝑥,𝑥∗⟩≤0−‖𝑥‖=−‖𝑥‖,⟨𝑥,𝑥∗⟩≤2‖𝑥‖−‖𝑥‖=‖𝑥‖;(2.13) hence, ⟨𝑥,𝑥∗⟩=‖𝑥‖.(2.14) On the other hand, using (2.14), we get ‖𝑦‖≥‖𝑥‖+⟨𝑦−𝑥,𝑥∗⟩=‖𝑥‖+⟨𝑦,𝑥∗⟩−⟨𝑥,𝑥∗⟩=⟨𝑦,𝑥∗⟩,∀𝑦∈𝑋,(2.15) and consequently, ‖𝑦‖=‖−𝑦‖≥‖𝑥‖+⟨−𝑦−𝑥,𝑥∗⟩=‖𝑥‖−⟨𝑦,𝑥∗⟩−⟨𝑥,𝑥∗⟩=−⟨𝑦,𝑥∗⟩;(2.16) that is, −‖𝑦‖≤⟨𝑦,𝑥∗⟩.(2.17) Equation (2.17) together with (2.15) implies that ||⟨𝑦,𝑥∗⟩||≤‖𝑦‖,∀𝑦∈𝑋;(2.18) hence, ‖𝑥∗‖≤1. Note that (2.14) implies that ‖𝑥∗‖≥⟨𝑥,𝑥∗⟩/‖𝑥‖=1; we assert that ‖𝑥∗‖=1.(2.19) Thus we have from (2.14) and (2.19) that 𝑥∗∈𝑋∗∶‖𝑥∗‖=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)∈ℝ𝑙, we have â€–ğ‘¥â€–âˆž=max1≤𝑗≤𝑙||𝑥𝑗||,‖𝑥‖1=𝑙𝑗=1||𝑥𝑗||.(2.21)

Corollary 2.3. In l-dimensional Euclidean space ℝ𝑙, there holds the following result: 𝜕‖𝑥‖1=î‚»î€½ğœ‰âˆˆâ„ğ‘™âˆ¶â€–ğœ‰â€–âˆž=1,⟨𝑥,𝜉⟩=‖𝑥‖1,𝑥≠0,ğœ‰âˆˆâ„ğ‘™âˆ¶â€–ğœ‰â€–âˆžî€¾=⎧⎪⎨⎪⎩≤1,𝑥=0,𝜉∈ℝ𝑙∶𝜉𝑖=𝑥𝑖||𝑥𝑖||,if𝑥𝑖≠0;𝜉𝑖∈[]−1,1,if𝑥𝑖=0,𝑥≠0,ğœ‰âˆˆâ„ğ‘™âˆ¶â€–ğœ‰â€–âˆžî€¾â‰¤1,𝑥=0.(2.22) Let 𝐻 be a Hilbert space and 𝑓∶𝐻→ℝ a functional. Recall that (i)𝑓is convex if 𝑓(𝜆𝑥+(1−𝜆)𝑦)≤𝜆𝑓(𝑥)+(1−𝜆)𝑓(𝑦), for all 0<𝜆<1, for all 𝑥,𝑦∈𝐻;(ii)𝑓 is strictly convex if 𝑓(𝜆𝑥+(1−𝜆)𝑦)<𝜆𝑓(𝑥)+(1−𝜆)𝑓(𝑦), for all 0<𝜆<1, for all 𝑥,𝑦∈𝐻 with 𝑥≠𝑦;(iii)𝑓 is coercive if 𝑓(𝑥)→∞ whenever â€–ğ‘¥â€–â†’âˆž. 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 𝑓∶𝐻→ℝ be a convex and subdifferentiable functional. Then 𝑥∈𝐶 is a solution of the problem min𝑥∈𝐶𝑓(𝑥)(2.23) if and only if there exists some 𝜉∈𝜕𝑓(𝑥) satisfying the following optimality condition: ⟨𝜉,𝑣−𝑥⟩≥0,∀𝑣∈𝐶.(2.24)

3. Main Results

It is well known that SFP (1.1) is equivalent to the minimization problem min𝑥∈𝐶‖‖𝐼−𝑃𝑄‖‖𝐴𝑥2.(3.1) Using the idea of Tikhonov's regularization method, Wang and Xu [14] studied the minimization problem in Hilbert spaces: min𝑥∈𝐶‖‖𝐼−𝑃𝑄‖‖𝐴𝑥2+𝛼‖𝑥‖2,(3.2) where 𝛼>0 is the regularization parameter.

In what follows, 𝐻1 and 𝐻2 in SFP (1.1) are restricted to ℝ𝑁 and ℝ𝑀, respectively, and ‖⋅‖ will stand for the usual 2-norm of any Euclidean space ℝ𝑙; that is, for any 𝑥=(𝑥1,𝑥2,…,𝑥𝑙)∈ℝ𝑙, ‖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: min𝑥∈𝐶12‖‖𝐼−𝑃𝑄‖‖𝐴𝑥2+𝛼‖𝑥‖1,(3.4) where 𝛼>0 is the regularization parameter. Let 𝑓𝛼1(𝑥)=2‖‖𝐼−𝑃𝑄‖‖𝐴𝑥2+𝛼‖𝑥‖1.(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 𝐴∶ℝ2→ℝ2 is a bounded linear operator. Obviously, 𝑆𝛼={(𝑥,𝑦)∶𝑥+𝑦=1,𝑥≥0,𝑦≥0} and it contains more than one element.

Proposition 3.2. For any 𝛼>0, 𝑥𝛼∈𝑆𝛼 if and only if there exists some 𝜉∈𝜕(‖𝑥‖1) satisfying the following inequality: 𝐴∗𝐼−𝑃𝑄𝐴𝑥𝛼+𝛼𝜉,𝑣−𝑥𝛼≥0,∀𝑣∈𝐶.(3.7)

Proof. Let 1𝑓(𝑥)=2‖‖𝐼−𝑃𝑄‖‖𝐴𝑥2,(3.8) then 𝑓𝛼(𝑥)=𝑓(𝑥)+𝛼‖𝑥‖1.(3.9) Since 𝑓 is convex and differentiable with gradient ∇𝑓(𝑥)=𝐴∗𝐼−𝑃𝑄𝐴𝑥,(3.10)𝑓𝛼 is convex, coercive, and subdifferentiable with the subdifferential 𝜕𝑓𝛼(𝑥)=𝜕𝑓(𝑥)+𝛼𝜕‖𝑥‖1;(3.11) that is, 𝜕𝑓𝛼(𝑥)=𝐴∗𝐼−𝑃𝑄𝐴𝑥+𝛼𝜕‖𝑥‖1.(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)‖𝑥𝛼‖1 is decreasing for 𝛼∈(0,∞);(ii)‖(𝐼−𝑃𝑄)𝐴𝑥𝛼‖ is increasing for 𝛼∈(0,∞).

Proof. Let 𝛼>𝛽>0, for any 𝑥𝛼∈𝑆𝛼, 𝑥𝛽∈𝑆𝛽. We immediately obtain 12‖‖𝐼−𝑃𝑄𝐴𝑥𝛼‖‖2‖‖𝑥+𝛼𝛼‖‖1≤12‖‖𝐼−𝑃𝑄𝐴𝑥𝛽‖‖2‖‖𝑥+𝛼𝛽‖‖1,1(3.13)2‖‖𝐼−𝑃𝑄𝐴𝑥𝛽‖‖2‖‖𝑥+𝛽𝛽‖‖1≤12‖‖𝐼−𝑃𝑄𝐴𝑥𝛼‖‖2‖‖𝑥+𝛽𝛼‖‖1.(3.14) Adding up (3.13) and (3.14) yields 𝛼‖‖𝑥𝛼‖‖1‖‖𝑥+𝛽𝛽‖‖1‖‖𝑥≤𝛼𝛽‖‖1‖‖𝑥+𝛽𝛼‖‖1,(3.15) which implies ‖𝑥𝛼‖1≤‖𝑥𝛽‖1. Hence (i) holds.
Using (3.14) again, we have 12‖‖𝐼−𝑃𝑄𝐴𝑥𝛽‖‖2≤12‖‖𝐼−𝑃𝑄𝐴𝑥𝛼‖‖2‖‖𝑥+𝛽𝛼‖‖1−‖‖𝑥𝛽‖‖1,(3.16) which together with (i) implies ‖‖𝐼−𝑃𝑄𝐴𝑥𝛽‖‖2≤‖‖𝐼−𝑃𝑄𝐴𝑥𝛼‖‖2,(3.17) and hence (ii) holds.

Let ℱ=𝐶∩𝐴−1(𝑄), where 𝐴−1(𝑄)={𝑥∈ℝ𝑁∶𝐴𝑥∈𝑄}. In what follows, we assume that ℱ≠∅; that is, the solution set of SFP (1.1) is nonempty. The fact that ℱ 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†∈ℱ is said to be the minimum-norm solution of SFP (1.1) in the sense of norm ‖⋅‖ if ‖𝑥†‖=inf𝑥∈ℱ‖𝑥‖. In other words, 𝑥† is the projection of the origin onto the solution set ℱ 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∈ℱ is said to be a minimum-norm solution of SFP (1.1) in the sense 𝑜𝑓 1-norm if ‖̃𝑥‖1=inf𝑥∈ℱ‖𝑥‖1. We use ℱ1 to stand for all minimum-norm solutions of SFP (1.1) in the sense of 1-norm and ℱ1 is called the minimum-norm solution set of SFP (1.1) in the sense of 1-norm.
Obviously, ℱ1 is a closed convex subset of ℱ. Moreover, it is easy to see that ℱ1≠∅. Indeed, taking a sequence {𝑥𝑛}⊂ℱ such that ‖𝑥𝑛‖1→inf𝑥∈ℱ‖𝑥‖1 as ğ‘›â†’âˆž, then {𝑥𝑛} is bounded. There exists a convergent subsequence {𝑥𝑛𝑘} of {𝑥𝑛}. Set −𝑥=limğ‘˜â†’âˆžğ‘¥ğ‘›ğ‘˜, then −𝑥∈ℱ since ℱ is closed. On the other hand, using lower semicontinuity of the norm, we have ‖‖−𝑥‖‖≤limğ‘˜â†’âˆžâ€–â€–ğ‘¥ğ‘›ğ‘˜â€–â€–=inf𝑥∈ℱ‖𝑥‖1,(3.18) and this implies that −𝑥∈ℱ1.
However, ℱ1 may contain more than one elements, in general (see Example 3.1, ℱ1={(𝑥,𝑦)∶𝑥+𝑦=1,𝑥,𝑦≥0}).

Theorem 3.6. Let 𝛼>0 and 𝑥𝛼∈𝑆𝛼. Then 𝜔(𝑥𝛼)⊂ℱ1, where 𝜔(𝑥𝛼)={𝑥∶∃{𝑥𝛼𝑘}⊂{𝑥𝛼},ğ‘¥ğ›¼ğ‘˜â†’ğ‘¥ğ‘¤ğ‘’ğ‘Žğ‘˜ğ‘™ğ‘¦}.

Proof. Taking ̃𝑥∈ℱ1 arbitrarily, for any 𝛼∈(0,∞), we always have 12‖‖𝐼−𝑃𝑄𝐴𝑥𝛼‖‖2‖‖𝑥+𝛼𝛼‖‖1≤12‖‖𝐼−𝑃𝑄‖‖𝐴̃𝑥2+𝛼‖̃𝑥‖1.(3.19) Since ̃𝑥 is a solution of SFP (1.1), ‖(𝐼−𝑃𝑄)𝐴̃𝑥‖=0. This implies that 12‖‖(𝐼−𝑃𝑄)𝐴𝑥𝛼‖‖2‖‖𝑥+𝛼𝛼‖‖1≤𝛼‖̃𝑥‖1,(3.20) then, ‖‖𝑥𝛼‖‖1≤‖̃𝑥‖1;(3.21) thus {𝑥𝛼} is bounded.
Take 𝜔∈𝜔(𝑥𝛼) arbitrarily, then there exists a sequence {𝛼𝑛} such that 𝛼𝑛→0 and 𝑥𝛼𝑛→𝜔 as ğ‘›â†’âˆž. Put 𝑥𝛼𝑛=𝑥𝑛. By Proposition 3.2, we deduce that there exists some 𝜉𝑛∈𝜕(‖𝑥𝑛‖1) such that 𝐴∗𝐼−𝑃𝑄𝐴𝑥𝑛+𝛼𝑛𝜉𝑛,̃𝑥−𝑥𝑛≥0.(3.22) This implies that 𝐼−𝑃𝑄𝐴𝑥𝑛,𝐴̃𝑥−𝑥𝑛≥𝛼𝑛⟨𝜉𝑛,𝑥𝑛−̃𝑥⟩.(3.23) Since 𝐴̃𝑥∈𝑄, the characterizing inequality (2.2) gives 𝐼−𝑃𝑄𝐴𝑥𝑛,𝐴̃𝑥−𝑃𝑄𝐴𝑥𝑛≤0,(3.24) then, ‖‖𝐼−𝑃𝑄𝐴𝑥𝑛‖‖2≤𝐼−𝑃𝑄𝐴𝑥𝑛𝑥,𝐴𝑛.−̃𝑥(3.25) Combining (3.23) and (3.25), we have ‖‖𝐼−𝑃𝑄𝐴𝑥𝑛‖‖2≤𝛼𝑛⟨𝜉𝑛,Ìƒğ‘¥âˆ’ğ‘¥ğ‘›âŸ©â‰¤ğ›¼ğ‘›â€–â€–ğœ‰ğ‘›â€–â€–âˆžâ€–â€–Ìƒğ‘¥âˆ’ğ‘¥ğ‘›â€–â€–1≤2𝛼𝑛‖̃𝑥‖1.(3.26) Consequently, we get limğ‘›â†’âˆžâ€–â€–î€·ğ¼âˆ’ğ‘ƒğ‘„î€¸ğ´ğ‘¥ğ‘›â€–â€–=0.(3.27) Furthermore, noting the fact that 𝑥𝑛→𝜔 and 𝐼−𝑃𝑄 and 𝐴 are all continuous operators, we have (𝐼−𝑃𝑄)𝐴𝜔=0; that is, 𝐴𝜔∈𝑄; thus, 𝜔∈ℱ. Since ̃𝑥 is a minimum-norm solution of SFP (1.1) in the sense of 1-norm, using (3.21) again, we get ‖𝜔‖1≤liminfğ‘›â†’âˆžâ€–â€–ğ‘¥ğ‘›â€–â€–1≤‖̃𝑥‖1=min‖𝑥‖1.∶𝑥∈ℱ(3.28) Thus we can assert that 𝜔∈ℱ1 and this completes the proof.

Corollary 3.7. If ℱ1 contains only one element ̃𝑥, then 𝑥𝛼→̃𝑥, (𝛼→0).

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 ℱ1! The following simple example shows this fact.

Example 3.9. Let 𝐶={(𝑥,𝑦)∶𝑥+2𝑦≥2,𝑥≥0,𝑦≥0},  𝑄={(𝑥,𝑦)∶𝑥+𝑦=1,𝑥≥0,𝑦≥0}, and ⎛⎜⎜⎝1𝐴=20⎞⎟⎟⎠01.(3.29) It is not hard to see that 𝐴∶ℝ2→ℝ2 is a bounded linear operator and 𝐴(𝑥,𝑦)𝑇=((1/2)𝑥,𝑦)𝑇, for  all (𝑥,𝑦)∈𝐶. Obviously, ℱ={(𝑥,𝑦)∶𝑥+2𝑦=2,𝑥≥0,𝑦≥0}, 𝑥†=(2/5,4/5), but ℱ1={(0,1)}. Hence, 𝑥†∈ℱ⧵ℱ1.

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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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.


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