# Some Proximal Methods for Solving Mixed Variational Inequalities

Authors: Muhammad Aslam Noor; Zhenyu Huang
Journal: Abstract and Applied Analysis
 (2012)

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## Abstract

It is well known that the mixed variational inequalities are equivalent to the fixed point problem. We use this alternative  equivalent formulation to suggest some new proximal  point methods  for solving the mixed variational inequalities. These new methods include the explicit, the implicit, and the extragradient method as special cases. The convergence analysis of these new methods is considered  under some suitable conditions. Our method of constructing these iterative methods is very  simple. Results proved in this  paper may stimulate further research in this direction.


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## Body

## 1. Introduction

Variational inequalities are being used to study a wide class of diverse unrelated problems arising in various branches of pure and applied sciences in a unified framework. Several extensions and generalizations of the variational inequalities have been considered using novel and innovative techniques. A useful and significant generalization of the variational inequalities is known as the mixed variational inequality involving a nonlinear term. Due to the presence of the nonlinear term, the projection and its variant forms cannot be applied to establish the equivalence between the mixed variational inequalities and the fixed point problem. It is well known that the nonlinear term in the mixed variational inequality is a proper, convex, and lower semicontinuous, then one can establish the equivalence between the mixed variational inequality and the fixed point problem. This equivalence has been used to study the existence of a solution of the mixed variational inequalities as well as to develop numerical methods. We use this alternative equivalent formulation to suggest and analyze a wide class of proximal point methods for solving the mixed variational inequalities, which includes the implicit and explicit resolvent method as special cases. This is the main motivation of this paper. We also consider its convergence criteria under suitable conditions. Our method of constructing these methods is very simple as compared with other methods. We have only included very iterative methods. Readers are encouraged to construct other methods using the technique of this paper for solving other kinds of variational inequalities and related optimization problems. We hope that the ideas and techniques of this paper may stimulate further research in this area of pure and applied sciences.


## 2. Preliminaries

LetH be a real Hilbert space, whose inner product and norm are denoted by 〈·,·〉 and ∥·∥, respectively. Let K be a nonempty, closed, and convex set in H, and let φ(·):H→R∪{+∞} be a continuous function.For a given operatorT:H→H, we consider the problem of finding u∈H such that(2.1)〈Tu,v-u〉+φ(v)-φ(u)≥0,∀v∈H,
which is called the mixed variational inequality or variational inequality of the second kind.For the applications, formulations, numerical methods, and other aspects of the mixed variational inequalities, see [1–15] and the references therein.If the operatorT is linear, positive, and symmetric and the function φ(·) is convex function, then the minimum of the functional I[v], defined as(2.2)I[v]=〈Tv,v〉+gφ(v)
on the convex set, can be characterized by the mixed variational inequality (2.1).If the functionφ(·) is proper, convex, and semicontinuous, then problem (2.1) is equivalent to find u∈H such that(2.3)0∈Tu+∂φ(u),
where ∂φ is the subdifferential. The problem (2.3) is called the variational inclusion problem or problem of finding the zero of the sum of two (more) monotone operators. It is known that a wide class of problems with applications in industry, physical, regional and engineering sciences can be studied by the problems (2.1) and (2.3), see, for example, [1–15] and the references therein.If the functionφ(·) is the indictor function of a closed and convex set K in the Hilbert space H, then problem (2.1) is equivalent to finding u∈K such that(2.4)〈Tu,v-u〉≥0,∀v∈K,
which is known as the classical variational inequality, introduced and studied by Stampacchia [15] in 1964. For the applications, formulations, generalizations, numerical results, and other aspects of the variational inequalities, see [1–15].We now recall some well-known results and concepts.Definition 2.1 (see [1]).
LetT be a maximal monotone operator, then the resolvent operator associated with T is defined as
(2.5)JT(u)=(I+ρT)-1(u),∀u∈H,
where ρ>0 is a constant and I is the identity operator.It is well known that the subdifferential∂φ(·) is a maximal monotone operator, and we can define its resolvent as
(2.6)Jφ(u)=(I+ρ∂φ)-1(u).
The resolvent operator Jφ defined by (2.6) has the following useful characterization.Lemma 2.2 (see [1]).
For a givenz∈H,u∈H satisfies the inequality
(2.7)〈u-z,v-u〉+ρφ(v)-ρφ(u)≥0,∀v∈H,
if and only if
(2.8)u=Jφz,
where Jφ(u)=(I+ρ∂φ)-1(u) is the resolvent operator.It is well known that the resolvent operatorJφ is nonexpansive, that is,
(2.9)‖Jφu-Jφv‖≤‖u-v‖,∀u,v∈H.
Using Lemma 2.2, one can easily show that the mixed variational inequality (2.1) is equivalent to finding u∈H such that
(2.10)u=Jφ[u-ρTu],
where ρ>0 is a constant.Lemma2.2 implies that variational inequality (2.1) and the fixed point problem (2.10) are equivalent. This alternative equivalent formulation has played a central role in the development of the mixed variational inequality theory. This equivalence formulation has been extensively used to develop several iterative methods for solving the variational inequalities, see, for example, [1–15] and the references therein.Definition 2.3.
An operatorT:H→H is said to be partially relaxed strongly monotone, if and only if, there exists a constant α>0 such that
(2.11)〈Tu-Tv,z-v〉≥α‖z-v‖2,∀u,v,z∈H
and pseudomontone with respect to the function φ if and only if
(2.12)〈Tu,v-u〉+φ(v)-φ(u)≥0⟹〈Tv,v-u〉+φ(v)-φ(u)≥0,∀u,v∈H.Definition 2.4.
An operatorT:H→H, is said to be partially relaxed strongly pseudomonotone if the operator is both relaxed strongly monotone and pseudomonotone.


