Some Proximal Methods for Solving Mixed Variational Inequalities

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.

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

Let 𝐻 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 operator 𝑇∶𝐻→𝐻, we consider the problem of finding 𝑢∈𝐻 such that⟨𝑇𝑢,𝑣−𝑢⟩+𝜑(𝑣)−𝜑(𝑢)≥0,∀𝑣∈𝐻, (2.1) 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 operator 𝑇 is linear, positive, and symmetric and the function 𝜑(⋅) is convex function, then the minimum of the functional 𝐼[𝑣], defined as𝐼[𝑣]=⟨𝑇𝑣,𝑣⟩+g𝜑(𝑣) (2.2) on the convex set, can be characterized by the mixed variational inequality (2.1).

If the function 𝜑(⋅) is proper, convex, and semicontinuous, then problem (2.1) is equivalent to find 𝑢∈𝐻 such that0∈𝑇𝑢+𝜕𝜑(𝑢), (2.3) 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 function 𝜑(⋅) 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⟨𝑇𝑢,𝑣−𝑢⟩≥0,∀𝑣∈𝐾, (2.4) 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]). Let 𝑇 be a maximal monotone operator, then the resolvent operator associated with 𝑇 is defined as 𝐽_𝑇(𝑢)=𝐼+𝜌𝑇⁻¹(𝑢),∀𝑢∈𝐻, (2.5) 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 𝐽_𝜑(𝑢)=𝐼+𝜌𝜕𝜑⁻¹(𝑢). (2.6) The resolvent operator 𝐽_𝜑 defined by (2.6) has the following useful characterization.

Lemma 2.2 (see [1]). For a given 𝑧∈𝐻,𝑢∈𝐻 satisfies the inequality ⟨𝑢−𝑧,𝑣−𝑢⟩+𝜌𝜑(𝑣)−𝜌𝜑(𝑢)≥0,∀𝑣∈𝐻, (2.7) if and only if 𝑢=𝐽_𝜑𝑧, (2.8) where 𝐽_𝜑(𝑢)=(𝐼+𝜌𝜕𝜑)⁻¹(𝑢) is the resolvent operator.

It is well known that the resolvent operator 𝐽_𝜑 is nonexpansive, that is, 𝐽_𝜑𝑢−𝐽_𝜑𝑣≤‖𝑢−𝑣‖,∀𝑢,𝑣∈𝐻. (2.9) Using Lemma 2.2, one can easily show that the mixed variational inequality (2.1) is equivalent to finding 𝑢∈𝐻 such that 𝑢=𝐽_𝜑𝑢−𝜌𝑇𝑢, (2.10) where 𝜌>0 is a constant.

Lemma 2.2 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 operator 𝑇∶𝐻→𝐻 is said to be partially relaxed strongly monotone, if and only if, there exists a constant 𝛼>0 such that ⟨𝑇𝑢−𝑇𝑣,𝑧−𝑣⟩≥𝛼‖𝑧−𝑣‖²,∀𝑢,𝑣,𝑧∈𝐻 (2.11) and pseudomontone with respect to the function 𝜑 if and only if ⟨𝑇𝑢,𝑣−𝑢⟩+𝜑(𝑣)−𝜑(𝑢)≥0⟹⟨𝑇𝑣,𝑣−𝑢⟩+𝜑(𝑣)−𝜑(𝑢)≥0,∀𝑢,𝑣∈𝐻. (2.12)

Definition 2.4. An operator 𝑇∶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 given 𝑢₀∈𝐻, find the approximate solution 𝑢_𝑛+1 by the iterative scheme 𝑢_𝑛+1=𝐽_𝜑𝑢_𝑛−𝜌𝑇𝑢_𝑛,𝑛=0,1,2,…. (3.1)

Algorithm 3.1 is known as the projection iterative method. For the convergence analysis of Algorithm 3.1, see Noor et al. [6].

For a given 𝜆∈[0,1], we can rewrite (2.10) as𝑢=𝐽_𝜑𝑢−𝜌𝑇𝑢+𝜆𝜌(𝑇𝑢−𝑇𝑢). (3.2) 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 given 𝑢₀∈𝐻, find the approximate solution 𝑢_𝑛+1 by the iterative scheme 𝑢_𝑛+1=𝐽_𝜑𝑢_𝑛−𝜌𝑇𝑢_𝑛+1+𝜆𝜌𝑇𝑢_𝑛+1−𝑇𝑢_𝑛,𝑛=0,1,2,…. (3.3) 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 given 𝑢₀∈𝐻, find the approximate solution 𝑢_𝑛+1 by the iterative scheme 𝑢_𝑛+1=𝐽_𝜑𝑢_𝑛−𝜌𝑇𝑢_𝑛+1,𝑛=0,1,2,…. (3.4) 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 given 𝑢₀∈𝐻, find the approximate solution 𝑢_𝑛+1 by the iterative schemes: 𝑦_𝑛=𝐽_𝜑𝑢_𝑛−𝜌𝑇𝑢_𝑛,𝑢_𝑛+1=𝐽_𝜑𝑢_𝑛−𝜌𝑇𝑦_𝑛+𝜆𝜌𝑇𝑦_𝑛−𝑇𝑢_𝑛,𝑛=0,1,2,…. (3.5) Algorithm 3.3 is a new two-step iterative method for solving the variational inequality (2.1).

