Abstract
In this study, we introduce and study a generalized complementarity problem involving XOR operation and three classes of generalized variational inequalities involving XOR operation. Under certain appropriate conditions, we establish equivalence between them. An iterative algorithm is defined for solving one of the three generalized variational inequalities involving XOR operation. Finally, an existence and convergence result is proved, supported by an example.
1. Introduction
It is well known that the many unrelated free boundary value problems related to mathematical and engineering sciences can be solved by using the techniques of variational inequalities. In a variational inequality formulation, the location of the free boundary becomes an intrinsic part of the solution, and no special devices are needed to locate it. Complementarity theory is an equally important area of operations research and application oriented. The linear as well as nonlinear programs can be distinguished by a family of complementarity problems. The complementarity theory have been elongated for the purpose of studying several classes of problems occurring in fluid flow through porous media, economics, financial mathematics, machine learning, optimization, and transportation equilibrium, for example, [1–5].
The correlations between the variational inequality problem and complementarity problem were recognized by Lions [6] and Mancino and Stampacchia [7]. However, Karamardian [8, 9] showed that both the problems are equivalent if the convex set involved is a convex cone. For more details on variational inequalities and complementarity problems, refer to [6, 10–12].
The exclusive “XOR,” sometimes also exclusive disjunction (short: XOR) or antivalence, is a Boolean operation which only outputs true if only exactly one of its both inputs is true (so, if both inputs differ). There are many applications of XOR terminology, that is, it is used in cryptography, gray codes, parity, and CRC checks. Commonly, the symbol is used to denote the XOR operation. Some problems related to variational inclusions involving XOR operation were studied by [13–16].
Influenced by the applications of all the above discussed concepts in this study, we introduce and study a generalized complementarity problem involving XOR operation with three classes of generalized variational inequalities involving XOR operation. Some equivalence relations are established between them. An existence and convergence result is proved for one of the three types of generalized variational inequalities involving XOR operation. For illustration, an example is provided.
2. Some Basic Concepts and Formulation of the Problem
Throughout this study, we assume to be real ordered Banach space with norm and be its dual space. Suppose that is the metric induced by the norm, (respectively, ) is the family of nonempty (respectively, closed and bounded) subsets of . The Hausdorff metric on is defined aswhere , and .
Let be a pointed closed convex positive cone in , and denotes the value of the linear continuous function at .
The following definitions and concepts are required to achieve the goal of this study, and most of them can be found in [17, 18].
Definition 1. The relation “” is called the partial order relation induced by the cone , that is, if and only if .
Definition 2. For arbitrary elements , if (or ) holds, then and are said to be comparable to each other (denoted by ).
Definition 3. For arbitrary elements , and mean the least upper bound and the greatest upper bound of the set . Suppose and exist, then some binary operations are defined as(i)(ii)(iii)(iv)The operations , , and are called OR, AND, XOR, and XNOR operations, respectively.
Proposition 1. Let be an XOR operation and be an XNOR operation. Then, the following relations hold:(i), (ii)if , then (iii), if (iv)If , then if and only if (v)(vi)(vii)(viii)If and , then (ix)If , then , for all and
Proposition 2. Let be a cone in ; then, for each , the following relations hold:(i)(ii)(iii)(iv)If , then
Definition 4. Let be a single-valued mapping, then(i) is said to be a comparison mapping, if , then , , and , for all (ii) is said to be a strongly comparison mapping, if is a comparison mapping and , if and only if , for any
Definition 5. Let be a proper functional. A vector is called subgradient of at , ifThe set of all subgradients of at is denoted by . The mapping defined byis called subdifferential of .
Definition 6. The resolvent operator associated with is given bywhere is a constant, and is the identity operator.
It is well known that the resolvent operator is single-valued as well as nonexpansive.