## 3. Main Results

In this section, we use the fixed point formulation (2.10) to suggest a new unified implicit method for solving the mixed variational inequalities (2.1), and this is the main motivation of this paper.Algorithm 3.1.
For a givenu0∈H, find the approximate solution un+1 by the iterative scheme
(3.1)un+1=Jφ[un-ρTun],n=0,1,2,….Algorithm3.1 is known as the projection iterative method. For the convergence analysis of Algorithm 3.1, see Noor et al.  [6].For a givenλ∈[0,1], we can rewrite (2.10) as(3.2)u=Jφ[u-ρTu+λρ(Tu-Tu)].
This fixed point formulation is used to suggest the following new proximal point iterative method for solving mixed variational inequality (2.1) as follows.Algorithm 3.2.
For a givenu0∈H, find the approximate solution un+1 by the iterative scheme
(3.3)un+1=Jφ[un-ρTun+1+λρ(Tun+1-Tun)],n=0,1,2,….
Note that Algorithm 3.2 is an implicit-type iterative method. It is clear that for λ=1, Algorithm 3.2 reduces to Algorithm 3.1. For λ=0, Algorithm 3.2 collapses to the following implicit iterative method for solving the mixed variational inequality (2.1).Algorithm 3.3.
For a givenu0∈H, find the approximate solution un+1 by the iterative scheme
(3.4)un+1=Jφ[un-ρTun+1],n=0,1,2,….
For the convergence analysis of Algorithm 3.3, see Noor [8] and the references therein.In order to implement Algorithm 3.2, we use the predictor-corrector technique. We use Algorithm 3.1 as the predictor and Algorithm 3.2 as the corrector. Consequently, we obtain the following two-step iterative method for solving the variational inequality (2.1).Algorithm 3.4.
For a givenu0∈H, find the approximate solution un+1 by the iterative schemes:
(3.5)yn=Jφ[un-ρTun],un+1=Jφ[un-ρTyn+λρ(Tyn-Tun)],n=0,1,2,….
Algorithm 3.3 is a new two-step iterative method for solving the variational inequality (2.1).Forλ=1, Algorithm 3.3 reduces to Algorithm 3.1. For λ=0, Algorithm 3.3 reduces to the following iterative method for solving the mixed variational inequality (2.1).Algorithm 3.5.
For a givenu0∈H, find the approximate solution un+1 by the iterative schemes:
(3.6)yn=Jφ[un-ρTun],un+1=Jφ[un-ρTyn],n=0,1,2,….
which is known as the extragradient method and is due to Korpelevič  [5]?Forλ=1/2, Algorithm 3.2 collapses to the following iterative method for solving the mixed variational inequality (2.1) and appears to be a new one.Algorithm 3.6.
For a givenu0∈H, find the approximate solution un+1 by the iterative scheme
(3.7)un+1=Jφ[un-ρTun+1+Tun2],n=0,1,2,….
This clearly shows that Algorithm 3.2 is a unified implicit method and includes the previously known extragradient method of  Korpelevič [5] and several new methods as special cases.
We now consider the convergence criteria of Algorithm3.2 and this is the main motivation of our next result.Theorem 3.7.
Let the operatorT be partially relaxed strongly pseudomonotone with constant α>0. Then
(3.8)‖u-un+1‖2≤‖u-un‖2-(1-2λρα)‖un-un+1‖2,
