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

Authors: Rais Ahmad; Arvind Kumar Rajpoot; Imran Ali; Ching-Feng Wen
Journal: Journal of Mathematics
 (2021)

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## 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.


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## Body

## 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 thatH is a real ordered positive Hilbert Space with its norm ⋅ and inner product ⋅,⋅, C⊆H is a closed convex pointed cone, d is the metric induced by the norm ⋅, C˜BH is the family of nonempty, closed, and bounded subsets of H, and D⋅,⋅ is the Hausdorff metric on C˜BH.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 elementsx and y of a set X are said to be comparable with respect to a binary operation ≤, if at least one of x≤y or y≤x is true. Comparable elements x and y are denoted by x∝y.Definition 3.
A partial order is any binary relation which is reflexive, antisymmetric, and transitive.Definition 4.
Supposelubx,y and glbx,y for the set x,y exist; then, XOR and XNOR operations denoted by ⊕ and ⊙ are defined as follows:(i)
x⊕y=x−y∨y−x(ii)
x⊙y=x−y∧y−x, where x∨y=lubx,y, x∧y=glbx,y, lub means the least upper bound, and glb means the greatest lower boundProposition 1 (see [29]).
Let⊕ be an XOR-operation and ⊙ be an XNOR operation. Then, the following axioms are true:(i)
x⊙x=0,x⊙y=y⊙x=−x⊕y=−y⊕x(ii)
x⊕x=0,x⊕y=y⊕x,0≤x⊕y(iii)
x⊕0=x(iv)
0≤x⊕y,ifx∝y(v)
Ifx∝y, then x⊕y=0 if and only if x=y(vi)
0⊕0=0(vii)
x⊕y≤x−y(viii)
Ifx∝y, then x⊕y=x−yDefinition 5.
LetN:H×H×H⟶H be a single-valued mapping and A:C⟶C˜BH be a multivalued mapping. Then(i)
N is said to be Lipschitz continuous in the first argument if there exists a constant λN1>0 such that(1)Nu1,⋅,⋅−Nu2,⋅,⋅≤λN1u1−u2,∀x1,x2∈C,u1∈Ax1andu2∈Ax2,(ii)
N is said to be Lipschitz continuous in the second argument if there exists a constant λN2>0 such that(2)N⋅,u1,⋅−N⋅,u2,⋅≤λN2u1−u2,∀x1,x2∈C,u1∈Ax1andu2∈Ax2.
Similarly, we can define the Lipschitz continuity ofN in the third argument.Definition 6.
A multivalued mappingA:C⟶C˜BH is said to be D-Lipschitz continuous if for any x,y∈C, there exists a constant λDA>0 such that(3)DAx,Ay≤λDAx−y.Definition 7.
(see [30]). Let ψ:H⟶ℝ∪+∞ be a proper convex functional. A vector w∈H is called subgradient of ψ at x∈domψ, if(4)w,y−x≤ψy−ψx,∀y∈H.
The set of all subgradients ofψ at x is denoted by ∂ψx. The mapping ∂ψ:H⟶2H defined by(5)∂ψx=w∈H:w,y−x≤ψy−ψx,∀y∈His called subdifferential of ψ.Definition 8.
LetP:C⟶C be a mapping and ψ:C⟶ℝ∪+∞ be a proper convex functional. The proximal operator Jρ∂ψ:C⟶C is defined by(6)Jρ∂ψx=P+ρ∂ψ−1x,∀x∈C,where ρ>0 is a constant.Definition 9.
The Yosida approximation operator ofψ is defined by(7)Yρ∂ψx=1ρI−Jρ∂ψx,∀x∈C,where ρ>0 is a constant.Furthermore, we prove some propositions related to proximal operator and Yosida approximation operator.Proposition 2.
