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Projection Methods and a New System of Extended General Regularized Nonconvex Set-Valued Variational Inequalities
A new system of extended general nonlinear regularized nonconvex set-valued variational inequalities is introduced, and the equivalence between the extended general nonlinear regularized nonconvex set-valued variational inequalities and the fixed point problems is verified. Then, by this equivalent formulation, a new perturbed projection iterative algorithm with mixed errors for finding a solution of the aforementioned system is suggested and analyzed. Also the convergence of the suggested iterative algorithm under some suitable conditions is proved.
Variational inequality theory, introduced by Stampacchia , has become a rich source of inspiration and motivation for the study of a large number of problems arising in economics, finance, transportation, network and structural analysis, elasticity, and optimization. Many research papers have been written lately, both on the theory and applications of this field. Important connections with main areas of pure and applied sciences have been made. (See, for example, [2–4] and the references cited therein.) The development of variational inequality theory can be viewed as the simultaneous pursuit of two different lines of research. On the one hand, it reveals the fundamental facts on the qualitative aspects of the solution to important classes of problems. On the other hand, it also enables us to develop highly efficient and powerful new numerical methods to solve, for example, obstacle, unilateral, free, moving and the complex equilibrium problems. One of the most interesting and important problems in variational inequality theory is the development of an efficient numerical method. There is a substantial number of numerical methods including projection method and its variant forms, Wiener-Holf (normal) equations, auxiliary principle, and descent framework for solving variational inequalities and complementarity problems. For applications on physical formulations, numerical methods and other aspects of variational inequalities, see [1–37] and the references therein.
Projection method and its variant forms represent important tool for finding the approximate solution of various types of variational and quasi-variational inequalities, the origin of which can be traced back to Lions and Stampacchia . The projection type methods were developed in 1970s and 1980s. The main idea in this technique is to establish the equivalence between the variational inequalities and the fixed point problem using the concept of projection. This alternate formulation enables us to suggest some iterative methods for computing the approximate solution; for example, see [5–7, 16–18, 29, 30, 35–37].
It should be pointed that almost all the results regarding the existence and iterative schemes for solving variational inequalities and related optimizations problems are being considered in the convexity setting. Consequently, all the techniques are based on the properties of the projection operator over convex sets, which may not hold in general, when the sets are nonconvex. It is known that the uniformly prox-regular sets are nonconvex and include the convex sets as special cases. For more details, see, for example, [12, 20, 21, 28]. In recent years, Bounkhel et al. , Moudafi , Noor [25, 26], and Pang et al.  have considered variational inequalities in the context of uniformly prox-regular sets.
In this paper, we introduce and consider a new system of extended general nonlinear regularized nonconvex set-valued variational inequalities. We establish the equivalence between the extended general nonlinear regularized nonconvex set-valued variational inequalities and the fixed point problems, and then, by this equivalent formulation, we suggest and analyze a new perturbed projection iterative algorithm with mixed errors for finding a solution of the aforementioned system. We also prove the convergence of the suggested iterative algorithm under some suitable conditions.
Throughout this paper, we will let be a real Hilbert space which is equipped with an inner product and corresponding norm . Let be a nonempty convex subset of , and, denote the family of all closed and bounded subsets of . We denote by or the usual distance function to the subset ; that is,. Let us recall the following well-known definitions and some auxiliary results of nonlinear convex analysis and nonsmooth analysis [19–21, 28].
Definition 2.1. Let be a point not lying in . A point is called a closest point or a projection of onto if . The set of all such closest points is denoted by ; that is,
Definition 2.2. The proximal normal cone of at a point with is given by Clarke et al. , in Proposition , give a characterization of as the following.
Lemma 2.3. Let be a nonempty closed subset in . Then if and only if there exists a constant such that for all .
The above inequality is called the proximal normal inequality. The special case in which is closed and convex is an important one. In Proposition of , the authors give the following characterization of the proximal normal cone, the closed and convex subset .
Lemma 2.4. Let be a nonempty, closed, and convex subset in . Then if and only if , for all .
Definition 2.5. Let be a real Banach space, and let be the Lipschitz with constant near a given point ; that is, for some , one has for all where denotes the open ball of radius and centered at . The generalized directional derivative of at in the direction , denoted as , is defined as follows: where is a vector in and is a positive scalar.
The generalized directional derivative defined earlier can be used to develop a notion of tangency that does not require to be smooth or convex.
