# Self-Adaptive Implicit Methods for Monotone Variant Variational Inequalities

## Abstract

The efficiency of the implicit method proposed by He (1999) depends on the parameter Open image in new window 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 Open image in new window 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 Open image in new window is a mapping from Open image in new window into itself.

Both Open image in new window and Open image in new window 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 Open image in new window , for example, [3, 4, 5, 6, 7, 8, 9, 10], and [11, 12, 13, 14, 15, 16] for Open image in new window .

It is observed that if Open image in new window is invertible, then by setting Open image in new window , the inverse mapping of Open image in new window can be reduced to Open image in new window . Thus, theoretically, all numerical methods for solving Open image in new window can be used to solve Open image in new window . However, in many practical applications, the inverse mapping Open image in new window 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 Open image in new window and recently, the Goldstein's type method was extended from solving Open image in new window to Open image in new window [12, 17].

When Open image in new window is the identity mapping, it reduces to Open image in new window and if Open image in new window is the identity mapping, it reduces to Open image in new window . He's implicit method is as follows.

(S0)Given Open image in new window , and a positive definite matrix Open image in new window .

with Open image in new window being the projection from Open image in new window onto Open image in new window , under the Euclidean norm.

In the above algorithm, there are two parameters Open image in new window and Open image in new window , which affect the efficiency of the algorithm. It was observed that nearly for all problems, Open image in new window close to Open image in new window is a better choice than smaller Open image in new window , while different problem has different *optimal* Open image in new window . A suitable parameter Open image in new window 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 Open image in new window , instead of a fixed parameter Open image in new window , 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 Open image in new window , 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 Open image in new window . By replacing the constant factor Open image in new window in (1.4) and (1.5) with a self-adaptive variable positive sequence Open image in new window , the efficiency of the algorithm can be improved greatly. Moreover, it is also robust to the initial choice of the parameter Open image in new window . Thus, for any given problems, we can choose a parameter Open image in new window arbitrarily, for example, Open image in new window or Open image in new window . 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 Open image in new window . 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 Open image in new window that satisfies (1.8).

Throughout this paper, we make the following assumptions.

Assumption A.

The solution set of Open image in new window , denoted by Open image in new window , 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 Open image in new window and a symmetric positive definite matrix Open image in new window , we denote Open image in new window as the Euclidean-norm and Open image in new window as the matrix-induced norm, that is, Open image in new window .

where Open image in new window 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 Open image in new window (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 Open image in new window , Open image in new window , Open image in new window and a positive definite matrix Open image in new window .

Set Open image in new window and go to Step??1.

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

Remark 3.1.

Hence, the sequence Open image in new window is bounded. Then, let Open image in new window and Open image in new window .

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, 22, 23, 24] for solving variational inequality problems, where the numerical results indicated its efficiency and robustness to the choice of the initial parameter Open image in new window . 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 Open image in new window generated by the proposed self-adaptive exact implicit method converges to a solution of Open image in new window .

Proof.

This, together with the monotonicity of the mapping Open image in new window , means that the generated sequence Open image in new window is bounded.

Thus, from Lemma 2.1, Open image in new window is a solution of Open image in new window .

which means that Open image in new window cannot be a cluster point of Open image in new window . Thus, Open image in new window has just one cluster point.

## 4. Inexact Implicit Method and Convergence Analysis

where Open image in new window is a nonnegative sequence with Open image in new window . 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 Open image in new window generated by the proposed self-adaptive inexact implicit method converges to a solution point of Open image in new window .

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].

Proposed method | He's method | Proposed method | He's method | |||||

It. no. | CPU | It. no. | CPU | It. no. | CPU | It. no. | CPU | |

25 | 0.3910 | 100 | 1.0780 | 34 | 50.4850 | — | — | |

20 | 0.3120 | 37 | 0.4850 | 25 | 39.8440 | 17 | 25.0940 | |

26 | 0.4060 | 350 | 5.8750 | 33 | 61.4070 | — | — |

Numerical results for VMCP with dimension Open image in new window .

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 Open image in new window .

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 Open image in new window .

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 Open image in new window .

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 Open image in new window , while He's method is efficient for proper choice of Open image in new window . 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 Open image in new window .

## 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 Open image in new window is monotone and there is at least one solution of the problem. The reported preliminary numerical results verified our assertion.

## Notes

### 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.

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