where un+1 is the approximate solution, which is obtained from Algorithm 3.2.Proof.
Letu∈H be a solution of (2.1). Then, by using the pseudomonotonicity of T, we have
(3.9)〈Tv,v-u〉+φ(v)-φ(u)≥0,∀u,v∈H.
Taking v=un+1 in (3.9) and (2.1), we have
(3.10)〈Tun+1,un+1-u〉+φ(un+1)-φ(u)≥0,〈Tu,un+1-u〉+φ(un+1)-φ(u)≥0.
Using Lemma 2.2, we can rewrite (3.3) in the following equivalent form:
(3.11)〈(1-λ)ρTun+1+λρTun+un+1-un,v-un+1〉+φ(v)-φ(un+1)≥0,∀v∈H.
Taking v=u in (3.11), we have
(3.12)〈(1-λ)ρTun+1+λρTun+un+1-un,u-un+1〉+φ(u)-φ(un+1)≥0.
Form (3.10) and (3.12), we have
(3.13)〈un+1-un,u-un+1〉≥〈2(1-λ)ρTun+1,un+1-u〉+2λρ〈Tun-Tu,un+1-u〉≥2λρ〈Tun-Tu,un+1-u〉.
Using the inequality 〈u,v〉=∥u-v∥2-∥u∥2-∥v∥2, for all u,v∈H, and partially relaxed strongly monotonicity of the operator T, form (3.13), we obtain
(3.14)‖u-un+1‖2≤‖u-un‖2-(1-2λρα)‖un-un+1‖2,
which is the required (3.3).Theorem 3.8.
LetH  be a finite dimensional space. If un+1 is the approximate solution obtained from Algorithm 3.2, and let u∈H be a solution of problem (2.1). If 0<ρ<1/2αλ, then limn→∞un=u.Proof.
Letu∈H be a solution of (2.1). For 0<ρ<1/2αλ, we see that the sequence {∥u-un∥} is nonincreasing and consequently {un}is bounded. Also from (3.8), we have
(3.15)∑n=0∞(1-2ρλα)‖un+1-un‖2≤‖u-u0‖2,
which implies that
(3.16)lim⁡n→∞‖un+1-un‖=0.
Let û be the cluster point of {un} and the subsequence {unj} of this sequence converges to û∈H. Replacing un by unj in (3.11) and taking the limit as nj→∞ and using (3.16), we have
(3.17)〈Tû,v-û〉+φ(v)-φ(û)≥0,∀v∈H,
which shows that û solves the mixed variational inequality (2.1) and
(3.18)‖un+1-û‖2≤‖un-û‖2.
Thus, it follows from the above inequality that the sequence {un} has exactly one cluster point and limn→∞un=û, the required result.For a givenλ∈[0,1], we can rewrite (2.10) as:(3.19)u=Jφ[(1-λ)u+λu-ρTu].
We use this alternative equivalent fixed point formulation to suggest and analyze the following iterative method for solving the mixed variational inequality (2.1) as.Algorithm 3.9.
For a givenu0∈H, find the approximate solution un+1 by the iterative scheme
(3.20)un+1=Jφ[λ(un-un+1)+un+1-ρTun+1],n=0,1,2,….
It is clear that Algorithm 3.9 is an implicit method. To implement Algorithm 3.9, we use the predictor-corrector technique. We use Algorithm 3.1 as the predictor and Algorithm 3.9 as the corrector. Consequently, we have the following predictor-corrector method for solving the mixed variational inequalities (2.1).Algorithm 3.10.
For a givenu0∈H, find the approximate solution un+1 by the iterative schemes
(3.21)yn=Jφ[un-ρTun],un+1=Jφ[λ(un-yn)+yn-ρTyn],n=0,1,2,….Algorithm3.10 is a new two-step implicit method for solving the mixed variational inequality (2.1).Forλ=1, Algorithm 3.9 collapses to the following algorithm.Algorithm 3.11.
For a givenu0∈H, find the approximate solution un+1 by the iterative schemes
(3.22)yn=Jφ[un-ρTun].un+1=Jφ[un-ρTyn],n=0,1,2,….
which is known as extraresolvent method and includes the extragradient method of Korpelevič [5] as special case.Forλ=0, Algorithm 3.9 reduces to the following algorithm.Algorithm 3.12.
For a givenu0∈H, find the approximate solution un+1 by the iterative schemes
(3.23)yn=Jφ[un-ρTun],un+1=Jφ[yn-ρTyn],n=0,1,2,….Algorithm3.12 is called the modified extraresolvent method, which is mainly due to Noor [8].From the above discussion, it is clear that Algorithm3.9 is a unified implicit resolvent method for solving the mixed variational inequalities. Algorithm 3.9 includes several new and previously known methods as special cases. Using the technique of Noor et al. [13], one can easily consider the convergence analysis of Algorithm 3.9.For a givenλ∈[0,1], one can rewrite (2.10) in the following form:(3.24)u=Jφ[u-ρT((1-λ)u+λu)].This fixed point formulation enables us to suggest and analyze a class of iterative methods for solving the mixed variational inequality (2.1).Algorithm 3.13.
For a givenu0∈H, find the approximate solution un+1 by the iterative scheme
(3.25)un+1=Jφ[un-ρT((1-λ)un+1+λun)],n=0,1,2,….
For different choice of the parameter λ, we can obtain the extraresoalvent method and Algorithm 3.1 as special cases. In particular, for λ=1/2, we have the following iterative method for solving the mixed variational inequality (2.1).Algorithm 3.14.
For a givenu0∈H, find the approximate solution un+1 by the iterative scheme
(3.26)un+1=Jφ[un-ρT(un+1+un2)],n=0,1,2,….
We remark that if T is a linear operator, then Algorithm 3.6 and Algorithm 3.14 are equivalent.Remark 3.15.
We remark that if the functionφ(·) is the indicator function of a closed and convex set K, then Jφ=(I+ρ∂φ)-1≡PK, the projection of H onto the convex and closed set K. Consequently, Algorithms 3.1–3.12 are reduced to the algorithms considered in [13].


## 4. Conclusion

In this paper, we have used the equivalence between the mixed variational inequality and the fixed point problem to suggest and analyze some new proximal point methods for solving the mixed variational inequality. We have also shown that these new implicit methods include the extraresolvent and the classical implicit resolvent methods as special cases. We have also discussed the convergence criteria of the proposed new iterative methods under some suitable conditions. We have also shown that this technique can be used to suggest several iterative methods for solving various classes of equilibrium and variational inequalities problems. The technique of constructing various iterative methods is very simple and contains a wealth of new ideas. The results proved in this paper can be extended for the multivalued mixed quasi-variational inequalities and related optimization problems. Comparison of these methods with other methods is an interesting problem for further research.



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