LetP:C⟶C and ψ:C⟶ℝ∪+∞ be linear mappings, then the proximal operator Jρ∂ψ is linear. That is(8)αJρ∂ψx=Jρ∂ψαx,provided(9)Jρ∂ψ−1Jρ∂ψx=Jρ∂ψ−1Jρ∂ψx=x,∀x∈Candα>0.Proof.
Using the definition ofJρ∂ψ, linearity of P and ψ, and Theorem 1.48 and Theorem 1.49 of [31], we have(10)αJρ∂ψx=Jρ∂ψJρ∂ψ−1αJρ∂ψx=Jρ∂ψP+ρ∂ψαJρ∂ψx=Jρ∂ψPαJρ∂ψx+ρ∂ψαJρ∂ψx=Jρ∂ψPαJρ∂ψx+ρ∂ψαJρ∂ψx=Jρ∂ψPαJρ∂ψx+ρ∂αψJρ∂ψx=Jρ∂ψαPJρ∂ψx+ρα∂ψJρ∂ψx=Jρ∂ψαP+ρ∂ψJρ∂ψx=Jρ∂ψαJρ∂ψ−1Jρ∂ψx=Jρ∂ψαJρ∂ψ−1Jρ∂ψx=Jρ∂ψαx.Proposition 3.
The Yosida approximation operatorYρ∂ψ:C⟶C is linear, that is,(11)Yρ∂ψαx=αYρ∂ψx,∀x∈C.Proof.
Using the definition ofYρ∂ψ and Proposition 2, we have(12)Yρ∂ψαx=1ρI−Jρ∂ψαx=1ραx−Jρ∂ψαx=1ραx−αJρ∂ψx=αρI−Jρ∂ψx=αYρ∂ψx,∀x∈C.Proposition 4.
The proximal operatorJρ∂ψ:C⟶C is Lipschitz continuous, provided P:C⟶C is strongly monotone with respect to Jρ∂ψ with constant μ>0,ψ is strongly convex with modulus λ>0, and Jρ∂ψ is strongly monotone with constant σ>0, where σ=2λ.Proof.
Letx,y∈C, then(13)Jρ∂ψx=P+ρ∂ψ−1x,(14)Jρ∂ψy=P+ρ∂ψ−1y.
Thus,(15)1ρx−PJρ∂ψx∈∂ψJρ∂ψx,(16)1ρy−PJρ∂ψy∈∂ψJρ∂ψy.
Asψ is strongly convex with modulus λ>0, then the proximal operator Jρ∂ψ is strongly monotone with constant σ>0, where σ=2λ (see [31]). Therefore,(17)σJρ∂ψx−Jρ∂ψy2≤1ρx−PJρ∂ψx−1ρy−PJρ∂ψy,x−y=1ρx−y−PJρ∂ψx−PJρ∂ψy,x−y=1ρx−y,x−y−PJρ∂ψx−PJρ∂ψy,x−y.
SinceP is strongly monotone with respect to Jρ∂ψ with constant μ>0, we have(18)σJρ∂ψx−Jρ∂ψy2≤1ρx−y2−μJρ∂ψx−Jρ∂ψy2,which implies that(19)Jρ∂ψx−Jρ∂ψy≤θx−y,whereθ=1σρ+μ.
That is,Jρ∂ψ is Lipschitz continuous.Proposition 5.
The Yosida approximation operator is strongly monotone if all the conditions of Proposition4 hold.Proof.
Using the Lipschitz continuity of proximal operatorJρ∂ψ, we have(20)Yρ∂ψx−Yρ∂ψy,x−y=1ρI−Jρ∂ψx−1ρI−Jρ∂ψy,x−y=1ρx−y,x−y−Jρ∂ψx−Jρ∂ψy,x−y≥1ρx−y2−Jρ∂ψx−Jρ∂ψyx−y≥1ρx−y2−θx−y2=1−θρx−y2=δyx−y2,whereδy=1−θρ.