Definition 7. A mapping is said to be(i)Positive homogeneous if, for all and , (ii)Convex, if and all
Definition 8. A multivalued mapping is said to be(i)Upper semicontinuous at if, for every open set containing , there exists an open set containing such that , where is equipped with topology(ii)Upper semicontinuous on if it is upper semicontinuous at every point of (iii)Upper hemicontinuous on if its restriction to line segments of is upper semicontinuous(iv)Monotone if, for every
Definition 9. A multivalued mapping is said to be -Lipschitz continuous, if there exists a constant such that
Definition 10. A multivalued mapping is said to be relaxed Lipschitz continuous, if there exists a constant such thatLet be a multivalued mapping with nonempty values and be a proper functional. We consider the following generalized complementarity problem involving XOR operation.
Find , such thatWe denote by the solution set of generalized complementarity problem involving XOR operation (9).
We mention some special cases of problem (9) as follows.(i)If we replace by and by , then problem (9) reduces to the problem of finding and such that Problem (10) is called generalized complementarity problem, introduced and studied by Huang et al. [19].(ii)If , then problems (9) as well as (10) reduce to the problem of finding and such that Problem (11) can be found in [20, 21].We remark that for suitable choices of operators involved in the formulation of (9), a number of known complementarity problems can be obtained easily, for example, [17, 22–24].
Simultaneously, we also study the following three types of generalized variational inequalities involving XOR operation.(1)Find such that(2)Find such that(3)Find such thatWe denote the solution set of (12) by , (13) by , and (14) by .
Many known variational inequality problems can be obtained from problems (12)–(14), for example, [25–29] and the references therein.
3. Equivalence Results
We establish the equivalence among problems (9), (12)–(14). First, we establish the equivalence between generalized complementarity problem involving XOR operation (9) and generalized variational inequality problem involving XOR operation (12).
Theorem 1. Let be a multivalued mapping with nonempty values and be a proper functional. Then, the following statements are true:(i)If , then (ii)If is positive homogeneous, then
Proof. (i)Let , then , and there exists such that Since , by (iv) of Proposition 1, we have which implies that By using (16) and (17), we have that is, which implies that . So, we have .(ii)Let , then , and there exists such thatSince is a pointed closed convex positive cone, clearly and . Putting in generalized variational inequality involving XOR operation (12) and using positive homogenity of , we getNow, putting in generalized variational inequality involving XOR operation ((12)) and using positive homogenity of , we getwhich implies thatthus,that is,Combining (21) and (25), we haveFrom generalized variational inequality involving XOR operations (12) and (16), we havewhich implies thatthus, we have . So, we have . That is, .
Theorem 2. The following statements are true.(i)(ii)If is monotone, then (iii)If is upper hemicontinuous and is convex, then
Proof. (i)Is trivial(ii)Let . Then, for all , there exists such that Since is monotone, for every , and using the above inequality, we have which implies that . Thus, .(iii)Suppose that the conclusion is not true. Then, there exists such that and . Then, for some and , we haveSince is upper hemicontinuous and is convex, setting and taking , we havewhich implies thatthus,which contradicts that . Thus, , and (iii) is true.
Remark 1. If we replace by and dropping the concepts related to operation, then with slight modification in Theorems 1 and 2, one can obtain some results of Huang et al. [19]. Additionally, for suitable choices of operators in Theorems 1 and 2, one can obtain some results of Farajzadeh and Harandi [30].
4. Existence and Convergence Result
In this section, we first establish the equivalence between the generalized variational inequality problem involving XOR operation (12) and a nonlinear equation. Based on this equivalence, we construct an iterative algorithm for solving generalized variational inequality problem involving XOR operation (12).
Lemma 1. The generalized variational inequality problem involving XOR operation (12) admits a solution , and , if and only if the following relation is satisfied:where is a constant, is the resolvent operator associated with , and is the identity operator.
Proof. From the definition of resolvent operator associated with and relation (35), we havewhich implies that , that is,By the definition of subdifferential operator and (37), we haveUsing (vi) of Proposition 1, we haveThus, the generalized variational inequality problem involving XOR operation (12) is satisfied.
Conversely, suppose that generalized variational inequality problem involving XOR operation (12) is satisfied. That is,that is, , which implies thatthat is, the relation (35) is satisfied.
Based on Lemma 1, we develop the following iterative algorithm for solving the generalized variational inequality problem involving XOR operation (12).
Iterative Algorithm 1. Let be a pointed closed convex positive cone. Suppose that , for . Let for , there exists , such thatSince , by Nadler [31], there exists , using (iv) of Proposition 2, and as , we haveContinuing this way, compute the sequences and by the following scheme:for , where , can be chosen arbitrarily, , is the Hausdorff metric on , and is a constant.
Now, we prove our main result.
Theorem 3. Let be a real ordered Banach space and be a pointed closed convex positive cone in with partial ordering “.” Let be a functional such that the resolvent operator associated with is strongly comparison and continuous. Suppose that is a multivalued mapping such that is the relaxed Lipschitz continuous with constant and -Lipschitz continuous with constant . Let and , where and , , such that for , the following condition is satisfied:
Then, the sequences and strongly converge to and , respectively, the solutions of generalized variational inequality problem involving XOR operation (12).
Proof. Since , for , using (iii) of Proposition 1, we evaluateFrom (47), it follows thatAs , , obviously, , for . Since the resolvent operator is strongly comparison, we haveUsing above facts, (iv) of Proposition 2 and nonexpansiveness of , (48) becomesSince the multivalued mapping is the relaxed Lipschitz continuous with constant , -Lipschitz continuous with constant , and using (45) of Iterative Algorithm 1, we havethus,where .
Combining (50) and (52), we havethus, we havewhere . Hence, for , we haveIt is clear from condition (46) that , and consequently, we have , as . Thus, is a Cauchy sequence in , and as is complete, , as . From (45) of Iterative Algorithm 1, we havethus, is also a Cauchy sequence in such that , as . Now, we will show that is a solution of generalized variational inequality problem involving XOR operation (12). As , and resolvent operator is continuous, we can writeThus, the relation (35) is satisfied. It remains to show that . Since , we haveHence , as . By Lemma 1, , is a solution of generalized variational inequality problem involving XOR operation (12). This completes the proof.
Remark 2. Combining Theorems 1 and 3, we assert that the solution , of generalized variational inequality involving XOR operation (12) is also a solution of generalized complementarity problem involving XOR operation (9).
5. Numerical Example
In this section, we construct a numerical example in support of Theorem 3. Finally, the convergence graphs and the computation tables are provided for the sequences generated by Iterative Algorithm 1.
Example 1. Let with the usual inner product and norm. Let be a pointed closed convex positive cone in . Let be a functional, be the subdifferential of , be a multivalued mapping, and be the resolvent operator associated with such thatThen,One can easily verify that the resolvent operator is a strongly comparison mapping and continuous.
For , , and , we havethat is,Thus, is the relaxed Lipschitz continuous with constant .
Also,that is,Thus, is the -Lipschitz continuous with constant .
Let us take , then for and , the condition (46)is satisfied.
Furthermore, for and , we obtain the sequences and generated by the Iterative Algorithm 1 aswhere , and thus, . It is clear that the sequence converges to , and consequently, the sequence also converges to .
For initial values , and 15, we have the following convergence graphs, which ensure that the sequences and converge to 0. Two computation tables are provided for the iterations (Tables 1 and 2) of the sequences and (Figures 1, and 2).
6. Conclusion
In this study, we introduce and study a generalized complementarity problem involving XOR operation with three classes of generalized variational inequalities involving XOR operation. Some equivalence relations are established between them. Finally, a generalized variational inequality problem involving XOR operation (12) is solved in real ordered Banach spaces. A numerical example is constructed with convergence graphs and computation tables for illustration of our main result.
We remark that our results may be further extended using other tools of functional analysis.
Data Availability
The data used to support the findings of this study are included within the article.
Conflicts of Interest
The authors declare that they have no conflicts of interest.
Acknowledgments
This work was supported by the National Natural Science Foundation of China (Grant no. 11671365) and the Natural Science Foundation of Zhejiang Province (Grant no. LY14A010011).