Exact Penalization and Necessary Optimality Conditions for Multiobjective Optimization Problems with Equilibrium Constraints

Zhu, Shengkun; Li, Shengjie

doi:https://doi.org/10.1155/2014/630547

Abstract and Applied Analysis

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Research Article | Open Access

Volume 2014 | Article ID 630547 | https://doi.org/10.1155/2014/630547

Exact Penalization and Necessary Optimality Conditions for Multiobjective Optimization Problems with Equilibrium Constraints

Shengkun Zhu^1,2and Shengjie Li²

Academic Editor: Geraldo Botelho

Received01 Dec 2013

Accepted16 Mar 2014

Published15 May 2014

Abstract

A calmness condition for a general multiobjective optimization problem with equilibrium constraints is proposed. Some exact penalization properties for two classes of multiobjective penalty problems are established and shown to be equivalent to the calmness condition. Subsequently, a Mordukhovich stationary necessary optimality condition based on the exact penalization results is obtained. Moreover, some applications to a multiobjective optimization problem with complementarity constraints and a multiobjective optimization problem with weak vector variational inequality constraints are given.

1. Introduction

In this paper, we consider a general multiobjective optimization problem with equilibrium constraints as follows:

where , , , , , , and are vector-valued maps, is a set-valued map, and is a nonempty and closed subset of . As usual, we denote by the interior of and by the graph of . Moreover, denotes the nonnegative quadrant in , and and denote, respectively, the origins of and . Throughout this paper, we assume that is locally Lipschitz, , , and are continuously Fréchet differentiable, is closed (i.e., is closed in ), and the feasible set of (MOPEC) is nonempty. Obviously, is a closed subset of . Recall that a point is said to be an efficient (resp. weak efficient) solution for (MOPEC) if and only if A point is said to be a local efficient (resp. local weak efficient) solution for (MOPEC) if and only if there exists a neighborhood of such that

During the past few decades, there have been a lot of papers devoted to study the scalar optimization problem (i.e., the case ) with equilibrium constraints, which plays an important role in engineering design, economic equilibria, operations research, and so on. It is well recognized that the scalar optimization problem with equilibrium constraints covers various classes of optimization-related problems and models arisen in practical applications, such as mathematical programs with geometric constraints, mathematical programs with complementarity constraints, and mathematical programs with variational inequality constraints. For more details, we refer to [1–4]. It is worth noting that when is a general closed set-valued map, even if is a fixed closed subset of for all , the general constraint system (1) fails to satisfy the standard linear independence constraint qualification and Mangasarian-Fromovitz constraint qualification at any feasible point [5]. Thus, it is a hard work to establish Karush-Kuhn-Tucker (in short, KKT) necessary optimality conditions for (MOPEC). Recently, by virtue of advanced tools of variational analysis and various coderivatives for set-valued maps developed in [6–8] and references therein, some necessary optimality conditions including the strong, Mordukhovich, Clarke, and Bouligand stationary conditions are obtained by using different reformulations under some generalized constraint qualifications. Simultaneously, Ye and Zhu [3] claimed that the Mordukhovich stationary (in short, M-stationary) condition is the strongest stationary condition except the strong stationary condition which is equivalent to the classical KKT condition, and proposed some new constraint qualifications for M-stationary conditions to hold.

It is well known that the penalization method is a very important and effective tool for dealing with optimization theories and numerical algorithms of constrained extremum problems. In scalar optimization with equality and inequality constraints, the classical exact penalty function with order 1 was extensively used to investigate optimality conditions and convergence analysis; see [6, 9, 10] and references therein. Clarke [6] derived some Fritz-John necessary optimality conditions for a constrained mathematical programming problem on a Banach space by virtue of exact penalty functions with order 1. Moreover, Burke [9] showed that the existence of an exact penalization function is equivalent to a calmness condition involving with the objective function and the equality and inequality constrained system. Subsequently, Flegel and Kanzow [4] demonstrated that the corresponding relationships still held in a generalized bilevel programming problem and a mathematical programming problem with complementarity constraints, respectively. Simultaneously, they obtained some KKT necessary optimality conditions by using exact penalty formulations and nonsmooth analysis. Recently, the classical penalization theory has been widely generalized by various kinds of Lagrangian functions, especially the augmented Lagrangian function, introduced by Rockafellar and Wets [7], and the nonlinear Lagrangian function, proposed by Rubinov et al. [11]. It has also been proved that the exactness of these types of penalty functions is equivalent to some generalized calmness conditions; see more details in [11, 12].

