Yosida Complementarity Problem with Yosida Variational Inequality Problem and Yosida Proximal Operator Equation Involving XOR-Operation

Ahmad, Rais; Rajpoot, Arvind Kumar; Ali, Imran; Wen, Ching-Feng

doi:https://doi.org/10.1155/2021/9362630

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Fixed Points and Computational Optimization

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Volume 2021 | Article ID 9362630 | https://doi.org/10.1155/2021/9362630

Yosida Complementarity Problem with Yosida Variational Inequality Problem and Yosida Proximal Operator Equation Involving XOR-Operation

Rais Ahmad,¹Arvind Kumar Rajpoot,¹Imran Ali,²and Ching-Feng Wen^3,4

Academic Editor: Antonio Di Crescenzo

Received03 Sept 2021

Revised07 Oct 2021

Accepted01 Nov 2021

Published27 Nov 2021

Abstract

Due to the importance of Yosida approximation operator, we generalized the variational inequality problem and its equivalent problems by using Yosida approximation operator. The aim of this work is to introduce and study a Yosida complementarity problem, a Yosida variational inequality problem, and a Yosida proximal operator equation involving XOR-operation. We prove an existence result together with convergence analysis for Yosida proximal operator equation involving XOR-operation. For this purpose, we establish an algorithm based on fixed point formulation. Our approach is based on a proximal operator technique involving a subdifferential operator. As an application of our main result, we provide a numerical example using the MATLAB program R2018a. Comparing different iterations, a computational table is assembled and some graphs are plotted to show the convergence of iterative sequences for different initial values.

1. Introduction

Stampacchia [1] and Ficchera [2] originated the study of variational inequalities, separately. Variational inequalities are mathematical models for many problems occurring in physics, engineering sciences, transportation planning, financial problems, and in many industrial strategies, etc. (see, for example, [3–11]). In 1968, Cottle and Dantzig [12] proposed linear complementarity problem which appear continually in computational mechanics. It is interesting to note that finding the solution of linear complementarity problem is associated with minimizing some quadratic function. However, in 1964, Cottle [13] in his Ph. D thesis introduced nonlinear complementarity problem which is closely related to Hartman and Stampacchia variational inequality problem. The proximal operator technique is useful to establish equivalence between variational inequalities and proximal operator equations. The proximal operator equation approach is used to solve variational inequalities and related optimization problems.

XOR is a logical operation and represents the inequality function, that is, the output is true if the inputs are not alike; otherwise, the output is false. An easy way to remember XOR is “must have one or the other but not both.” It is important to note that XOR does not leak information about the original plain text. The inner XOR is the encryption and the outer XOR is the decryption, that is, the exact XOR function can be used for both encryption and decryption. Consider a string of binary digits 10101 and XOR the string 10111 with it to get 00010. That is, the original string is encoded and the second string becomes key; if we XOR our key with our encoded string, we get our original string back. XOR allows to easily encrypt and decrypt a string; the other logical operations do not.

The possible strategy of solving stochastic notion of multivalued differential equation in finite dimensional space is based on Yosida approximation approach. The existence of multivalued stochastic differential equation in finite dimensional space with a time-independent, deterministic maximal monotone operator through Yosida approximation approach was first discussed by Petterson [14]. Yosida approximation operators are used to solve wave equations, heat equations, etc. For more details and recent past developments about complementarity problems, variational inequalities, proximal operator equations, Yosida approximation operator, and related topics, we refer to [15–28] and references therein.

Motivated by all the above discussed concepts, in this paper, we consider and study a Yosida complementarity problem, a Yosida variational inequality problem, and a Yosida proximal operator equation involving XOR-operation. Some equivalence results are proved. To obtain the solution of Yosida proximal operator equation involving XOR-operation, we define an algorithm based on fixed point formulation. Convergence criteria are also discussed. In support of our main result, an example is provided using MATLAB program R2018a. A comparison of different iterations is assembled in the form of a computational table, and the convergence of the iterative sequences is shown by some graphs for different initial values.

2. Preliminaries and Basic Results

We suppose that is a real ordered positive Hilbert Space with its norm and inner product , is a closed convex pointed cone, is the metric induced by the norm , is the family of nonempty, closed, and bounded subsets of , and is the Hausdorff metric on .

The following definitions, concepts, and results are required for the presentation of this paper.

Definition 1. A convex cone is a subset of a vector space over an ordered field that is closed under linear combinations with positive coefficients.

Definition 2. Two elements and of a set are said to be comparable with respect to a binary operation , if at least one of or is true. Comparable elements and are denoted by .

Definition 3. A partial order is any binary relation which is reflexive, antisymmetric, and transitive.

Definition 4. Suppose and for the set exist; then, XOR and XNOR operations denoted by and are defined as follows:(i)(ii), where , , means the least upper bound, and means the greatest lower bound

Proposition 1 (see [29]). Let be an XOR-operation and be an XNOR operation. Then, the following axioms are true:(i)(ii)(iii)(iv)(v)If , then if and only if (vi)(vii)(viii)If , then

Definition 5. Let be a single-valued mapping and be a multivalued mapping. Then(i) is said to be Lipschitz continuous in the first argument if there exists a constant such that(ii) is said to be Lipschitz continuous in the second argument if there exists a constant such thatSimilarly, we can define the Lipschitz continuity of in the third argument.

Definition 6. A multivalued mapping is said to be -Lipschitz continuous if for any , there exists a constant such that

Definition 7. (see [30]). Let be a proper convex 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 8. Let be a mapping and be a proper convex functional. The proximal operator is defined bywhere is a constant.

Definition 9. The Yosida approximation operator of is defined bywhere is a constant.

Furthermore, we prove some propositions related to proximal operator and Yosida approximation operator.

Proposition 2. Let and be linear mappings, then the proximal operator is linear. That isprovided

Proof. Using the definition of , linearity of and , and Theorem 1.48 and Theorem 1.49 of [31], we have

Proposition 3. The Yosida approximation operator is linear, that is,

Proof. Using the definition of and Proposition 2, we have

Proposition 4. The proximal operator is Lipschitz continuous, provided is strongly monotone with respect to with constant is strongly convex with modulus , and is strongly monotone with constant , where .

Proof. Let , thenThus,As is strongly convex with modulus , then the proximal operator is strongly monotone with constant , where (see [31]). Therefore,Since is strongly monotone with respect to with constant , we havewhich implies thatThat is, is Lipschitz continuous.

Proposition 5. The Yosida approximation operator is strongly monotone if all the conditions of Proposition 4 hold.

Proof. Using the Lipschitz continuity of proximal operator , we have

3. Description of the Problems and Equivalence Lemmas

Let be a real ordered positive Hilbert space and be a closed convex pointed cone. Let be the multivalued mappings and be a single-valued mapping. Suppose is a proper, convex functional and is the Yosida approximation operator. We consider the following Yosida complementarity problem involving XOR-operation.

Find such that

From problem (21), one can easily obtain the complementarity problems studied by Huang et al. [32], Yin and Xu [33], Flores-Bazán and López [34], Isac [35, 36] and Farajzadeh and Harandi [37], etc.

In connection with Yosida complementarity problem involving XOR-operation (21), we mention the following Yosida variational inequality problem involving XOR-operation.

Find such that

In acquaintance with Yosida variational inequality problem involving XOR-operation (22), we mention the following Yosida proximal operator equation involving XOR-operation.

Find such thatwhere is a constant, is the proximal operator, is a mapping, and .

The equivalence between problem (21) and problem (22) and the equivalence between problem (22) and problem (23) are given as follows.

Lemma 1. Let be the multivalued mappings and be a single-valued mapping. Suppose is a linear, proper functional. Let be the Yosida approximation operator. If , for all , then the Yosida complementarity problem involving XOR-operation (21) and the Yosida variational inequality problem involving XOR-operation (22) are equivalent.

Proof. Let the Yosida complementarity problem involving XOR-operation (21) holds. We haveSince , using of Proposition 1, we haveAlso, , we haveBy and of Proposition 1, we haveUsing the properties of inner product, we can writeApplying (25) and (27), (28) becomeswhich is the Yosida variational inequality problem involving XOR-operation (22).
On the other hand, let the Yosida variational inequality problem (22) holds. That is, such thatAs is a closed convex pointed cone, as well as . Putting and and using linearity of and Proposition 3, we haveThus, we haveAdding (31) and (32), we haveSincewe haveUsing (25), from the above inequality, we haveit follows thatUsing of Proposition 1, we haveCombination of (33) and (39) is the required Yosida complementarity problem involving XOR-operation (21).

The following Lemma guarantees the equivalence between the Yosida variational inequality problem involving XOR-operation (22) and a fixed point equation.

Lemma 2. Let be a mapping, then the Yosida variational inequality problem involving XOR-operation (22) has a solution , if and only if it satisfies the equation:where is a constant.

Proof. Let satisfy equation (40), that is,Using the definition of the proximal operator and from the above equation, we haveApplying the definition of subdifferential operator, the above inclusion holds if and only ifUsing of Proposition 1, we haveIt follows thatwhich is the required Yosida variational inequality problem involving XOR-operation (22).

The Lemma mentioned below ensures the equivalence between the Yosida variational inequality problem involving XOR-operation (22) and the Yosida proximal operator equation involving XOR-operation (23).

Lemma 3. Suppose and is a one-one mapping. Then is the solution of the Yosida variational inequality problem involving XOR-operation (22) if and only if satisfy the Yosida proximal operator equation involving XOR-operation (23), where , in which is the proximal operator and .

Proof. Let be the solution of the Yosida variational inequality problem involving XOR-operation (22). Then by Lemma 2, it satisfies the equation:Let thenUsing of Proposition 1, we haveThus, we havewhich is the required Yosida proximal operator equation involving XOR-operation (23).
Conversely, let be the solution of Yosida proximal operator equation involving XOR-operation (23).
That is, we haveUsing of Proposition 1, definition of and comparability of with , we obtainFrom above, we haveSince is a one-one mapping, we obtainApplying Lemma 2, we conclude that is the solution of Yosida variational inequality problem involving XOR-operation (22).

4. Algorithm and Existence Results

Invoking Lemmas 2 and 3, we suggest the following algorithm for solving Yosida proximal operator equation involving XOR-operation (23).

Algorithm 1. For any , we letTake any such thatSince , by Nadler’s theorem [38], there exist , , using of Proposition 1 and comparability of , we havewhere is the Hausdorff metric on .
Let and take any such thatContinuing the above procedure, we compute the sequences and by the schemes given below:where is a constant and .

Theorem 1. Let be a real ordered positive Hilbert Space and be a closed convex pointed cone. Let be the -Lipschitz continuous mappings with constants , and respectively. Let be a single-valued mapping such that is Lipschitz continuous in first, second, and third arguments with constants , and , respectively. Let be the Yosida approximation operator such that is strongly monotone with constant and be the proximal operator such that is Lipschitz continuous with constant . Suppose be a Lipschitz continuous mapping with constant , strongly monotone with respect to with constant and be a strongly convex, subdifferentiable, proper functional satisfying . Suppose that , for and if the following condition is satisfied:then there exists satisfying the Yosida proximal operator equation involving XOR-operation (23) and the sequences and generated by Algorithm 1 converge strongly to , and , respectively.

Proof. Using (x) of Algorithm 1 and of Proposition 1, we haveIt follows from (60) thatSince and using and of Proposition 1, from (61), we obtainSince is Lipschitz continuous in all the three arguments with constants , and , respectively, and are -Lipschitz continuous mappings with constants , respectively, and using of Algorithm 1, we haveUsing strong monotonicity of the Yosida approximation operator with constant and Lipschitz continuity of the proximal operator with constant , we haveIt follows thatwhere .
Combining (62) and (63), using Lipschitz continuity of and of Algorithm 1, we havewhere and .
Letting , where , it follows that as . From (59), we have and . Consequently, we conclude from (65) and (67) that and both are Cauchy sequences. Since is complete and is a closed convex subset of and thus is also complete, we may assume that and . From , and of Algorithm 1, it follows that , and are also Cauchy sequences such that and , as .
It can be shown easily by using the techniques of [28] that , and . By Lemma 3, we conclude that , and is the solution of Yosida proximal operator equation involving XOR-operation (23).

We provide the following numerical example using MATLAB program R2018a along with a computational table and a convergence graphs for different initial values in support of Algorithm 1 and Theorem 1.

Example 1. Suppose . Let and be the mappings such that for ,Since Hence, is strongly convex with modulus .
For , the proximal operator is given byIt is simple to see that is Lipschitz continuous with constant , strongly monotone with respect to with constant , and is Lipschitz continuous with constant .
In view of proximal operator calculated above, the Yosida approximation operator is given byAlso,Hence, is strongly monotone with constant .
Let us consider the mappings and such thatwhere .that is, .
Thus, A is D-Lipschitz continuous with constant . Similarly, we can obtain that B and C are D-Lipschitz continuous with constants and , respectively.
N is Lipschitz continuous in all the three arguments with constants .Hence,Below we show that condition (59) is satisfied.
For , and . Hence, and . That is, condition (59) is satisfied.
For, , the Yosida proximal operator equation involving XOR-operator (23) is fulfilled.Furthermore, we obtain the sequences and generated by iterative Algorithm 1 asFrom (77) and (78), we haveClearly, the sequence converges to 0, and consequently, the sequence also converges to 0.
It is shown in Figures 1–3 that, for initial values , and 5, the sequence converges to 0. A consolidated graph using Figures 1–3 is provided in Figure 4. In Table 1, comparing different initial values of and for different iterations, it is obtained that the sequence converges to 0.

5. Conclusion

In this work, we introduce and study three new problems, that is, a Yosida complementarity problem, a Yosida variational inequality problem, and a Yosida proximal operator equation involving XOR-operation. It is shown that Yosida complementarity problem involving XOR-operation is equivalent to a Yosida variational inequality problem involving XOR-operation and Yosida variational inequality problem involving XOR-operation is equivalent to a Yosida proximal operator equation involving XOR-operation. An algorithm is established to obtain the solution of Yosida proximal operator equation involving XOR-operation. Finally, an existence and convergence result is proved. A numerical example is given in support of our main result.

It is still an open and interesting problem that how to establish equivalence between Yosida complementarity problem involving XOR-operation and Yosida proximal operator equation problem involving XOR-operation.

Data Availability

No data were used to support this study.

Disclosure

A variant form of Yosida variational inequality and Yosida proximal operator equation involving XOR-operation was considered in “Some Problems Concerning Generalized Variational Inequalities”, Ph. D Thesis, (2009), AMU Aligarh [39]. In this variant form, neither the concept of Yosida approximation operator nor the concept of XOR-operation was used. Moreover, no complementarity problem was considered in the abovementioned thesis.

Conflicts of Interest

The authors declare that they have no conflicts of interest.

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