## 3. Description of the Problems and Equivalence Lemmas

LetH be a real ordered positive Hilbert space and C⊆H be a closed convex pointed cone. Let A,B,C:C⟶C˜BH be the multivalued mappings and N:H×H×H⟶H be a single-valued mapping. Suppose ψ:C⟶ℝ∪+∞ is a proper, convex functional and Yρ∂ψ:C⟶C is the Yosida approximation operator. We consider the following Yosida complementarity problem involving XOR-operation.Findx∈C,u∈Ax,v∈Bx,w∈Cx such that(21)Nu,v,w,Yρ∂ψx⊕ψYρ∂ψx=0,Nu,v,w,Yρ∂ψy⊕ψYρ∂ψy≥0,∀y∈C.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.Findx∈C,u∈Ax,v∈Bx,w∈Cx such that(22)Nu,v,w,Yρ∂ψy−Yρ∂ψx⊕ψYρ∂ψy−ψYρ∂ψx≥0,∀y∈C.In acquaintance with Yosida variational inequality problem involving XOR-operation (22), we mention the following Yosida proximal operator equation involving XOR-operation.Findx,z∈C,u∈Ax,v∈Bx,w∈Cx such that(23)Nu,v,w⊕ρ−1Rρ∂ψz=0,where ρ>0 is a constant, Rρ∂ψ=I−PJρ∂ψ,Jρ∂ψ is the proximal operator, P:C⟶C is a mapping, and z=PYρ∂ψx+ρNu,v,w.The equivalence between problem (21) and problem (22) and the equivalence between problem (22) and problem (23) are given as follows.Lemma 1.
LetA,B,C:C⟶C˜BH be the multivalued mappings and N:H×H×H⟶H be a single-valued mapping. Suppose ψ:C⟶ℝ∪+∞ is a linear, proper functional. Let Yρ∂ψ:C⟶C be the Yosida approximation operator. If Nu,v,w,Yρ∂ψ∝ψYρ∂ψx, for all x∈C,u∈Ax,v∈Bx,w∈Cx, 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 have(24)Nu,v,w,Yρ∂ψx⊕ψYρ∂ψx=0,Nu,v,w,Yρ∂ψy⊕ψYρ∂ψy≥0,∀x,y∈C,u∈Ax,v∈Bx,w∈Cx.
SinceNu,v,w,Yρ∂ψx∝ψYρ∂ψx, using v of Proposition 1, we have(25)Nu,v,w,Yρ∂ψx=ψYρ∂ψx.
Also,Nu,v,w,Yρ∂ψy⊕ψYρ∂ψy≥0, we have(26)Nu,v,w,Yρ∂ψy⊕ψYρ∂ψy⊕ψYρ∂ψy≥ψYρ∂ψy.
Byii and iii of Proposition 1, we have(27)Nu,v,w,Yρ∂ψy≥ψYρ∂ψy.
Using the properties of inner product, we can write(28)Nu,v,w,Yρ∂ψy−Yρ∂ψx=Nu,v,w,Yρ∂ψy−Nu,v,w,Yρ∂ψx.
Applying (25) and (27), (28) becomes(29)Nu,v,w,Yρ∂ψy−Yρ∂ψx≥ψYρ∂ψy−ψYρ∂ψx,Nu,v,w,Yρ∂ψy−Yρ∂ψx⊕ψYρ∂ψy−ψYρ∂ψx≥0,which is the Yosida variational inequality problem involving XOR-operation (22).
On the other hand, let the Yosida variational inequality problem (22) holds. That is, x∈C,u∈Ax,v∈Bx,w∈Cx such that(30)Nu,v,w,Yρ∂ψy−Yρ∂ψx⊕ψYρ∂ψy−ψYρ∂ψx≥0,∀y∈C.
