SelfAdaptive Implicit Methods for Monotone Variant Variational Inequalities
 Zhili Ge^{1} and
 Deren Han^{1}Email author
DOI: 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 selfadaptive implicit method, especially the inexact version, is efficient and robust.
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 selfadaptively. 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 selfadaptive 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 selfadaptively, 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 Euclideannorm and as the matrixinduced 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. SelfAdaptive Exact Implicit Method
(S0)Given , , and a positive definite matrix .
Set and go to Step??1.
We refer to the above method as the selfadaptive 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 selfadaptive 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 selfadaptive 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 selfadaptive 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
with
Comparison of the proposed method and He's method [11].

 

Proposed method  He's method  Proposed method  He's method  
It. no.  CPU  It. no.  CPU  It. no.  CPU  It. no.  CPU  
0.5  25  0.3910  100  1.0780  34  50.4850  —  — 
0.05  20  0.3120  37  0.4850  25  39.8440  17  25.0940 
0.01  26  0.4060  350  5.8750  33  61.4070  —  — 
Numerical results for VMCP with dimension .
 Proposed method  He's method  

It. no.  CPU  It. no.  CPU  
 69  0.0780  —  — 
 65  0.1250  7335  6.1250 
 61  0.0790  485  0.4530 
 59  0.0620  60  4.0780 
10  60  0.0780  315  0.3280 
1  66  0.0110  2672  2.500 
 70  0.0940  22541  21.0320 
 73  0.0780  —  — 
Numerical results for VMCP with dimension .
 Proposed method  He's method  

It. no.  CPU  It. no.  CPU  
 82  1.6090  —  — 
 74  1.4850  1434  28.3750 
 64  1.2660  199  3.8910 
 63  1.2500  174  3.4060 
10  68  1.3500  1486  30.4840 
1  75  1.4850  —  — 
 75  1.5000  —  — 
 86  1.7030  —  — 
Numerical results for VMCP with dimension .
 Proposed method  He's method  

It. no.  CPU  It. no.  CPU  
 61  0.0620  —  — 
 61  0.0940  3422  3.7190 
 60  0.0790  684  0.6410 
 67  0.0780  59  0.0620 
10  65  0.0940  309  0.2970 
1  69  0.0940  2637  2.3750 
 72  0.0940  21949  18.9220 
 75  0.1250  —  — 
Numerical results for VMCP with dimension .
 Proposed method  He's method  

It. no.  CPU  It. no.  CPU  
 61  1.2500  —  — 
 64  1.2810  1527  29.8750 
 64  1.2660  150  2.9220 
 64  1.2810  222  4.3440 
10  89  1.7920  1922  37.6250 
1  70  1.3910  —  — 
 88  1.7340  —  — 
 84  1.6560  —  — 
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 selfadaptive implicit method for solving monotone variant variational inequalities. The proposed selfadaptive adjusting rule avoids the difficult task of choosing a "suitable" parameter, which makes the method efficient for initial parameter. Our selfadaptive 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
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