However, to the best of our knowledge, there are only a few papers devoted to study the penalty method for constrained multiobjective optimization problems, especially, for (MOPEC). Huang and Yang [13] first introduced a vector-valued nonlinear Lagrangian and penalty functions for multiobjective optimization problems with equality and inequality constraints and obtained some relationships between the exact penalization property and a generalized calmness-type condition. Moreover, Mordukhovich [8] and Bao et al. [14] investigated some more general optimization problems with equilibrium constraints by methods of modern variational analysis. It is worth noting that the standard Mangasarian-Fromovitz constraint qualification and error bound condition for a nonlinear programming problem with equality and inequality constraints implies the calmness condition; see [6, 15] for details. Taking into account this fact, it is necessary to further investigate the calmness condition and the penalty method for constrained multiobjective optimization problems.

The main motivation of this work is that there has been no study on the penalization method and M-stationary condition for (MOPEC) by using an appropriate calmness condition associated with the objective function and the constraint system. Although there have been many papers dealing with constrained multiobjective optimization problems, for example, [3, 8] and references therein, the KKT necessary optimality conditions are obtained under some generalized qualification conditions only involved with the constraint system. Inspired by the ideas reported in [3, 4, 6, 8, 13], we introduce a so-called (MOPEC-) calmness condition with order at a local efficient (weak efficient) solution associated with the objective function and the constraint system for (MOPEC) and show that the (MOPEC-) calmness condition can be implied by an error bound condition of the constraint system. Moreover, we establish some equivalent relationships between the exact penalization property with order and the (MOPEC-) calmness condition. Simultaneously, we apply a nonlinear scalar technical to obtain a KKT necessary optimality condition for (MOPEC) by using Mordukhovich generalized differentiation and the (MOPEC-) calmness condition with order 1.

The organization of this paper is as follows. In Section 2, we recall some basic concepts and tools generally used in variational analysis and set-valued analysis. In Section 3, we introduce a (MOPEC-) calmness condition for (MOPEC) and establish some relationships between the exact penalization property and the (MOPEC-) calmness condition. Moreover, we obtain a KKT necessary optimality condition under the (MOPEC-) calmness condition with order 1. In Section 4, we apply the obtained results to a multiobjective optimization problem with complementarity constraints and a multiobjective optimization problem with weak vector variational inequality constraints, respectively.

2. Notations and Preliminaries

Throughout this paper, all vectors are viewed as column vectors. Since all the norms on finite dimensional spaces are equivalent, we take specially the sum norm on and the product space for simplicity; that is, for all , , and, for all , . As usual, we denote by the transposition of and by the inner product of vectors and , respectively. For a given map and a vector , the function is defined by for all . In general, we denote by the closed unit ball in and by the open ball with center at and radius for any .

The main tools for our study in this paper are the Mordukhovich generalized differentiation notions which are generally used in variational analysis and set-valued analysis; see more details in [6–8, 16] and references therein. Recall that is said to be Fréchet differentiable at if and only if there exists a matrix such that Obviously, is uniquely determined by . As usual, is called the Fréchet derivative of at and denoted by . If is Fréchet differentiable at every , then is said to be Fréchet differentiable on . is said to be continuously Fréchet differentiable at if and only if the map is continuous at . Specially, we denote by the adjoint operator of ; that is, for all and . Moreover, is said to be strictly differentiable at if and only if Obviously, if is continuously Fréchet differentiable at , then is strictly differentiable at .

For a nonempty subset , the indicator function is defined by , and , and the distance function is defined by for all , respectively. Given a point , recall that the Fréchet normal cone of at , which is a convex, closed subset of and consisted of all the Fréchet normals, has the form where means and . The Mordukhovich (or basic, limiting) normal cone of at is Specially, if is convex, then we have Let be an extended real-valued function and let , where denotes the domain of . The Fréchet subdifferential of at is defined in the geometric form by , or, equivalently, is defined in the analytical form by The Mordukhovich (or basic, limiting) subdifferential and singular subdifferential of at are defined, respectively, by and . Clearly, we have and where means and . Specially, for any , it follows that and . Furthermore, if is a convex function, then we have Recall that the Fréchet coderivative and the Mordukhovich (or basic, limiting) coderivative of the set-valued map at are the set-valued maps from to defined, respectively, by

Next, we collect some useful and important propositions and definitions for this paper.

Proposition 1 (see [8]). For every nonempty subset and every , we have(i) and .
In addition, if is closed, then we get(ii) and .

The following necessary optimality condition, called generalized Fermat rule, for a function to attain its local minimum is useful for our analysis.

Proposition 2 (see [7, 8]). Let be a proper lower semicontinuous function. If attains a local minimum at , then and .

We recall the following sum rule for the Mordukhovich subdifferential which is important in the sequel.

Proposition 3 (see [8]). Let be proper lower semicontinuous functions and . Suppose that the qualification condition is fulfilled. Then one has Specially, if either or is locally Lipschitz around , then one always has

The following propositions of the scalarization of Mordukhovich coderivatives and the chain rule of Mordukhovich subdifferentials are important for this paper.

Proposition 4 (see [8, 16]). Let be continuous around . Then If in addition is locally Lipschitz around , then

Proposition 5 (see [8, 16]). Let the vector-valued map be locally Lipschitz and let be lower semicontinuous. If then Moreover, if is strictly differentiable and is locally Lipschitz, then one always has

Finally in this section, we recall the following useful concept called nonlinear scalar function and some of its properties. For more details, we refer to [17–20].

Lemma 6. Given , the nonlinear scalar function , defined by is convex, strictly -monotone, -monotone, nonnegative homogeneous, globally Lipschitz with modulus . Simultaneously, for every , it follows that , Furthermore, for every , Specially, one has

3. Exact Penalization, Calmness Condition, and Necessary Optimality Condition for (MOPEC)

In this section, we focus our attention on establishing some equivalent properties between a multiobjective exact penalization and a calmness condition, called (MOPEC-) calmness, for (MOPEC). Simultaneously, we show that a local error bound condition associated merely with the constraint system, equivalently, a calmness condition of the parametric constraint system, implies the (MOPEC-) calmness condition. Subsequently, we apply a nonlinear scalar method to obtain a M-stationary necessary optimality condition under the (MOPEC-) calmness condition.

Consider the following parametric form of the feasible set with parameter : Denote the corresponding feasible set by Obviously, for the set-valued map , we have .

We are now in the position to introduce a (MOPEC-) calmness concept for (MOPEC).

Definition 7. Given and being a local efficient (resp. local weak efficient) solution for (MOPEC), then (MOPEC) is said to be (MOPEC-) calm with order at if and only if there exist and such that, for all and all , one has

Remark 8. Given and being a local efficient (resp. local weak efficient) solution for (MOPEC), we can also characterize the (MOPEC-) calmness condition by means of sequences. It is easy to verify that (MOPEC) is (MOPEC-) calm with order at if and only if there exists such that, for every sequence with and every sequence satisfying , , and , it holds that

Note that the (MOPEC-) calmness condition depends on not only the objective function but also the constraint system. In order to make up this deficiency, we propose the following local error bound notion for (MOPEC) associated merely with the constraint system.

Definition 9. Given and , then the constraint system of (MOPEC) is said to have a local error bound with order at if and only if there exist and such that, for all and all , one has

Next, we show that the local error bound implies the (MOPEC-) calmness.

Theorem 10. Let be a local efficient (resp. local weak efficient) solution for (MOPEC). If the constraint system of (MOPEC) has a local error bound with order at , then (MOPEC) is (MOPEC-) calm with order at .

Proof. Since is a local efficient (resp. local weak efficient) solution for (MOPEC) and , it immediately follows that holds for all with and sufficiently small . Thus, we only need to prove the case . Assume that (MOPEC) is not (MOPEC-) calm with order at . Then, for every , there exist and such that Since is nonempty and closed, there exists a projection of onto such that for all . Note that and . Then it follows that Together with being a local efficient (resp. local weak efficient) solution for (MOPEC), there exists some such that Moreover, since is locally Lipschitz, there exist and such that By (31) and (33), we have for all Together with and (34), we can conclude that This is a contradiction to the assumption that (MOPEC) has a local error bound with order at since , , , , and .

Remark 11. Specially, if we consider the case for every given and , then Definition 9 reduces to the fact that there exist and such that, for all and all , one has It is worth noting that this condition is essentially sufficient and necessary for the situation . Clearly, the necessity holds. In fact, since , it follows that for all and all . Moreover, is nonnegative homogeneous and . By Lemma 6, we have , which implies . For the sufficiency, since is continuous and is compact, there exists some such that Obviously, and ; then we have . Thus, we get from the nonnegative homogeneity of that, for all , all and all , By Lemma 6, we have which implies Furthermore, recall that a set-valued map is said to be calm with order at if and only if there exist neighborhoods of and of and a real number such that Then we can immediately obtain the following characterization of local error bounds for the constraint system of (MOPEC) based on the arguments in Remark 11.

Proposition 12. Given and , then the following assertions are equivalent.(i) The constraint system of (MOPEC) has a local error bound with order at .(ii) There exist and such that, for all and all ,(iii) The set-valued map , defined by (26), is calm with order at .

Proof. As discussed in Remark 11, (i) is equivalent to (ii). We only need to prove the equivalence of (ii) and (iii). In fact, it follows from the definition of calmness for a set-valued map that (ii) is obviously equivalent to the calmness with order at of the set-valued map .

As we know, there have been many papers devoted to investigate the calmness of a set-valued map (which is equivalent to the metric subregularity of its converse ). For more details, we refer to [21–24] and references therein. It has been shown in Remark 11 and Proposition 12 that there have been no differences between the scalar () and the multiobjective () settings when we only consider the calmness or the local error bound for the constraint system of (MOPEC). However, if we pay attention to the weaker (MOPEC-) calmness, we cannot negative the differences between them.

We now give the following equivalent characterizations of two classes of multiobjective penalty problems and the (MOPEC-) calmness condition.

Theorem 13. Let be a local efficient (resp. local weak efficient) solution for (MOPEC). Then the following assertions are equivalent.(i)(MOPEC) is (MOPEC-) calm with order at .(ii)There exists some such that, for any , is a local efficient (resp. local weak efficient) solution for the following multiobjective penalty problem with order : where , , .(iii)There exists some such that, for any , is a local efficient (resp. local weak efficient) solution for the following multiobjective penalty problem with order :

Proof. We only prove the case for being a local weak efficient solution since the proof of the case for being a local efficient solution is similar.
(i)⇒(ii). Suppose to the contrary that, for every , there exists with and such that Take and . Then it follows that and . Together with and , we get for all . Moreover, by (47), we have Note that , , , and and are continuously Fréchet differentiable. Then it follows that and . Together with and (48), this is a contradiction to the (MOPEC-) calmness with order of (MOPEC) at .
(ii)⇒(i). Suppose that (MOPEC) is not (MOPEC-) calm with order at . Then, for every , there exist and such that (31) holds. Since , it follows that , , and ; that is, for all . Thus, we have which implies , . Together with (31), we get This shows that the multiobjective penalty problem with order does not admit a local exact penalization at since , , and .
(i)⇒(iii). Assume that, for every , there exists such that Note that Thus, for every , there exists with such that Take , , , and . Then it follows that , , and , which implies since , and Connecting , (51), and (54), we have for any Moreover, it follows from [25, Lemma 3.21] and (51) that for any with we get Note that is locally Lipschitz and . Then it follows that , which implies from (54). Together with , , , and (55), this is a contradiction to the (MOPEC-) calmness with order of (MOPEC) at .
(iii)⇒(i). Assume that (MOPEC) is not (MOPEC-) calm with order at . Then, by the same argument to the proof of (ii)⇒(i), it follows that, for every , there exist and with , , and ; that is, such that (31) holds. Thus, we have Together with (31) and , we get which implies that the multiobjective penalty problem with order does not admit a local exact penalization at since the sequence and .

It is well known that a calmness condition with order 1 for standard nonlinear programming can lead to a KKT condition. In fact, we can also obtain a M-stationary condition for (MOPEC) under the (MOPEC-) calmness condition with order 1. To this end, we need the following generalized Fermat rule for a multiobjective optimization problem with an abstract constraint, which is established by applying the nonlinear scalar function in Lemma 6.

Lemma 14. Let be a locally Lipschitz vector-valued map, and let be a nonempty and closed subset. If is a local weak efficient solution for the multiobjective optimization problem then there exists some with such that

Proof. Define the function by Since is a local weak efficient solution, attains a local minimum at . Otherwise, there exists a sequence converging to such that since . Then we have and . Together with Lemma 6, we get This is a contradiction to being a local weak efficient solution since and . Note that is locally Lipschitz and is closed. It follows from Propositions 2, 3, and 5 and Lemma 6 that Therefore, there exists some with such that By Proposition 4, it follows that This completes the proof.