Definition 2.6. The tangent cone to at a point in is defined as follows:
Having defined a tangent cone, the likely candidate for the normal cone is the one obtained from by polarity. Accordingly, we define the normal cone of at by polarity with as follows:
Definition 2.7. The Clarke normal cone, denoted by , is given by , where means the closure of the convex hull of . It is clear that one always has . The converse is not true in general. Note that is always closed and convex cone, whereas is always convex but may not be closed (see [19, 20, 28]).
In 1995, Clarke et al.  introduced and studied a new class of nonconvex sets, called proximally smooth sets; subsequently, Poliquin et al. in  investigated the aforementioned sets, under the name of uniformly prox-regular sets. These have been successfully used in many nonconvex applications in areas such as optimizations, economic models, dynamical systems, differential inclusions, and so forth. For such applications see [9–11, 13]. This class seems particularly well suited to overcome the difficulties which arise due to the nonconvexity assumptions on . We take the following characterization proved in  as a definition of this class. We point out that the original definition was given in terms of the differentiability of the distance function (see ).
Definition 2.8. For any , a subset of is called normalized uniformly prox-regular (or uniformly -prox-regular ) if every nonzero proximal normal to can be realized by an -ball.
This means that, for all and all with ,
Obviously, the class of normalized uniformly prox-regular sets is sufficiently large to include the class of convex sets, -convex sets, and submanifolds (possibly with boundary) of , the images under a diffeomorphism of convex sets and many other nonconvex sets, see [14, 21].
Lemma 2.9 (see ). A closed set is convex if and only if it is proximally smooth of radius for every .
Proposition 2.10. Let , and let be a nonempty closed and uniformly -prox-regular subset of . Set . Then the following statements hold.(a)For all , one has .(b)For all , is Lipschitz continuous with constant on .(c)The proximal normal cone is closed as a set-valued mapping.
As a direct consequent of part (c) of Proposition 2.10, we have . Therefore, we will define for such a class of sets.
In order to make clear the concept of -prox-regular sets, we state the following concrete example. The union of two disjoint intervals and is -prox-regular with . The finite union of disjoint intervals is also -prox-regular, and depends on the distances between the intervals.
Definition 2.11. The single-valued operator is called(a)monotone if (b)-strongly monotone if, there exists a constant such that (c)-Lipschitz continuous if there exists a constant such that
Definition 2.12. Let be a set-valued operator, and let be a single-valued operator. Then is said to be(a)monotone if (b)-strongly monotone with respect to if there exists a constant such that
Definition 2.13. A two-variable set-valued operator is called --Lipschitz continuous in the first variable, if there exists a constant such that, for all , where is the Hausdorff pseudo-metric, that is, for any two nonempty subsets and of ,
It should be pointed that if the domain of is restricted to the family closed bounded subsets of (denoted by ), then is the Hausdorff metric.
3. System of Extended General Regularized Nonconvex Set-Valued Variational Inequalities
In this section, we introduce a new system of extended general nonlinear regularized nonconvex set-valued variational inequalities and a new system of extended general nonlinear set-valued variational inequalities in Hilbert spaces and investigate their relations.
Let be two nonlinear set-valued operators, and let be four nonlinear single-valued operators such that , for each . For any constants and , we consider the problem of finding and , such that and
The problem (3.1) is called the system of extended general nonlinear regularized nonconvex set-valued variational inequalities involving six different nonlinear operators.
Lemma 3.1. If is a uniformly prox-regular set, then the problem (3.1) is equivalent to that of finding and , such that and where denotes the -normal cone of at in the sense of nonconvex analysis.
Proof. Let with , , and , be a solution of the system (3.1). If , because the vector zero always belongs to any normal cone, then . If , then, for all with , one has
Now, by Lemma 2.3, one gets , and so
Similarly, one can establish
Conversely, if with , , and , is a solution of the system (3.2), then, in view of Definition 2.8, and , with are a solution of the system (3.1).
The problem (3.2) is called the extended general nonlinear nonconvex set-valued variational inclusion system associated with the system of extended general nonlinear regularized nonconvex set-valued variational inequalities (3.1).
Some special cases of the system (3.1) are as follows.
Case 1. If ; that is, , the convex set in , then the system (3.1) collapses to the following system.
Find and , such that and which is called the system of extended general nonlinear set-valued variational inequalities in the sense of convex analysis.
Case 2. If are two nonlinear single-valued operators, , the identity operator, and , then the system (3.6) reduces to the system of finding such that which has been considered and studied by Noor .
Case 3. If ; that is, , are two univariate nonlinear single-valued operators, and , then the system (3.1) changes into that of finding such that and which has been introduced and studied by Yang et al. .
Case 8. If , then the system (3.12) collapses to the following problem.
Find such that Inequality of type (3.13) is called variational inequality, which was introduced and studied by Stampacchia  in 1964.
4. Perturbed Projection Iterative Algorithms
In this section, by using the projection operator technique, we first verify the equivalence between the extended general nonlinear regularized nonconvex set-valued variational inequalities (3.1) and the fixed point problems. Then, by using the obtained fixed point formulation, we construct two new perturbed projection iterative algorithms with mixed errors for solving the systems (3.1) and (3.6).
Remark 4.2. The equality (4.1) can be written as follows: where are two constants.
The fixed point formulation (4.5) enables us to construct the following perturbed iterative algorithms with mixed errors.
Algorithm 4.3. Let , , , , and be the same as in the system (3.1) such that be an onto operator for . For arbitrary chosen initial point , compute the iterative sequence by using where initial points and are chosen arbitrary, is a parameter and , , and are four sequences in to take into account a possible inexact computation of the resolvent operator point satisfying the following conditions:
Algorithm 4.4. Let , , , , and be the same as in the system (3.6) such that is an onto operator for . For arbitrary chosen initial point , compute the iterative sequence by using where initial points and are chosen arbitrary, the parameter and the sequences , , , and are the same as in Algorithm 4.3.
Remark 4.5. It should be pointed that(a)when , for all , Algorithms 4.3 and 4.4 reduce to the perturbed iterative process with mean errors;(b)if , for all , then Algorithms 4.3 and 4.4 change into the perturbed iterative process without error.
5. Main Results
Theorem 5.1. Let , , , , and be the same as in the system (3.1) such that, for each ,(a) is -strongly monotone with respect to and --Lipschitz continuous in the first variable;(b) is -strongly monotone and -Lipschitz continuous;(c) is -Lipschitz continuous.
If the constants and satisfy the following conditions: where , then there exist with and , such that is a solution of the system (3.1), and the sequence generated by Algorithm 4.3 converges strongly to .
Proof. It follows from (4.6) that Since is -strongly monotone with respect to and --Lipschitz continuous in the first variable and is -Lipschitz continuous, we conclude that Substituting (5.3) in (5.2), we get Like the proof (5.4), by using (4.6), we can prove that On the other hand, by using (4.6) and Proposition 2.10, we find that From -strongly monotonicity and -Lipschitz continuity of , we have Substituting (5.7) in (5.6), we obtain which leads to In similar way to the proofs (5.6)–(5.9), we can prove that It follows from (5.4) and (5.10) that From (5.5) and (5.9), it follows that Now define on by , for all . It is obvious that is a Hilbert space. Applying (5.11) and (5.12), one has where Obviously, , as , where In view of the condition (5.1), we know that . Then, for , there exists such that for each . Thus, it follows from (5.13) that, for each , Hence, for any , we have Since , it follows from (4.7) and (5.17) that , as . Hence, and are both Cauchy sequences in , and so there exist and such that and , as . By the inequalities (5.9) and (5.10), it follows that the sequences and are both also Cauchy in . Thus, there exist such that and , as . Since for , is --Lipschitz continuous in the first variable, it follows from (4.6) that as . Hence, and are also both Cauchy sequences in and so there exist such that and , as . Further, noting , we have Since , like the proof (5.19), we obtain The right sides of the inequalities (5.19) and (5.20) tend to zero as . Hence, and . Since the operators and are continuous, it follows from (4.6) and (4.7) that Since the operators , , and are continuous, it follows from (4.6) and (5.21) that Now, Lemma 4.1 guarantees that is a solution of the system (3.1). This completes the proof.
Theorem 5.2. Let , , , , and be the same as in the system (3.6) such that, for each ,(a) is -strongly monotone with respect to and --Lipschitz continuous in the first variable;(b) is -strongly monotone and -Lipschitz continuous;(c) is -Lipschitz continuous.
If the constants and satisfy the following conditions: then there exist with and , such that is a solution of the system (3.6) and the sequence generated by Algorithm 4.4 converges strongly to .
The authors are very grateful to the referees for their comments and suggestions which improved the presentation of this paper. The work was supported by the Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education, Science and Technology (Grant number: 2011-0021821).
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