- Research Article
- Open Access

# Self-Adaptive Implicit Methods for Monotone Variant Variational Inequalities

- Zhili Ge
^{1}and - Deren Han
^{1}Email author

**2009**:458134

https://doi.org/10.1155/2009/458134

© Z. Ge and D. Han. 2009

**Received:**26 January 2009**Accepted:**24 February 2009**Published:**26 March 2009

## Abstract

The efficiency of the implicit method proposed by He (1999) depends on the parameter heavily; while it varies for individual problem, that is, different problem has different "suitable" parameter, which is difficult to find. In this paper, we present a modified implicit method, which adjusts the parameter automatically per iteration, based on the message from former iterates. To improve the performance of the algorithm, an inexact version is proposed, where the subproblem is just solved approximately. Under mild conditions as those for variational inequalities, we prove the global convergence of both exact and inexact versions of the new method. We also present several preliminary numerical results, which demonstrate that the self-adaptive implicit method, especially the inexact version, is efficient and robust.

## Keywords

- Variational Inequality
- Variational Inequality Problem
- Nonempty Closed Convex Subset
- Cluster Point
- Implicit Method

## 1. Introduction

where is a mapping from into itself.

Both and serve as very general mathematical models of numerous applications arising in economics, engineering, transportation, and so forth. They include some widely applicable problems as special cases, such as mathematical programming problems, system of nonlinear equations, and nonlinear complementarity problems, and so forth. Thus, they have been extensively investigated. We refer the readers to the excellent monograph of Faccinei and Pang [1, 2] and the references therein for theoretical and algorithmic developments on , for example, [3–10], and [11–16] for .

It is observed that if is invertible, then by setting , the inverse mapping of can be reduced to . Thus, theoretically, all numerical methods for solving can be used to solve . However, in many practical applications, the inverse mapping may not exist. On the other hand, even if it exists, it is not easy to find it. Thus, there is a need to develop numerical methods for and recently, the Goldstein's type method was extended from solving to [12, 17].

When is the identity mapping, it reduces to and if is the identity mapping, it reduces to . He's implicit method is as follows.

(S0)Given , and a positive definite matrix .

with being the projection from onto , under the Euclidean norm.

In the above algorithm, there are two parameters
and
, which affect the efficiency of the algorithm. It was observed that nearly for all problems,
close to
is a better choice than smaller
, while different problem has different *optimal*
. A suitable parameter
is thus difficult to find for an individual problem. For solving variational inequality problems, He et al. [18] proposed to choose a sequence of parameters
, instead of a fixed parameter
, to improve the efficiency of the algorithm. Under the same conditions as those in [11], they proved the global convergence of the algorithm. The numerical results reported there indicated that for any given initial parameter
, the algorithm can find a suitable parameter self-adaptively. This improves the efficiency of the algorithm greatly and makes the algorithm easy and robust to implement in practice.

In this paper, in a similar theme as [18], we suggest a general rule for choosing suitable parameter in the implicit method for solving . By replacing the constant factor in (1.4) and (1.5) with a self-adaptive variable positive sequence , the efficiency of the algorithm can be improved greatly. Moreover, it is also robust to the initial choice of the parameter . Thus, for any given problems, we can choose a parameter arbitrarily, for example, or . The algorithm chooses a suitable parameter self-adaptively, based on the information from the former iteration, which makes it able to add a little additional computational cost against the original algorithm with fixed parameter . To further improve the efficiency of the algorithm, we also admit approximate computation in solving the subproblem per iteration. That is, per iteration, we just need to find a vector that satisfies (1.8).

Throughout this paper, we make the following assumptions.

Assumption A.

The solution set of , denoted by , is nonempty.

Assumption B.

The rest of this paper is organized as follows. In Section 2, we summarize some basic properties which are useful in the convergence analysis of our method. In Sections 3 and 4, we describe the exact version and inexact version of the method and prove their global convergence, respectively. We report our preliminary computational results in Section 5 and give some final conclusions in the last section.

## 2. Preliminaries

For a vector and a symmetric positive definite matrix , we denote as the Euclidean-norm and as the matrix-induced norm, that is, .

where is an arbitrary positive constant. Then, we have the following lemma.

Lemma 2.1.

is the residual function of the projection equation (2.2).

Proof.

See [11, Theorem??1].

The following lemma summarizes some basic properties of the projection operator, which will be used in the subsequent analysis.

Lemma 2.2.

The following lemma plays an important role in convergence analysis of our algorithm.

Lemma 2.3.

Proof.

See [20] for a simple proof.

Lemma 2.4.

Proof.

where the last inequality follows from the monotonicity of (Assumption B). This completes the proof.

## 3. Exact Implicit Method and Convergence Analysis

We are now in the position to describe our algorithm formally.