Next, we show that the (MOPEC-) calmness condition with order 1 is sufficient to establish a M-stationary condition for (MOPEC).

Theorem 15. Suppose that is a local weak efficient solution for (MOPEC) and (MOPEC) is (MOPEC-) calm with order 1 at . Then is a M-stationary point for (MOPEC); that is, there exist with , , , , and with such that

Proof. Since is a local weak efficient solution for (MOPEC) and (MOPEC) is (MOPEC-) calm with order 1 at , it follows from Theorem 13 (i)(iii) that there exists some such that is a local weak efficient solution for the multiobjective penalty problem with order 1. For simplicity, let the real-valued function defined by Note that is locally Lipschitz, , , and are continuously Fréchet differentiable, and is closed. Then is locally Lipschitz and the penalty function is also locally Lipschitz. Together with the closedness of and Lemma 14, it follows that there exists some with such that Moreover, by using and Proposition 3, we have Note that , , and are continuously Fréchet differentiable and is closed. Then it follows from Propositions 1 (ii), 4, and 5 that, for all , for all , where denotes the identity map from to itself. Let be the set of active constraints of at . Then we can conclude from (70)–(72) that Together with (68) and (69), there exist with , , and ; that is, , such that Take with , , and , , with and . Then we have This completes the proof.

Combining Proposition 12 and Theorem 15, we immediately have the following corollary.

Corollary 16. Let be a local efficient solution for (MOPEC). Suppose that the constraint system of (MOPEC) has a local error bound with order 1 at , or, equivalently, the set-valued map , defined by (26), is calm with order 1 at . Then is a M-stationary point for (MOPEC).

Remark 17. Recently, Kanzow and Schwartz [26] discussed the enhanced Fritz-John conditions for a smooth scalar optimization problem with equilibrium constraints and proposed some new constraint qualifications for the enhanced M-stationary condition. In particular, they obtained some sufficient conditions for the existence of a local error bound for the constraint system and the exactness of penalty functions with order 1 by using an appropriate condition. Subsequently, Ye and Zhang [27] extended Kanzow and Schwartz’s results to the nonsmooth case. It is worth noting that the exactness of the penalty function with order 1 in [26, 27] was established by using various qualification conditions, which were actually sufficient for the local error bound property of the constraint system; see [28, 29] for more details. However, just as shown in Theorem 13, the exactness for the two types of multiobjective penalty functions with order is obtained by means of the equivalent (MOPEC-) calmness condition, which is associated with not only the objective function but also the constraint system. Simultaneously, it follows from Theorem 10 and Proposition 12 that the (MOPEC-) calmness condition is weaker than the local error bound property of the constraint system.

4. Applications

The main purpose of this section is to apply the obtained results for (MOPEC) to a multiobjective optimization problem with complementarity constraints (in short, (MOPCC)) and a multiobjective optimization problem with weak vector variational inequality constraints (in short, (MOPWVVI)) and establish corresponding calmness conditions and M-stationary conditions.

4.1. Applications to (MOPCC)

In this subsection, we consider the following multiobjective optimization problem with complementarity constraints:

(MOPCC) where is locally Lipschitz, are continuously Fréchet differentiable, and is a nonempty and closed subset of . As usual, we denote Obviously, the feasible set is a closed subset of . It is easy to verify that can be reformulated as a special case of (MOPEC) if we let , where . Note that is constant and equals to . Then the parametric form of with parameter is Clearly, for the set-valued map , one has .

Inspired by Definitions 7 and 9, we give the following concepts, called (MOPCC-) calm and local error bound, for (MOPCC).

Definition 18. Given and being a local efficient (resp. local weak efficient) solution for (MOPCC), then (MOPCC) is said to be (MOPCC-) calm with order at if and only if there exist and such that, for all and all , one has

Definition 19. Given and , then the constraint system of (MOPCC) is said to have a local error bound with order at if and only if there exist and such that, for all and all , one has

Similarly, it follows from Theorem 10 that if the constraint system of (MOPCC) has a local error bound with order at , then (MOPCC) is (MOPCC-) calm with order at . Moreover, by Proposition 12, the constraint system of (MOPCC) has a local error bound with order at if and only if the set-valued map is calm with order at . Specially, if we take and , then Definitions 18 and 19 reduce to Definitions 3.3 and 3.6 in [4], respectively. Simultaneously, the corresponding results to Propositions 3.4 and 3.7 in [4] also hold.