For 𝜆=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 given 𝑢₀∈𝐻, find the approximate solution 𝑢_𝑛+1 by the iterative schemes: 𝑦_𝑛=𝐽_𝜑𝑢_𝑛−𝜌𝑇𝑢_𝑛,𝑢_𝑛+1=𝐽_𝜑𝑢_𝑛−𝜌𝑇𝑦_𝑛,𝑛=0,1,2,…. (3.6) which is known as the extragradient method and is due to Korpelevič [5]?

For 𝜆=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 given 𝑢₀∈𝐻, find the approximate solution 𝑢_𝑛+1 by the iterative scheme 𝑢_𝑛+1=𝐽_𝜑𝑢_𝑛−𝜌𝑇𝑢_𝑛+1+𝑇𝑢_𝑛2,𝑛=0,1,2,…. (3.7) 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 Algorithm 3.2 and this is the main motivation of our next result.

Theorem 3.7. Let the operator 𝑇 be partially relaxed strongly pseudomonotone with constant 𝛼>0. Then 𝑢−𝑢_𝑛+1²≤𝑢−𝑢_𝑛²−1−2𝜆𝜌𝛼𝑢_𝑛−𝑢_𝑛+1², (3.8) where 𝑢_𝑛+1 is the approximate solution, which is obtained from Algorithm 3.2.

Proof. Let 𝑢∈𝐻 be a solution of (2.1). Then, by using the pseudomonotonicity of 𝑇, we have ⟨𝑇𝑣,𝑣−𝑢⟩+𝜑(𝑣)−𝜑(𝑢)≥0,∀𝑢,𝑣∈𝐻. (3.9) Taking 𝑣=𝑢_𝑛+1 in (3.9) and (2.1), we have 𝑇𝑢_𝑛+1,𝑢_𝑛+1−𝑢+𝜑𝑢_𝑛+1−𝜑(𝑢)≥0,𝑇𝑢,𝑢_𝑛+1−𝑢+𝜑𝑢_𝑛+1−𝜑(𝑢)≥0. (3.10) Using Lemma 2.2, we can rewrite (3.3) in the following equivalent form: (1−𝜆)𝜌𝑇𝑢_𝑛+1+𝜆𝜌𝑇𝑢_𝑛+𝑢_𝑛+1−𝑢_𝑛,𝑣−𝑢_𝑛+1+𝜑(𝑣)−𝜑𝑢_𝑛+1≥0,∀𝑣∈𝐻. (3.11) Taking 𝑣=𝑢 in (3.11), we have (1−𝜆)𝜌𝑇𝑢_𝑛+1+𝜆𝜌𝑇𝑢_𝑛+𝑢_𝑛+1−𝑢_𝑛,𝑢−𝑢_𝑛+1+𝜑(𝑢)−𝜑𝑢_𝑛+1≥0. (3.12) Form (3.10) and (3.12), we have 𝑢_𝑛+1−𝑢_𝑛,𝑢−𝑢_𝑛+1≥2(1−𝜆)𝜌𝑇𝑢_𝑛+1,𝑢_𝑛+1−𝑢+2𝜆𝜌𝑇𝑢_𝑛−𝑇𝑢,𝑢_𝑛+1−𝑢≥2𝜆𝜌𝑇𝑢_𝑛−𝑇𝑢,𝑢_𝑛+1−𝑢. (3.13) 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 𝑢−𝑢_𝑛+1²≤𝑢−𝑢_𝑛²−1−2𝜆𝜌𝛼𝑢_𝑛−𝑢_𝑛+1², (3.14) which is the required (3.3).

Theorem 3.8. Let 𝐻 be a finite dimensional space. If 𝑢_𝑛+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 lim_𝑛→∞𝑢_𝑛=𝑢.

Proof. Let 𝑢∈𝐻 be a solution of (2.1). For 0<𝜌<1/2𝛼𝜆, we see that the sequence {‖𝑢−𝑢_𝑛‖} is nonincreasing and consequently {𝑢_𝑛}is bounded. Also from (3.8), we have 1−2𝜌𝜆𝛼𝑢^∞_𝑛=0𝑛+1−𝑢_𝑛²≤𝑢−𝑢₀², (3.15) which implies that lim_𝑛→∞𝑢_𝑛+1−𝑢_𝑛=0. (3.16) Let 𝑢 be the cluster point of {𝑢_𝑛} and the subsequence {𝑢_𝑛𝑗} of this sequence converges to 𝑢∈𝐻. Replacing 𝑢_𝑛 by 𝑢_𝑛𝑗 in (3.11) and taking the limit as 𝑛_𝑗→∞ and using (3.16), we have ⟨𝑇𝑢,𝑣−𝑢⟩+𝜑(𝑣)−𝜑(𝑢)≥0,∀𝑣∈𝐻, (3.17) which shows that 𝑢 solves the mixed variational inequality (2.1) and 𝑢_𝑛+1−𝑢²≤𝑢_𝑛−𝑢². (3.18) Thus, it follows from the above inequality that the sequence {𝑢_𝑛} has exactly one cluster point and lim_𝑛→∞𝑢_𝑛=𝑢, the required result.

For a given 𝜆∈[0,1], we can rewrite (2.10) as:𝑢=𝐽_𝜑(1−𝜆)𝑢+𝜆𝑢−𝜌𝑇𝑢. (3.19) 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 given 𝑢₀∈𝐻, find the approximate solution 𝑢_𝑛+1 by the iterative scheme 𝑢_𝑛+1=𝐽_𝜑𝜆𝑢_𝑛−𝑢_𝑛+1+𝑢_𝑛+1−𝜌𝑇𝑢_𝑛+1,𝑛=0,1,2,…. (3.20) 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 given 𝑢₀∈𝐻, find the approximate solution 𝑢_𝑛+1 by the iterative schemes 𝑦_𝑛=𝐽_𝜑𝑢_𝑛−𝜌𝑇𝑢_𝑛,𝑢_𝑛+1=𝐽_𝜑𝜆𝑢_𝑛−𝑦_𝑛+𝑦_𝑛−𝜌𝑇𝑦_𝑛,𝑛=0,1,2,…. (3.21)

Algorithm 3.10 is a new two-step implicit method for solving the mixed variational inequality (2.1).

For 𝜆=1, Algorithm 3.9 collapses to the following algorithm.

Algorithm 3.11. For a given 𝑢₀∈𝐻, find the approximate solution 𝑢_𝑛+1 by the iterative schemes 𝑦_𝑛=𝐽_𝜑𝑢_𝑛−𝜌𝑇𝑢_𝑛.𝑢_𝑛+1=𝐽_𝜑𝑢_𝑛−𝜌𝑇𝑦_𝑛,𝑛=0,1,2,…. (3.22) which is known as extraresolvent method and includes the extragradient method of Korpelevič [5] as special case.

For 𝜆=0, Algorithm 3.9 reduces to the following algorithm.

Algorithm 3.12. For a given 𝑢₀∈𝐻, find the approximate solution 𝑢_𝑛+1 by the iterative schemes 𝑦_𝑛=𝐽_𝜑𝑢_𝑛−𝜌𝑇𝑢_𝑛,𝑢_𝑛+1=𝐽_𝜑𝑦_𝑛−𝜌𝑇𝑦_𝑛,𝑛=0,1,2,…. (3.23)

Algorithm 3.12 is called the modified extraresolvent method, which is mainly due to Noor [8].

From the above discussion, it is clear that Algorithm 3.9 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 given 𝜆∈[0,1], one can rewrite (2.10) in the following form:𝑢=𝐽_𝜑𝑢−𝜌𝑇((1−𝜆)𝑢+𝜆𝑢). (3.24)

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 given 𝑢₀∈𝐻, find the approximate solution 𝑢_𝑛+1 by the iterative scheme 𝑢_𝑛+1=𝐽_𝜑𝑢_𝑛−𝜌𝑇(1−𝜆)𝑢_𝑛+1+𝜆𝑢_𝑛,𝑛=0,1,2,…. (3.25) 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 given 𝑢₀∈𝐻, find the approximate solution 𝑢_𝑛+1 by the iterative scheme 𝑢_𝑛+1=𝐽_𝜑𝑢_𝑛𝑢−𝜌𝑇^𝑛+1+𝑢_𝑛2,𝑛=0,1,2,…. (3.26) 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 function 𝜑(⋅) is the indicator function of a closed and convex set 𝐾, then 𝐽_𝜑=(𝐼+𝜌𝜕𝜑)⁻¹≡𝑃_𝐾, 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.

Acknowledgment

The authors are grateful to Dr. S. M. Junaid Zaidi, Rector, COMSATS Institute of Information Technology, Pakistan for providing the excellent research facilities. The research of Prof. Dr. Z. Y. Huang is supported by National Natural Science Foundation of China (NSFC Grant No. 10871092), supported by the Fundamental Research Funds for the Central University of China (Grant No.1113020301 and Grant No.1116020301), and supported by A Project Funded by the Priority Academic Program Development of Jiangsu Higher Education Institutions (PAPD Grant).