AsC is a closed convex pointed cone, y=2x∈C as well as y=1/2x∈C. Putting y=2x and y=1/2x and using linearity of ψ and Proposition 3, we have(31)Nu,v,w,Yρ∂ψ2x−Yρ∂ψx⊕ψYρ∂ψ2x−ψYρ∂ψx≥0Nu,v,w,2Yρ∂ψx−Yρ∂ψx⊕ψ2Yρ∂ψx−ψYρ∂ψx≥0Nu,v,w,Yρ∂ψx⊕2ψYρ∂ψx−ψYρ∂ψx≥0Nu,v,w,Yρ∂ψx⊕ψYρ∂ψx≥0,Nu,v,w,Yρ∂ψ12x−Yρ∂ψx⊕ψYρ∂ψ12x−ψYρ∂ψx≥0Nu,v,w,−12Yρ∂ψx⊕−12ψYρ∂ψx≥0Nu,v,w,−12Yρ∂ψx≥−12ψYρ∂ψxNu,v,w,Yρ∂ψx≤ψYρ∂ψx.
Thus, we have(32)Nu,v,w,Yρ∂ψx⊕ψYρ∂ψx≤0.
Adding (31) and (32), we have(33)Nu,v,w,Yρ∂ψx⊕ψYρ∂ψx=0.
Since(34)Nu,v,w,Yρ∂ψy−Yρ∂ψx⊕ψYρ∂ψy−ψYρ∂ψx≥0,we have(35)Nu,v,w,Yρ∂ψy−Nu,v,w,Yρ∂ψx≥ψYρ∂ψy−ψYρ∂ψx.
Using (25), from the above inequality, we have(36)Nu,v,w,Yρ∂ψy−ψYρ∂ψx≥ψYρ∂ψy−ψYρ∂ψx,it follows that(37)Nu,v,w,Yρ∂ψy≥ψYρ∂ψy.
Usingii of Proposition 1, we have(38)Nu,v,w,Yρ∂ψy⊕ψYρ∂ψy≥ψYρ∂ψy⊕ψYρ∂ψy,(39)Nu,v,w,Yρ∂ψy⊕ψYρ∂ψy≥0.
Combination 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.
LetP:C⟶C be a mapping, then the Yosida variational inequality problem involving XOR-operation (22) has a solution x∈C,u∈Ax,v∈Bx,w∈Cx, if and only if it satisfies the equation:(40)Yρ∂ψx=Jρ∂ψPYρ∂ψx+ρNu,v,w,where ρ>0 is a constant.Proof.
Letx∈C,u∈Ax,v∈Bx,w∈Cx satisfy equation (40), that is,(41)Yρ∂ψx=Jρ∂ψPYρ∂ψx+ρNu,v,w.
Using the definition of the proximal operatorJρ∂ψ and from the above equation, we have(42)Yρ∂ψx=P+ρ∂ψ−1PYρ∂ψx+ρNu,v,wPYρ∂ψx+ρ∂ψYρ∂ψx=PYρ∂ψx+ρNu,v,wthat is,Nu,v,w∈∂ψYρ∂ψx.which gives us
Applying the definition of subdifferential operator, the above inclusion holds if and only if(43)ψYρ∂ψy−ψYρ∂ψx≥Nu,v,w,ψYρ∂ψy−ψYρ∂ψx.
Usingii of Proposition 1, we have(44)Nu,v,w,ψYρ∂ψy−ψYρ∂ψx⊕ψYρ∂ψy−ψYρ∂ψx≥Nu,v,w,ψYρ∂ψy−ψYρ∂ψx⊕Nu,v,w,ψYρ∂ψy−ψYρ∂ψx.
It follows that(45)Nu,v,w,Yρ∂ψy−Yρ∂ψx⊕ψYρ∂ψy−ψYρ∂ψx≥0,∀y∈C,which 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.
SupposeNu,v,w∝Rρ∂ψz and P:C⟶C is a one-one mapping. Then x∈C,u∈Ax,v∈Bx,w∈Cx is the solution of the Yosida variational inequality problem involving XOR-operation (22) if and only if x,z∈C,u∈Ax,v∈Bx,w∈Cx satisfy the Yosida proximal operator equation involving XOR-operation (23), where Rρ∂ψ=I−PJρ∂ψ, in which Jρ∂ψ is the proximal operator and PJρ∂ψz=PJρ∂ψz.Proof.
