Journal of Inequalities and Applications
Volume 2009 (2009), Article ID 458134, 20 pages
doi:10.1155/2009/458134
Research Article

Self-Adaptive Implicit Methods for Monotone Variant Variational Inequalities

Zhili Ge and Deren Han

Institute of Mathematics, School of Mathematics and Computer Science, Nanjing Normal University, Nanjing 210097, China

Received 26 January 2009; Accepted 24 February 2009

Academic Editor: Ram U. Verma

Copyright © 2009 Zhili Ge and Deren Han. 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.

Abstract

The efficiency of the implicit method proposed by He (1999) depends on the parameter 𝛽 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 𝛽 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.

1. Introduction

Let Ω be a closed convex subset of ℛ 𝑛 and let 𝐹 be a mapping from ℛ 𝑛 into itself. The so-called finite-dimensional variant variational inequalities, denoted by V V I ( Ω , 𝐹 ) , is to find a vector 𝑢 ∈ ℛ 𝑛 , such that 𝐹 ( 𝑢 ) ∈ Ω , ( 𝑣 − 𝐹 ( 𝑢 ) ) ⊤ 𝑢 ≥ 0 , ∀ 𝑣 ∈ Ω , ( 1 . 1 ) while a classical variational inequality problem, abbreviated by V I ( Ω , 𝑓 ) , is to find a vector 𝑥 ∈ Ω , such that 𝑥 − 𝑥 ⊤ 𝑓 ( 𝑥 ) ≥ 0 , ∀ 𝑥 ∈ Ω , ( 1 . 2 ) where 𝑓 is a mapping from ℛ 𝑛 into itself.

Both V V I ( Ω , 𝐹 ) and V I ( Ω , 𝑓 ) 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 V I ( Ω , 𝑓 ) , for example, [3–10], and [11–16] for V V I ( Ω , 𝐹 ) .

It is observed that if 𝐹 is invertible, then by setting 𝑓 = 𝐹 − 1 , the inverse mapping of 𝐹 , V V I ( Ω , 𝐹 ) can be reduced to V I ( Ω , 𝑓 ) . Thus, theoretically, all numerical methods for solving V I ( Ω , 𝑓 ) can be used to solve V V I ( Ω , 𝐹 ) . However, in many practical applications, the inverse mapping 𝐹 − 1 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 V V I ( Ω , 𝐹 ) and recently, the Goldstein's type method was extended from solving V I ( Ω , 𝑓 ) to V V I ( Ω , 𝐹 ) [12, 17].

In [11], He proposed an implicit method for solving general variational inequality problems. A general variational inequality problem is to find a vector 𝑢 ∈ ℛ 𝑛 , such that 𝐹 ( 𝑢 ) ∈ Ω , ( 𝑣 − 𝐹 ( 𝑢 ) ) ⊤ 𝐺 ( 𝑢 ) ≥ 0 , ∀ 𝑣 ∈ Ω . ( 1 . 3 ) When 𝐺 is the identity mapping, it reduces to V V I ( Ω , 𝐹 ) and if 𝐹 is the identity mapping, it reduces to V I ( Ω , 𝐺 ) . He's implicit method is as follows.

(S0)Given 𝑢 0 ∈ 𝑅 𝑛 , 𝛽 > 0 , 𝛾 ∈ ( 0 , 2 ) , and a positive definite matrix 𝑀 .(S1)Find 𝑢 𝑘 + 1 via 𝜃 𝑘 ( 𝑢 ) = 0 , ( 1 . 4 ) where 𝜃 𝑘 𝑢 ( 𝑢 ) = 𝐹 ( 𝑢 ) + 𝛽 𝐺 ( 𝑢 ) − 𝐹 𝑘 𝑢 − 𝛽 𝐺 𝑘 𝑢 + 𝛾 𝜌 𝑘 𝑀 , 𝑀 , 𝛽 − 1 𝑒 𝑢 𝑘 , 𝜌 𝑢 , 𝛽 ( 1 . 5 ) 𝑘 = ‖ ‖ , 𝑀 , 𝛽 𝑒 ( 𝑢 𝑘 ‖ ‖ , 𝛽 ) 2 𝑒 ( 𝑢 𝑘 , 𝛽 ) ⊤ 𝑀 − 1 𝑒 𝑢 𝑘 , , 𝛽 𝑒 ( 𝑢 , 𝛽 ) ∶ = 𝐹 ( 𝑢 ) − 𝑃 Ω [ ] , 𝐹 ( 𝑢 ) − 𝛽 𝐺 ( 𝑢 ) ( 1 . 6 ) with 𝑃 Ω being the projection from ℛ 𝑛 onto Ω , under the Euclidean norm.

He's method is attractive since it solves the general variational inequality problem, which is essentially equivalent to a system of nonsmooth equations 𝑒 ( 𝑢 , 𝛽 ) = 0 , ( 1 . 7 ) via solving a series of smooth equations (1.4). The mapping in the subproblem is well conditioned and many efficient numerical methods, such as Newton's method, can be applied to solve it. Furthermore, to improve the efficiency of the algorithm, He [11] proposed to solve the subproblem approximately. That is, at Step  1, instead of finding a zero of 𝜃 𝑘 , it only needs to find a vector 𝑢 𝑘 + 1 satisfying ‖ ‖ 𝜃 𝑘 𝑢 𝑘 + 1 ‖ ‖ ≤ 𝜂 𝑘 ‖ ‖ 𝑒 𝑢 𝑘 ‖ ‖ , 𝛽 , ( 1 . 8 ) where { 𝜂 𝑘 } is a nonnegative sequence. He proved the global convergence of the algorithm under the condition that the error tolerance sequence { 𝜂 𝑘 } satisfies ∞ 𝑘 = 0 𝜂 𝑘 2 < + ∞ . ( 1 . 9 )

In the above algorithm, there are two parameters 𝛽 > 0 and 𝛾 ∈ ( 0 , 2 ) , which affect the efficiency of the algorithm. It was observed that nearly for all problems, 𝛾 close to 2 is a better choice than smaller 𝛾 , while different problem has different optimal 𝛽 . A suitable parameter 𝛽 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 { 𝛽 𝑘 } , instead of a fixed parameter 𝛽 , 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 𝛽 0 , 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 V V I ( Ω , 𝐹 ) . By replacing the constant factor 𝛽 in (1.4) and (1.5) with a self-adaptive variable positive sequence { 𝛽 𝑘 } , the efficiency of the algorithm can be improved greatly. Moreover, it is also robust to the initial choice of the parameter 𝛽 0 . Thus, for any given problems, we can choose a parameter 𝛽 0 arbitrarily, for example, 𝛽 0 = 1 or 𝛽 0 = 0 . 1 . 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 𝛽 . 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 𝑢 𝑘 + 1 that satisfies (1.8).

Throughout this paper, we make the following assumptions.

Assumption A. The solution set of V V I ( Ω , 𝐹 ) , denoted by Ω ∗ , is nonempty.

Assumption B. The operator 𝐹 is monotone, that is, for any 𝑢 , 𝑣 ∈ ℛ 𝑛 , ( 𝑢 − 𝑣 ) ⊤ ( 𝐹 ( 𝑢 ) − 𝐹 ( 𝑣 ) ) ≥ 0 . ( 1 . 1 0 )

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 𝑥 ∈ ℛ 𝑛 and a symmetric positive definite matrix 𝑀 ∈ ℛ 𝑛 × 𝑛 , we denote √ ‖ 𝑥 ‖ = 𝑥 ⊤ 𝑥 as the Euclidean-norm and ‖ 𝑥 ‖ 𝑀 as the matrix-induced norm, that is, ‖ 𝑥 ‖ 𝑀 ∶ = ( 𝑥 ⊤ 𝑀 𝑥 ) 1 / 2 .

Let Ω be a nonempty closed convex subset of ℛ 𝑛 , and let 𝑃 Ω ( ⋅ ) denote the projection mapping from ℛ 𝑛 onto Ω , under the matrix-induced norm. That is, 𝑃 Ω ( 𝑥 ) ∶ = a r g m i n ‖ 𝑥 − 𝑦 ‖ 𝑀 , 𝑦 ∈ Ω . ( 2 . 1 ) It is known [12, 19] that the variant variational inequality problem (1.1) is equivalent to the projection equation 𝐹 ( 𝑢 ) = 𝑃 Ω 𝐹 ( 𝑢 ) − 𝛽 𝑀 − 1 𝑢 , ( 2 . 2 ) where 𝛽 is an arbitrary positive constant. Then, we have the following lemma.

Lemma 2.1. 𝑢 ∗ is a solution of V V I ( Ω , 𝐹 ) if and only if 𝑒 ( 𝑢 , 𝛽 ) = 0 for any fixed constant 𝛽 > 0 , where 𝑒 ( 𝑢 , 𝛽 ) ∶ = 𝐹 ( 𝑢 ) − 𝑃 Ω 𝐹 ( 𝑢 ) − 𝛽 𝑀 − 1 𝑢 ( 2 . 3 ) 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. Let Ω be a closed convex set in ℛ 𝑛 and let 𝑃 Ω denote the projection operator onto Ω under the matrix-induced norm, then one has ( 𝑤 − 𝑃 Ω ( 𝑣 ) ) ⊤ 𝑀 𝑣 − 𝑃 Ω ( 𝑣 ) ≤ 0 , ∀ 𝑣 ∈ ℛ 𝑛 , ∀ 𝑤 ∈ Ω , ( 2 . 4 ) ‖ ‖ 𝑃 Ω ( 𝑢 ) − 𝑃 Ω ‖ ‖ ( 𝑣 ) 𝑀 ≤ ‖ 𝑢 − 𝑣 ‖ 𝑀 , ∀ 𝑢 , 𝑣 ∈ ℛ 𝑛 . ( 2 . 5 )

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

Lemma 2.3. For a given 𝑢 ∈ ℛ 𝑛 , let ̃ 𝛽 ≥ 𝛽 > 0 . Then it holds that ‖ ‖ 𝑒 ̃ 𝛽 ‖ ‖ 𝑢 , 𝑀 ≥ ‖ 𝑒 ( 𝑢 , 𝛽 ) ‖ 𝑀 . ( 2 . 6 )

Proof. See [20] for a simple proof.

Lemma 2.4. Let 𝑢 ∗ ∈ Ω ∗ , then for all 𝑢 ∈ ℛ 𝑛 and 𝛽 > 0 , one has { [ 𝐹 ( 𝑢 ) − 𝐹 ( 𝑢 ∗ ) ] + 𝛽 𝑀 − 1 ( 𝑢 − 𝑢 ∗ ) } ⊤ 𝑀 𝑒 ( 𝑢 , 𝛽 ) ≥ ‖ 𝑒 ( 𝑢 , 𝛽 ) ‖ 2 𝑀 . ( 2 . 7 )

Proof. It follows from the definition of V V I ( Ω , 𝐹 ) (see (1.1) that { 𝑃 Ω [ 𝐹 ( 𝑢 ) − 𝛽 𝑀 − 1 𝑢 ] − 𝐹 ( 𝑢 ∗ ) } ⊤ 𝛽 𝑢 ∗ ≥ 0 . ( 2 . 8 ) By setting 𝑣 ∶ = 𝐹 ( 𝑢 ) − 𝛽 𝑀 − 1 𝑢 and 𝑤 ∶ = 𝐹 ( 𝑢 ∗ ) in (2.4), we obtain { 𝑃 Ω [ 𝐹 ( 𝑢 ) − 𝛽 𝑀 − 1 𝑢 ] − 𝐹 ( 𝑢 ∗ ) } ⊤ 𝑀 𝑒 ( 𝑢 , 𝛽 ) − 𝛽 𝑀 − 1 𝑢 ≥ 0 . ( 2 . 9 ) Adding (2.8) and (2.9), and using the definition of 𝑒 ( 𝑢 , 𝛽 ) in (2.3), we get { 𝐹 ( 𝑢 ) − 𝐹 ( 𝑢 ∗ ) − 𝑒 ( 𝑢 , 𝛽 ) } ⊤ 𝑀 𝑒 ( 𝑢 , 𝛽 ) − 𝛽 𝑀 − 1 𝑢 − 𝑢 ∗ ≥ 0 , ( 2 . 1 0 ) that is, ( 𝐹 ( 𝑢 ) − 𝐹 ( 𝑢 ∗ ) + 𝛽 𝑀 − 1 ( 𝑢 − 𝑢 ∗ ) ) ⊤ 𝑀 𝑒 ( 𝑢 , 𝛽 ) ≥ ‖ 𝑒 ( 𝑢 , 𝛽 ) ‖ 2 𝑀 + 𝛽 ( 𝐹 ( 𝑢 ) − 𝐹 ( 𝑢 ∗ ) ) ⊤ 𝑢 − 𝑢 ∗ ≥ ‖ 𝑒 ( 𝑢 , 𝛽 ) ‖ 2 𝑀 , ( 2 . 1 1 ) where the last inequality follows from the monotonicity of 𝐹 (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 𝛾 ∈ ( 0 , 2 ) , 𝛽 0 > 0 , 𝑢 0 ∈ ℛ 𝑛 and a positive definite matrix 𝑀 .(S1)Compute 𝑢 𝑘 + 1 such that 𝐹 𝑢 𝑘 + 1 + 𝛽 𝑘 𝑀 − 1 𝑢 𝑘 + 1 𝑢 − 𝐹 𝑘 − 𝛽 𝑘 𝑀 − 1 𝑢 𝑘 𝑢 + 𝛾 𝑒 𝑘 , 𝛽 𝑘 = 0 . ( 3 . 1 ) (S2)If the given stopping criterion is satisfied, then stop; otherwise choose a new parameter 𝛽 𝑘 + 1 ∈ [ 1 / ( 1 + 𝜏 𝑘 ) 𝛽 𝑘 , ( 1 + 𝜏 𝑘 ) 𝛽 𝑘 ] , where 𝜏 𝑘 satisfies ∞ 𝑘 = 0 𝜏 𝑘 < + ∞ , 𝜏 𝑘 ≥ 0 . ( 3 . 2 ) Set 𝑘 ∶ = 𝑘 + 1 and go to Step  1.

From (3.1), we know that 𝑢 𝑘 + 1 is the (exact) unique zero of 𝜃 𝑘 ( 𝑢 ) ∶ = 𝐹 ( 𝑢 ) + 𝛽 𝑘 𝑀 − 1 𝑢 𝑢 − 𝐹 𝑘 − 𝛽 𝑘 𝑀 − 1 𝑢 𝑘 𝑢 + 𝛾 𝑒 𝑘 , 𝛽 𝑘 . ( 3 . 3 ) We refer to the above method as the self-adaptive exact implicit method.

Remark 3.1. According to the assumption 𝜏 𝑘 ≥ 0 and ∑ ∞ 𝑘 = 0 𝜏 𝑘 < + ∞ , we have ∏ ∞ 𝑘 = 0 ( 1 + 𝜏 𝑘 ) < + ∞ . Denote 𝑆 𝜏 ∶ = ∞ 𝑘 = 0 1 + 𝜏 𝑘 . ( 3 . 4 ) Hence, the sequence { 𝛽 𝑘 } ⊂ [ ( 1 / 𝑆 𝜏 ) 𝛽 0 , 𝑆 𝜏 𝛽 0 ] is bounded. Then, let i n f { 𝛽 𝑘 } ∞ 𝑘 = 0 ∶ = 𝛽 𝐿 > 0 and s u p { 𝛽 𝑘 } ∞ 𝑘 = 0 ∶ = 𝛽 𝑈 < + ∞ .

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

Lemma 3.2. Let { 𝑢 𝑘 } be the sequence generated by the proposed self-adaptive exact implicit method. Then for any 𝑢 ∗ ∈ Ω ∗ and 𝑘 > 0 , one has ‖ ‖ ( 𝐹 ( 𝑢 𝑘 + 1 ) − 𝐹 ( 𝑢 ∗ ) ) + 𝛽 𝑘 𝑀 − 1 ( 𝑢 𝑘 + 1 − 𝑢 ∗ ) ‖ ‖ 2 𝑀 ≤ ‖ ‖ ( 𝐹 ( 𝑢 𝑘 ) − 𝐹 ( 𝑢 ∗ ) ) + 𝛽 𝑘 𝑀 − 1 ( 𝑢 𝑘 − 𝑢 ∗ ) ‖ ‖ 2 𝑀 ‖ ‖ − 𝛾 ( 2 − 𝛾 ) 𝑒 ( 𝑢 𝑘 , 𝛽 𝑘 ) ‖ ‖ 2 𝑀 . ( 3 . 5 )

Proof. Using (3.1), we get ‖ ‖ ( 𝐹 ( 𝑢 𝑘 + 1 ) − 𝐹 ( 𝑢 ∗ ) ) + 𝛽 𝑘 𝑀 − 1 ( 𝑢 𝑘 + 1 − 𝑢 ∗ ) ‖ ‖ 2 𝑀 = ‖ ‖ [ ( 𝐹 ( 𝑢 𝑘 ) − 𝐹 ( 𝑢 ∗ ) ) + 𝛽 𝑘 𝑀 − 1 ( 𝑢 𝑘 − 𝑢 ∗ ) ] − 𝛾 𝑒 ( 𝑢 𝑘 , 𝛽 𝑘 ) ‖ ‖ 2 𝑀 ≤ ‖ ‖ ( 𝐹 ( 𝑢 𝑘 ) − 𝐹 ( 𝑢 ∗ ) ) + 𝛽 𝑘 𝑀 − 1 ( 𝑢 𝑘 − 𝑢 ∗ ) ‖ ‖ 2 𝑀 ‖ ‖ − 2 𝛾 𝑒 ( 𝑢 𝑘 , 𝛽 𝑘 ) ‖ ‖ 2 𝑀 + 𝛾 2 ‖ ‖ 𝑒 ( 𝑢 𝑘 , 𝛽 𝑘 ) ‖ ‖ 2 𝑀 = ‖ ‖ ( 𝐹 ( 𝑢 𝑘 ) − 𝐹 ( 𝑢 ∗ ) ) + 𝛽 𝑘 𝑀 − 1 ( 𝑢 𝑘 − 𝑢 ∗ ) ‖ ‖ 2 𝑀 ‖ ‖ − 𝛾 ( 2 − 𝛾 ) 𝑒 ( 𝑢 𝑘 , 𝛽 𝑘 ) ‖ ‖ 2 𝑀 , ( 3 . 6 ) where the inequality follows from (2.7). This completes the proof.

Since 0 < 𝛽 𝑘 + 1 ≤ ( 1 + 𝜏 𝑘 ) 𝛽 𝑘 and 𝐹 is monotone, it follows that ‖ ‖ ( 𝐹 ( 𝑢 𝑘 + 1 ) − 𝐹 ( 𝑢 ∗ ) ) + 𝛽 𝑘 + 1 𝑀 − 1 ( 𝑢 𝑘 + 1 − 𝑢 ∗ ) ‖ ‖ 2 𝑀 = ‖ ‖ ( 𝐹 ( 𝑢 𝑘 + 1 ) − 𝐹 ( 𝑢 ∗ ) ) + 𝛽 𝑘 𝑀 − 1 ( 𝑢 𝑘 + 1 − 𝑢 ∗ ) + ( 𝛽 𝑘 + 1 − 𝛽 𝑘 ) 𝑀 − 1 ( 𝑢 𝑘 + 1 − 𝑢 ∗ ) ‖ ‖ 2 𝑀 = ‖ ‖ ( 𝐹 ( 𝑢 𝑘 + 1 ) − 𝐹 ( 𝑢 ∗ ) ) + 𝛽 𝑘 𝑀 − 1 ( 𝑢 𝑘 + 1 − 𝑢 ∗ ) ‖ ‖ 2 𝑀 + ( 𝛽 𝑘 + 1 − 𝛽 𝑘 ) 2 ‖ ‖ ( 𝑢 𝑘 + 1 − 𝑢 ∗ ) ‖ ‖ 2 𝑀 𝛽 + 2 𝑘 + 1 − 𝛽 𝑘 ( 𝑢 𝑘 + 1 − 𝑢 ∗ ) ⊤ 𝐹 𝑢 𝑘 + 1 𝑢 − 𝐹 ∗ + 𝛽 𝑘 𝑀 − 1 𝑢 𝑘 + 1 − 𝑢 ∗ ≤ ( 1 + 𝜏 𝑘 ) 2 ‖ ‖ ( 𝐹 ( 𝑢 𝑘 + 1 ) − 𝐹 ( 𝑢 ∗ ) ) + 𝛽 𝑘 𝑀 − 1 ( 𝑢 𝑘 + 1 − 𝑢 ∗ ) ‖ ‖ 2 𝑀 , ( 3 . 7 ) where the inequality follows from the monotonicity of the mapping 𝐹 . Combining (3.5) and (3.7), we have ‖ ‖ ( 𝐹 ( 𝑢 𝑘 + 1 ) − 𝐹 ( 𝑢 ∗ ) ) + 𝛽 𝑘 + 1 𝑀 − 1 ( 𝑢 𝑘 + 1 − 𝑢 ∗ ) ‖ ‖ 2 𝑀 ≤ ( 1 + 𝜏 𝑘 ) 2 ‖ ