- Open Access
Some remarks on regularized nonconvex variational inequalities
© Balooee and Kim; licensee Springer. 2013
- Received: 16 June 2013
- Accepted: 27 September 2013
- Published: 11 November 2013
In this paper, we investigate and analyze the nonconvex variational inequalities introduced by Noor in (Optim. Lett. 3:411-418, 2009) and (Comput. Math. Model. 21:97-108, 2010) and prove that the algorithms and results in the above mentioned papers are not valid. To overcome the problems in the above cited papers, we introduce and consider a new class of variational inequalities, named regularized nonconvex variational inequalities, instead of the class of nonconvex variational inequalities introduced in the above mentioned papers. We also consider a class of nonconvex Wiener-Hopf equations and establish the equivalence between the regularized nonconvex variational inequalities and the fixed point problems as well as the nonconvex Wiener-Hopf equations. By using the obtained equivalence formulations, we prove the existence of a unique solution for the regularized nonconvex variational inequalities and propose some projection iterative schemes for solving the regularized nonconvex variational inequalities. We also study the convergence analysis of the suggested iterative schemes under some certain conditions.
MSC:47H05, 47J20, 49J40, 90C33.
- regularized nonconvex variational inequalities
- nonconvex sets
- iterative algorithm
- convergence analysis
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 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-Hopf (normal) equations, auxiliary principle, and descent framework for solving variational inequalities and complementarity problems. For the applications, physical formulations, numerical methods and other aspects of variational inequalities, see [1–22] and the references therein.
The projection method and its variant forms represent an important tool for finding an 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 problems using the concept of projection. This alternative formulation enables us to suggest some iterative methods for computing an approximate solution.
It is worth mentioning that most of the results regarding the existence and iterative approximation of solutions to variational inequality problems have been investigated and considered so far to the case where the underlying set is a convex set. Recently, the concept of convex set has been generalized in many directions, which has potential and important applications in various fields. It is well known that the uniformly prox-regular sets are nonconvex and include the convex sets as special cases. This class of uniformly prox-regular sets has played an important part in many nonconvex applications such as optimization, dynamic systems and differential inclusions. For more details, see, for example, [9, 10, 13, 14, 21, 23, 24].
Very recently, Noor [25, 26] has introduced and considered a new class of variational inequalities, the so-called nonconvex variational inequalities (NVI) on the uniformly prox-regular sets. He has also introduced a class of Wiener-Hopf equations in . The author has asserted that NVI (2.1) from [25, 26] is equivalent to the fixed point problem (3.1) from [25, 26] as well as the Wiener-Hopf equation (2.5) from . Then, he used the fixed point formulation (3.1) from [25, 26] and the equivalence formulations (4.1) and (4.2) from , and suggested some iterative schemes for solving NVI (2.1) from [25, 26]. He also studied the convergence analysis of the suggested iterative methods under certain conditions.
In this paper, we establish that the equivalence formulation (3.1), used by Noor in [25, 26], is not correct. That is, Lemma 3.1 in [25, 26], which is the main tool to suggest the algorithms and to prove the strong convergence of the sequences generated by the proposed iterative algorithms in [25, 26], is incorrect. Consequently, the algorithms and results in [25, 26] are not valid. To overcome these problems in [25, 26], we introduce and consider a new class of variational inequalities, termed the regularized nonconvex variational inequalities (RNVI), instead of the class of NVI (2.1) from [25, 26]. We also consider a class of nonconvex Wiener-Hopf equations (NWHE) and establish the equivalence between RNVI and the fixed point problems as well as NWHE. By using the obtained equivalence formulations, we prove the existence of a unique solution for RNVI and propose some projection iterative schemes for solving RNVI. We also study the convergence analysis of the suggested iterative schemes under some certain conditions.
Throughout this article, we let ℋ be a real Hilbert space which is equipped with an inner product and the corresponding norm and K be a nonempty and closed subset of ℋ. We denote by or the usual distance function to the subset K, i.e., . Let us recall the following well-known definitions and some auxiliary results of nonlinear convex analysis and nonsmooth analysis [12–14, 21].
Clarke et al. , in Proposition 1.1.5, give characterization of as follows.
where y is a vector in X and t is a positive scalar.
The generalized directional derivative defined earlier can be used to develop a notion of tangency that does not require K to be smooth or convex.
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 optimization, economic models, dynamical systems, differential inclusions, etc. For such applications, see [6–8, 10]. This class seems particularly well suited to overcome the difficulties which arise due to the nonconvexity assumptions on K. 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 .
Obviously, the class of normalized uniformly prox-regular sets is sufficiently large to include the class of convex sets, p-convex sets, submanifolds (possibly with boundary) of ℋ, the images under a diffeomorphism of convex sets and many other nonconvex sets, see [11, 14].
Lemma 2.2 
A closed set is convex if and only if it is proximally smooth of radius r for every .
If , then in view of Definition 2.5 and Lemma 2.2, the uniform r-prox-regularity of is equivalent to the convexity of , which makes this class of great importance. For the case of , we set .
For all , ;
For all , is Lipschitz continuous with constant on .
In order to make clear the concept of r-prox-regular sets, we state the following concrete example: The union of two disjoint intervals and is r-prox-regular with , see [9, 13, 21]. The finite union of disjoint intervals is also r-prox-regular and r depends on the distances between the intervals.
where denotes the P-normal cone to at s in the sense of nonconvex analysis. However, this claim is not true in general.
Remark 3.1 Every solution of problem (1) is a solution of problem (2), but the converse is not necessarily true.
The converse of the above statement does not hold in general. Indeed, suppose that inclusion (5) holds for some . Then, Lemma 2.1 implies that inequality (4) holds for some . However, by using inequality (4), we cannot deduce inequality (3). □
The following example illustrates that problem (4) does not imply problem (3).
cannot hold for all .
where is a constant and is the projection of ℋ onto the uniformly r-prox-regular set .
Therefore, , that is, the set is nonempty and singleton. Hence, in the statement of Lemma 3.1, the constant ρ should be satisfied for some . Secondly, we note that the author [25, 26] used the nonconvex variational inclusion (2) as an equivalence formulation of the nonconvex variational inequality (1). However, in view of Remark 3.1 and Example 3.1, the two problems (1) and (2) are not necessarily equivalent.
where is a constant, and I is the identity operator. Problem (9) is called the nonconvex Winer-Hopf equation (NWHE).
Noor  claimed that problem (9) is equivalent to problem (1).
Lemma 3.2 (, Lemma 4.1)
where is a constant.
By a careful reading, we discovered that Lemma 3.1 is the main tool to establish the statement of Lemma 3.2. As it is shown, the statement of Lemma 3.1 is not necessarily true. Consequently, the statement of Lemma 3.2 is not necessarily true.
Since Lemmas 3.1 and 3.2 are the main tools to suggest algorithms and to obtain the results in  and , in view of the above remarks, the results in [25, 26] and the papers where the same technique and method are used, are not valid.
Problem (11) is called the regularized nonconvex variational inequality (RNVI). We prove the equivalence between RNVI (11) and problem (2) as well as fixed point problem (8).
An inequality of type (12) is called the variational inequality, which was introduced and studied by Stampacchia  in 1964.
In the next proposition, the equivalence between nonconvex variational inclusion (2) and regularized nonconvex variational inequality (11) is established.
Proposition 4.1 If is a uniformly prox-regular set, then problem (11) is equivalent to problem (2).
Conversely, if is a solution of problem (2), then Definition 2.5 guarantees that is a solution of problem (11). □
Problem (2) is called the nonconvex variational inclusion associated with RNVI (11). Now, by using the projection operator technique, we establish the equivalence between problem (11) and fixed point problem (8).
Lemma 4.1 Let T be the same as in problem (11). Then is a solution of problem (11) if and only if u satisfies equation (8), provided that for some .
where I is the identity operator and we have used the well-known fact that . □
- (a)monotone if and only if
- (b)κ-strongly monotone if and only if there exists a constant such that
- (c)γ-Lipschitz continuous if and only if there exists a constant such that
In the next theorem, the existence and uniqueness of a solution for problem (11) are discussed.
where , then problem (11) admits a unique solution.
Condition (14) implies that . From inequality (18), we infer that F is a contraction mapping. According to the Banach fixed point theorem, there exists a unique point such that . It follows from (15) that . Now, Lemma 4.1 guarantees that is a solution of problem (11). This completes the proof. □
Noor  proposed the Mann iteration process for solving problem (1) as follows.
Algorithm 4.1 (, Algorithm 3.1)
where for all .
Noor  suggested the following two-step and three-step iterative methods for solving problem (1).
Algorithm 4.2 (, Algorithm 3.4)
where , for all .
Algorithm 4.3 (, Algorithm 3.5)
where , for all .
Remark 4.1 It should be pointed out that in the context of Algorithms 3.4 and 3.5 from , there are minor mistakes. In fact, in iterative processes (3.7) and (3.8) from Algorithm 3.4 in , must be replaced by , as we have done in Algorithm 4.2. Meanwhile, in the context of Algorithm 3.5 from , must be replaced by , as we have done in Algorithm 4.3.
By a careful reading, we found that Algorithms 4.1-4.3 do not work. Indeed, in a way similar to the argument of Remark 3.2, the points , and () do not belong necessarily to .
By utilizing Lemma 4.1, we suggest and analyze the following explicit projection iterative methods for solving problem (11).
We now study the convergence analysis of Algorithm 4.4 and this is the main motivation of our next result.
Theorem 4.2 Let the operator T be the same as in Theorem 4.1, and let all the conditions of Theorem 4.1 hold. Then the iterative sequence generated by Algorithm 4.4 converges strongly to the unique solution of problem (11).
where θ is the same as in (19). Since , it follows that the right-hand side of the above inequality tends to zero as , whence we deduce that as . This completes the proof. □
In a similar way to the proof of Theorem 4.2, one can prove the strong convergence of the iterative sequence generated by Algorithms 4.5 and 4.6.
In this section, by utilizing Lemma 4.1 and the projection method, the equivalence between NWHE (9) and RNVI (11) is established. By using the obtained equivalence formulation, some iterative algorithms for solving RNVI (11) are suggested and analyzed. The convergence analysis of the proposed iterative algorithms under some certain conditions is studied.
Let the operator T be the same as in problem (11). Related to problem (11), we consider the problem of finding satisfying (9).
Remark 5.1 It was shown that the Wiener-Hopf equations had played an important and significant role in developing several numerical techniques for solving variational inequalities and related optimizations problems (see, for example, [18, 19, 22] and the references therein).
In the next lemma, the equivalence between RNVI (11) and NWHE (9) is proved.
Lemma 5.1 Let T be the same as in problem (11). Then is a solution of RNVI (11) if and only if NWHE (9) has a solution satisfying (10), provided that for some .
where , that is, is a solution of NWHE (9).
then Lemma 4.1 implies that is a solution of RNVI (11). This completes the proof. □
Noor  used the equivalence formulation between the two problems (1) and (9) and suggested the following iterative methods for solving problem (1).
Algorithm 5.1 (, Algorithm 4.1)
where , for all and .
Algorithm 5.2 (, Algorithm 4.2)
where , for all and .
Algorithm 5.3 (, Algorithm 4.3)
where , for all and .
Remark 5.2 As it is pointed out, the two problems (1) and (9) are not necessarily equivalent. Hence, the equivalence between problems (1) and (9) cannot be used for suggesting Algorithms 5.1-5.3 to approximate the solution of problem (1). Even without considering the mentioned fact, we note that Algorithms 5.1-5.3 do not work. Indeed, in a way similar to the argument of Remark 3.2, iterative scheme (25) is well defined provided that for each , the point belongs to for some . Accordingly, must be taken in for some . However, for a given , iterative scheme (26) does not guarantee that for each , because is not necessarily convex.
The following example illustrates that for any given uniformly r-prox-regular set in ℋ and , the set in ℋ is not necessarily convex.
which is clearly nonconvex.
By using NWHE (9) and Lemma 5.1, we get a fixed point formulation to construct a new projection iterative algorithm for solving RNVI (11).
This fixed point formulation enables us to construct the following iterative algorithm for solving RNVI (11).
We now apply Lemma 5.1 and study the convergence analysis of Algorithm 5.4.
Theorem 5.1 Let the operator T be the same as in Theorem 4.1 and suppose that all the conditions of Theorem 4.1 hold. Assume further that is a constant satisfying conditions (13) and (14). Then there exists such that z is a solution of problem (9) and the sequence generated by Algorithm 5.4 converges strongly to z.
Condition (14) implies that . Since , it follows that the right-hand side of the above inequality tends to zero as , which implies that the sequence generated by Algorithm 5.4 converges strongly to z. This completes the proof. □
Can the existence of a solution for nonconvex variational inequality (1) be proved?
Can the Mann iteration process for solving nonconvex variational inequality (11) be presented?
This work was supported by the Basic Science Research Program through the National Research Foundation (NRF) Grant funded by the Ministry of Education of the Republic of Korea (2013R1A1A2054617). The authors thank the anonymous referees for their constructive comments which contributed to the improvement of the present paper.
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