Letx∈C,u∈Ax,v∈Bx,w∈Cx be the solution of the Yosida variational inequality problem involving XOR-operation (22). Then by Lemma 2, it satisfies the equation:(46)Yρ∂ψx=Jρ∂ψPYρ∂ψx+ρNu,v,w.
Letz=PYρ∂ψx+ρNu,v,w, then(47)Yρ∂ψx=Jρ∂ψz,Asz=PJρ∂ψz+ρNu,v,wz−PJρ∂ψz=ρNu,v,wI−PJρ∂ψz=ρNu,v,wRρ∂ψz=ρNu,v,wρ−1Rρ∂ψz=Nu,v,w,wherePJρ∂ψz=PJρ∂ψz.
Usingii of Proposition 1, we have(48)Nu,v,w⊕ρ−1Rρ∂ψz=Nu,v,w⊕Nu,v,w=0.
Thus, we have(49)Nu,v,w⊕ρ−1Rρ∂ψz=0,which is the required Yosida proximal operator equation involving XOR-operation (23).
Conversely, letx,z∈C,u∈Ax,v∈Bx,w∈Cx be the solution of Yosida proximal operator equation involving XOR-operation (23).
That is, we have(50)Nu,v,w⊕ρ−1Rρ∂ψz=0.
Usingv of Proposition 1, definition of Rρ∂ψ and comparability of Nu,v,w with Rρ∂ψz, we obtain(51)ρNu,v,w=Rρ∂ψz=I−PJρ∂ψz=z−PJρ∂ψz=PYρ∂ψx+ρNu,v,w−PJρ∂ψPYρ∂ψx+ρNu,v,w.
From above, we have(52)PYρ∂ψx=PJρ∂ψPYρ∂ψx+ρNu,v,w.
SinceP is a one-one mapping, we obtain(53)Yρ∂ψx=Jρ∂ψPYρ∂ψx+ρNu,v,w.
Applying Lemma2, we conclude that x∈C,u∈Ax,v∈Bx,w∈Cx is the solution of Yosida variational inequality problem involving XOR-operation (22).


## 4. Algorithm and Existence Results

Invoking Lemmas2 and 3, we suggest the following algorithm for solving Yosida proximal operator equation involving XOR-operation (23).Algorithm 1.
For anyx0,z0∈C,u0∈Ax0,v0∈Bx0,w0∈Cx0, we let(54)z1=PYρ∂ψx0+ρNu0,v0,w0.
Take anyx1∈C such that(55)Yρ∂ψx1=Jρ∂ψz1.
Sinceu0∈Ax0,v0∈Bx0,w0∈Cx0, by Nadler’s theorem [38], there exist u1∈Ax1,v0∈Bx1, w0∈Cx1, using viii of Proposition 1 and comparability of u0,u1;v0,v1andw0,w1, we have(56)u0⊕u1=u0−u1≤1+1DAYρ∂ψx0,AYρ∂ψx1,v0⊕v1=v0−v1≤1+1DBYρ∂ψx0,BYρ∂ψx1,w0⊕w1=w0−w1≤1+1DCYρ∂ψx0,CYρ∂ψx1,where D⋅,⋅ is the Hausdorff metric on C˜BH.
Letz2=PYρ∂ψx1+ρNu1,v1,w1 and take any x2∈C such that(57)Yρ∂ψx2=Jρ∂ψz2.
Continuing the above procedure, we compute the sequencesxn,un,vn and zn by the schemes given below:(58)Yρ∂ψxn=Jρ∂ψzn,un∈Axn,un+1∈Axn+1such thatun∝un+1,un⊕un+1=un−un+1≤1+1n+1DAYρ∂ψxn,AYρ∂ψxn+1,vn∈Bxn,vn+1∈Bxn+1such thatvn∝vn+1,vn⊕vn+1=vn−vn+1≤1+1n+1DBYρ∂ψxn,BYρ∂ψxn+1,wn∈Cxn,wn+1∈Cxn+1such thatwn∝wn+1,wn⊕wn+1=wn−wn+1≤1+1n+1DCYρ∂ψxn,CYρ∂ψxn+1,zn+1=PYρ∂ψxn+ρNun,vn,wn,where ρ>0 is a constant and n=0,1,2,3,⋯.Theorem 1.
LetH be a real ordered positive Hilbert Space and C⊆H be a closed convex pointed cone. Let A,B,C:C⟶C˜BH be the D-Lipschitz continuous mappings with constants λDA,λDB, and λDC, respectively. Let N:H×H×H⟶H be a single-valued mapping such that N is Lipschitz continuous in first, second, and third arguments with constants λN1,λN2, and λN3, respectively. Let Yρ∂ψ:C⟶C be the Yosida approximation operator such that Yρ∂ψ is strongly monotone with constant δy and Jρ∂ψ:C⟶C be the proximal operator such that Jρ∂ψ is Lipschitz continuous with constant θ. Suppose P:C⟶C be a Lipschitz continuous mapping with constant λP, strongly monotone with respect to Jρ∂ψ with constant μ and ψ:H⟶ℝ∪+∞ be a strongly convex, subdifferentiable, proper functional satisfying Yρ∂ψx∈dom∂ψ. Suppose that zn+1∝zn, for n=0,1,2,⋯ and if the following condition is satisfied:(59)θ<2δy,ξθ<1−λPθρθ,whereξθ=λN1λDA+λN2λDB+λN3λDCandθ=1σρ+μ,then there exists x,z∈C,u∈Ax,v∈Bx,w∈Cx satisfying the Yosida proximal operator equation involving XOR-operation (23) and the sequences xn,zn,un,vn and wn generated by Algorithm 1 converge strongly to x,z,u,v, and w, respectively.Proof.
Using (x) of Algorithm1 and ii of Proposition 1, we have(60)0≤zn+1⊕zn=PYρ∂ψxn+ρNun,vn,wn⊕PYρ∂ψxn+ρNun−1,vn−1,wn−1=PYρ∂ψxn⊕PYρ∂ψxn−1+ρNun,vn,wn⊕Nun−1,vn−1,wn−1.
It follows from (60) that(61)zn+1⊕zn≤PYρ∂ψxn⊕PYρ∂ψxn−1+ρNun,vn,wn⊕Nun−1,vn−1,wn−1.
Sincezn+1∝zn and using vii and viii of Proposition 1, from (61), we obtain(62)zn+1−zn≤PYρ∂ψxn−PYρ∂ψxn−1+ρNun,vn,wn−Nun−1,vn−1,wn−1.
SinceN is Lipschitz continuous in all the three arguments with constants λN1,λN2, and λN3, respectively, and A,B,C are D-Lipschitz continuous mappings with constants λDA,λDB,λDC, respectively, and using vii,viii,ix of Algorithm 1, we have(63)Nun,vn,wn−Nun−1,vn−1,wn−1=Nun,vn,wn−Nun−1,vn,wn+Nun−1,vn,wn−Nun−1,vn−1,wn+Nun−1,vn−1,wn−Nun−1,vn−1,wn−1≤λN1un−un−1+λN2vn−vn−1λN3wn−wn−1≤λN11+1nDAYρ∂ψxn,AYρ∂ψxn−1+λN21+1nDBYρ∂ψxn,BYρ∂ψxn−1+λN31+1nDCYρ∂ψxn,CYρ∂ψxn−1≤λN11+1nλDAYρ∂ψxn−Yρ∂ψxn−1+λN21+1nλDBYρ∂ψxn−Yρ∂ψxn−1+λN31+1nλDCYρ∂ψxn−Yρ∂ψxn−1=λN1λDA+λN2λDB+λN3λDC1+1nYρ∂ψxn−Yρ∂ψxn−1.
Using strong monotonicity of the Yosida approximation operatorYρ∂ψ with constant δY and Lipschitz continuity of the proximal operator Jρ∂ψ with constant θ, we have(64)xn−xn−12=Jρ∂ψzn−Jρ∂ψzn−1−Yρ∂ψxn−xn−Yρ∂ψxn−1−xn−12≤Jρ∂ψzn−Jρ∂ψzn−12−2Yρ∂ψxn−Yρ∂ψxn−1,xn−xn−1+xn−xn−12≤θ2zn−zn−12−2δyxn−xn−12+xn−xn−12.
It follows that(65)2δyxn−xn−12≤θ2zn−zn−12,xn−xn−1≤θ22δyzn−zn−1,xn−xn−1≤ξyzn−zn−1,where ξy=θ2/2δy.
Combining (62) and (63), using Lipschitz continuity of P,Jρ∂ψ and vi of Algorithm 1, we have(66)zn+1−zn≤λPYρ∂ψxn−Yρ∂ψxn−1+ρλN1λDA+λN2λDB+λN3λDC1+1nYρ∂ψxn−Yρ∂ψxn−1,(67)zn+1−zn≤λP+ρξnθYρ∂ψxn−Yρ∂ψxn−1≤λP+ρξnθJρ∂ψzn−Jρ∂ψzn−1=λP+ρξnθθzn−zn−1=λPθ+ρξnθθzn−zn−1=Snθzn−zn−1,where Snθ=λPθ+ρξnθθ,θ=1/σρ+μ and ξnθ=λN1λDA+λN2λDB+λN3λDC1+1/n.
LettingSθ=λPθ+ρξθθ, where ξθ=λN1λDA+λN2λDB+λN3λDC, it follows that Snθ⟶Sθ as n⟶∞. From (59), we have ξy<1 and Sθ<1. Consequently, we conclude from (65) and (67) that xn and zn both are Cauchy sequences. Since H is complete and C⊆H is a closed convex subset of H and thus C is also complete, we may assume that xn⟶x∈C and zn⟶z∈C. From vii,viii, and ix of Algorithm 1, it follows that un,vn, and wn are also Cauchy sequences such that un⟶u,vn⟶v and wn⟶w, as n⟶∞.
It can be shown easily by using the techniques of [28] that u∈Ax,v∈Bx, and w∈Cx. By Lemma 3, we conclude that x,z∈C,u∈Ax,v∈Bx, and w∈Cx 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 Algorithm1 and Theorem 1.Example 1.
SupposeC=H=0,∞. Let P:C⟶C and ψ:C⟶ℝ∪+∞ be the mappings such that for x∈C,(68)Px=x2,ψx=x2,Then∂ψx=2x,the sub differential ofψ.
Sinceψ′′x=2>0. Hence, ψ is strongly convex with modulus λ=2.
Forρ=1, the proximal operator Jρ∂ψ is given by(69)Jρ∂ψ=P+ρ∂ψ−1x=2x5,whereP+ρ∂ψx=5x2.
It is simple to see thatP is Lipschitz continuous with constant λp=11/10, strongly monotone with respect to Jρ∂ψ with constant μ=1/3, and Jρ∂ψ is Lipschitz continuous with constant θ=2/3.
In view of proximal operator calculated above, the Yosida approximation operator is given by(70)Yρ∂ψx=1ρI−Jρ∂ψx=3x5.
Also,(71)Yρ∂ψx−Yρ∂ψy,x−y=3x5−3y5,x−y=35x−y,x−y=35x−y,x−y=35x−y2≥25x−y2.
Hence,Yρ∂ψ is strongly monotone with constant δy=2/5.