### 3.1. Self-Adaptive Exact Implicit Method

(S0)Given , , and a positive definite matrix .

We refer to the above method as *the self-adaptive exact implicit method.*

Remark 3.1.

Hence, the sequence is bounded. Then, let and .

Now, we analyze the convergence of the algorithm, beginning with the following lemma.

Lemma 3.2.

Proof.

where the inequality follows from (2.7). This completes the proof.

Such a self-adaptive strategy was adopted in [18, 21–24] for solving variational inequality problems, where the numerical results indicated its efficiency and robustness to the choice of the initial parameter . Here we adopted it for solving variant variational inequality problems.

We are now in the position to give the convergence result of the algorithm, the main result of this section.

Theorem 3.3.

The sequence generated by the proposed self-adaptive exact implicit method converges to a solution of .

Proof.

This, together with the monotonicity of the mapping , means that the generated sequence is bounded.

Thus, from Lemma 2.1, is a solution of .

which means that cannot be a cluster point of . Thus, has just one cluster point.

## 4. Inexact Implicit Method and Convergence Analysis

where
is a nonnegative sequence with
. If (3.1) is replaced by (4.1), the modified method is called *inexact implicit method*.

We now analyze the convergence of the inexact implicit method.

Lemma 4.1.

Proof.

Substituting (4.6) and (4.9) into (4.5), we complete the proof.

Now, we prove the convergence of the inexact implicit method.

Theorem 4.2.

The sequence generated by the proposed self-adaptive inexact implicit method converges to a solution point of .

Proof.

are finite. The rest of the proof is similar to that of Theorem 3.3 and is thus omitted here.

## 5. Computational Results

Comparison of the proposed method and He's method [11].

As the results in Table 1, the results in Tables 2 to 5 indicate that the number of iterations and CPU time are rather insensitive to the initial parameter , while He's method is efficient for proper choice of . The results also show that the proposed method, as well as He's method, is very stable and efficient to the choice of the initial point .

## 6. Conclusions

In this paper, we proposed a self-adaptive implicit method for solving monotone variant variational inequalities. The proposed self-adaptive adjusting rule avoids the difficult task of choosing a "suitable" parameter, which makes the method efficient for initial parameter. Our self-adaptive rule adds only a tiny amount of computation than the method with fixed parameter, while the efficiency is enhanced greatly. To make the method more efficient and practical, an approximate version of the algorithm was proposed. The global convergence of both the exact version and the inexact version of the new algorithm was proved under mild assumptions; that is, the underlying mapping of is monotone and there is at least one solution of the problem. The reported preliminary numerical results verified our assertion.

## Declarations

### Acnowledgments

This research was supported by the NSFC Grants 10501024, 10871098, and NSF of Jiangsu Province at Grant no. BK2006214. D. Han was also supported by the Scientific Research Foundation for the Returned Overseas Chinese Scholars, State Education Ministry.