As mentioned in the introduction, there have been various stationary concepts proposed for (MOPCC). Here we only recall the notion of the M-stationary point.

Definition 20 (see [4]). A point is called a M-stationary point of (MOPCC) if and only if there exists a Lagrange multiplier with and such that

The following formula for the Mordukhovich normal cone of the set is useful in the sequel.

Lemma 21 (see [4]). For every , we have

We now apply Theorem 15 to establish a M-stationary condition for (MOPCC) by virtue of the (MOPCC-) calmness condition.

Theorem 22. Suppose that is a local weak efficient solution for (MOPCC) and (MOPCC) is (MOPCC-) calm with order 1 at . Then is a M-stationary point of (MOPCC).

Proof. As stated above, (MOPCC) is equivalent to (MOPCC) with and , given by (78). Note that the (MOPCC-) calmness of (MOPCC) implies the (MOPEC-) calmness of (MOPCC). Then it follows from Theorem 15 that there exist with , , , and with such that Note that for all . Then it follows that and Together with Lemma 21, we have for every with , Moreover, we get Taking with , , , , and , for all , and substituting (86) and (87) into (84), then we can conclude that is a M-stationary point of (MOPCC) with respect to the Lagrange multiplier .

4.2. Applications to (MOPWVVI)

Consider the following multiobjective optimization problem with weak vector variational inequality constraints:

(MOPWVVI) where is locally Lipschitz, and are continuously Fréchet differentiable. is the solution set of the weak vector variational inequality (in short, (WVVI)): find a vector such that where , are continuously Fréchet differentiable and is a nonempty, closed, and convex subset of . In the sequel, we denote by the feasible set of (MOPWVVI). Then it is clear that is closed.

Take . Then it follows from Lemma 6 that is a solution of (WVVI) if and only if Moreover, since is nonempty, closed, and convex, and is a convex function, we can conclude from Theorem 8.15 in [7] and Proposition 5 that is a solution of (WVVI) if and only if

This shows that there exists some with such that

Given being a local efficient (resp. local weak efficient) solution for (MOPWVVI), then is a solution of (MOPWVVI). Next, we define the concept of (MOPWVVI-) calmness with order at for (MOPWVVI) with respect to the corresponding with satisfying (92).

Definition 23. Given , being a local efficient (resp. local weak efficient) solution for (MOPWVVI) and with satisfying (92), then (MOPWVVI) is said to be (MOPWVVI-) calm with order at if and only if there exists such that, for every sequence with and every sequence satisfying , , , and , it holds that

Obviously, we can define a similar local error bound condition at a local weak efficient solution for (MOPWVVI) with respect to with satisfying (92). Moreover, we can obtain a corresponding relationship between the (MOPWVVI-) calmness condition and the local error bound condition. However, we omit the details here for simplicity.

Next, we establish a M-stationary condition for (MOPWVVI) under the (MOPWVVI-) calmness with order 1 assumption.

Theorem 24. Suppose that is a local weak efficient solution for (MOPWVVI) and with satisfy (92). If in addition (MOPWVVI) is (MOPWVVI-) calm with order 1 at , then there exist with , , , , and with such that where the set-valued map is defined by for all .

Proof. Consider the problem (MOPEC) with and for all . Obviously, is a feasible point of (MOPEC) and the feasible set of (MOPEC) is contained in . By assumption, is a local weak efficient solution for (MOPEC). Moreover, it is easy to verify that the (MOPWVVI-) calmness of (MOPWVVI) with order 1 at implies the (MOPEC-) calmness of (MOPEC) with order 1 at . Thus, together with , being continuously Fréchet differentiable and , we immediately complete the proof by Theorem 15.

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

Acknowledgments

The authors are grateful to the anonymous referee for his/her valuable comments and suggestions, which helped to improve the paper. This research was supported by the National Natural Science Foundation of China (Grant no. 11171362) and the Fundamental Research Funds for the Central Universities (Grant no. CDJXS12100021).

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Copyright © 2014 Shengkun Zhu and Shengjie Li. 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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