Let us consider the mappingsN:H×H×H⟶H and A,B,C:C⟶C˜BH such that(72)Ax=x7,Bx=x5,Cx=x6,Nu,v,w=u2+v2+w2,where x∈C,u∈Ax,v∈Bx,andw∈Cx.(73)Clearly,DAx,Ay=maxsupx∈Axdx,Fy,supy∈FydFx,y≤maxx7−y7,y7−x7=17maxx−y,y−x≤15x−y,that is, DAx,Ay≤1/5x−y.
Thus,A is D-Lipschitz continuous with constant λDA=1/5. Similarly, we can obtain that B and C are D-Lipschitz continuous with constants λDB=1/3 and λDC=1/3, respectively.
N is Lipschitz continuous in all the three arguments with constants λN1=λN2=λN3=1.(74)Then,Nu,v,w=x14+x10+x12=107x420,Sincez=PYρ∂ψx+ρNu,v,w=Yρ∂ψx2+Nu,v,w=12⋅3x5+Nu,v,w,z=3x10+Nu,v,w,Jρ∂ψz=6x50+2Nu,v,w5.
Hence,(75)Rρ∂ψz=I−Jρ∂ψz=353x10+Nu,v,w.
Below we show that condition (59) is satisfied.
ForλP=11/10,ρ=1,δy=2/5,μ=1/3,λ=2,σ=2λ=4,θ=2/3, λN1=λN2=λN3=1,λDA=1/5,λDB=1/3,λDC=1/3,ξθ=λN1λDA+λN2λDB+λN3λDC=0.86 and 1−λPθ/ρθ=1.021. Hence, θ<2δy and ξθ<1−λPθ/ρθ. That is, condition (59) is satisfied.
For,x=0, the Yosida proximal operator equation involving XOR-operator (23) is fulfilled.(76)That is,Nu,v,w⊕ρ−1Rρ∂ψz=Nu,v,w⊕353x10+Nu,v,w=0.
Furthermore, we obtain the sequencesxn and zn generated by iterative Algorithm 1 as(77)zn+1=PYρ∂ψxn+ρNun,vn,wn=Yρ∂ψxn2+107xn420=12⋅3xn5+107xn420=3xn10+107xn420,also,Y∂ψρxn=Jρ∂ψzn3xn5=2zn5xn=2zn5.
From (77) and (78), we have(78)zn+1=310+1074202zn5,zn+1=2331050zn.
Clearly, the sequencezn converges to 0, and consequently, the sequence xn also converges to 0.
It is shown in Figures1–3 that, for initial values z0=−5,2.5, and 5, the sequence zn converges to 0. A consolidated graph using Figures 1–3 is provided in Figure 4. In Table 1, comparing different initial values of zn and for different iterations, it is obtained that the sequence zn converges to 0.Figure 1
Case I: for the initial valuez0=−5.Figure 2
Case II: for the initial valuez0=2.5.Figure 3
Case III: for the initial valuez0=5.Figure 4
By combining all the above graphs, we get the following conjoining graph of convergence.Table 1
The values ofzn with initial values z0=−5, z0=2.5, and z0=5.
No. of iterationsForz0=−5Forz0=2.5Forz0=5znznznn = 1−52.55n = 2−1.109523809523810.5547619047619051.10952380952381n = 3−0.2462086167800450.1231043083900230.246208616780045n = 4−0.05463486448547670.02731743224273840.0546348644854767n = 5−0.01212373659534860.006061868297674320.0121237365953486n = 10−6.52333379900596e − 063.26166689950298e − 066.52333379900596e − 06n = 15−3.50996440070956e − 091.75498220035478e − 093.50996440070956e − 09n = 20−1.88858189291582e − 129.44290946457911e − 131.88858189291582e − 12n = 25−1.01617599469911e − 155.08087997349556e − 161.01617599469911e − 15n = 27−5.00382572119910e − 172.50191286059955e − 175.00382572119910e − 17n = 29−2.46397001884969e − 181.23198500942484e − 182.46397001884969e − 18n = 30−5.46766680373311e − 192.73383340186656e − 195.46766680373311e − 19


## 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.



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