## Authors’ Affiliations

## References

- Facchinei F, Pang JS:
*Finite-Dimensional Variational Inequalities and Complementarity Problems. Vol. I, Springer Series in Operations Research*. Springer, New York, NY, USA; 2003:xxxiv+624+I69.MATHGoogle Scholar - Facchinei F, Pang JS:
*Finite-Dimensional Variational Inequalities and Complementarity Problems, Vol. II, Springer Series in Operations Research*. Springer, New York, NY, USA; 2003:i-xxxiv, 625–1234 and II1–II57.MATHGoogle Scholar - Bertsekas DP, Gafni EM: Projection methods for variational inequalities with application to the traffic assignment problem. Mathematical Programming Study 1982, (17):139–159.Google Scholar
- Rachunková I, Tvrdý M:
**Nonlinear systems of differential inequalities and solvability of certain boundary value problems.***Journal of Inequalities and Applications*2000,**6**(2):199–226.MathSciNetGoogle Scholar - Agarwal RP, Elezovic N, Pecaric J:
**On some inequalities for beta and gamma functions via some classical inequalities.***Journal of Inequalities and Applications*2005,**2005**(5):593–613. 10.1155/JIA.2005.593MathSciNetView ArticleMATHGoogle Scholar - Dafermos S:
**Traffic equilibrium and variational inequalities.***Transportation Science*1980,**14**(1):42–54. 10.1287/trsc.14.1.42MathSciNetView ArticleGoogle Scholar - Verma RU:
**A class of projection-contraction methods applied to monotone variational inequalities.***Applied Mathematics Letters*2000,**13**(8):55–62. 10.1016/S0893-9659(00)00096-3MathSciNetView ArticleMATHGoogle Scholar - Verma RU:
**Projection methods, algorithms, and a new system of nonlinear variational inequalities.***Computers & Mathematics with Applications*2001,**41**(7–8):1025–1031. 10.1016/S0898-1221(00)00336-9MathSciNetView ArticleMATHGoogle Scholar - Ceng LC, Mastroeni G, Yao JC:
**An inexact proximal-type method for the generalized variational inequality in Banach spaces.***Journal of Inequalities and Applications*2007,**2007:**-14.Google Scholar - Chidume CE, Chidume CO, Ali B:
**Approximation of fixed points of nonexpansive mappings and solutions of variational inequalities.***Journal of Inequalities and Applications*2008,**2008:**-12.Google Scholar - He BS:
**Inexact implicit methods for monotone general variational inequalities.***Mathematical Programming*1999,**86**(1):199–217. 10.1007/s101070050086MathSciNetView ArticleMATHGoogle Scholar - He BS:
**A Goldstein's type projection method for a class of variant variational inequalities.***Journal of Computational Mathematics*1999,**17**(4):425–434.MathSciNetMATHGoogle Scholar - Noor MA:
**Quasi variational inequalities.***Applied Mathematics Letters*1988,**1**(4):367–370. 10.1016/0893-9659(88)90152-8MathSciNetView ArticleMATHGoogle Scholar - Outrata JV, Zowe J:
**A Newton method for a class of quasi-variational inequalities.***Computational Optimization and Applications*1995,**4**(1):5–21. 10.1007/BF01299156MathSciNetView ArticleMATHGoogle Scholar - Pang JS, Qi LQ:
**Nonsmooth equations: motivation and algorithms.***SIAM Journal on Optimization*1993,**3**(3):443–465. 10.1137/0803021MathSciNetView ArticleMATHGoogle Scholar - Pang JS, Yao JC:
**On a generalization of a normal map and equation.***SIAM Journal on Control and Optimization*1995,**33**(1):168–184. 10.1137/S0363012992241673MathSciNetView ArticleMATHGoogle Scholar - Li M, Yuan XM:
**An improved Goldstein's type method for a class of variant variational inequalities.***Journal of Computational and Applied Mathematics*2008,**214**(1):304–312. 10.1016/j.cam.2007.02.032MathSciNetView ArticleMATHGoogle Scholar - He BS, Liao LZ, Wang SL:
**Self-adaptive operator splitting methods for monotone variational inequalities.***Numerische Mathematik*2003,**94**(4):715–737.MathSciNetView ArticleMATHGoogle Scholar - Eaves BC:
**On the basic theorem of complementarity.***Mathematical Programming*1971,**1**(1):68–75. 10.1007/BF01584073MathSciNetView ArticleMATHGoogle Scholar - Zhu T, Yu ZQ:
**A simple proof for some important properties of the projection mapping.***Mathematical Inequalities & Applications*2004,**7**(3):453–456.MathSciNetView ArticleMATHGoogle Scholar - He BS, Yang H, Meng Q, Han DR:
**Modified Goldstein-Levitin-Polyak projection method for asymmetric strongly monotone variational inequalities.***Journal of Optimization Theory and Applications*2002,**112**(1):129–143. 10.1023/A:1013048729944MathSciNetView ArticleMATHGoogle Scholar - Han D, Sun W:
**A new modified Goldstein-Levitin-Polyak projection method for variational inequality problems.***Computers & Mathematics with Applications*2004,**47**(12):1817–1825. 10.1016/j.camwa.2003.12.002MathSciNetView ArticleMATHGoogle Scholar - Han D:
**Inexact operator splitting methods with selfadaptive strategy for variational inequality problems.***Journal of Optimization Theory and Applications*2007,**132**(2):227–243. 10.1007/s10957-006-9060-5MathSciNetView ArticleMATHGoogle Scholar - Han D, Xu W, Yang H:
**An operator splitting method for variational inequalities with partially unknown mappings.***Numerische Mathematik*2008,**111**(2):207–237. 10.1007/s00211-008-0181-7MathSciNetView ArticleMATHGoogle Scholar - Dembo RS, Eisenstat SC, Steihaug T:
**Inexact Newton methods.***SIAM Journal on Numerical Analysis*1982,**19**(2):400–408. 10.1137/0719025MathSciNetView ArticleMATHGoogle Scholar - Pang JS:
**Inexact Newton methods for the nonlinear complementarity problem.***Mathematical Programming*1986,**36**(1):54–71. 10.1007/BF02591989MathSciNetView ArticleMATHGoogle Scholar - Harker PT, Pang JS:
**A damped-Newton method for the linear complementarity problem.**In*Computational Solution of Nonlinear Systems of Equations (Fort Collins, CO, 1988), Lectures in Applied Mathematics*.*Volume 26*. American Mathematical Society, Providence, RI, USA; 1990:265–284.Google Scholar

## Copyright

This article is published under license to BioMed Central Ltd. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.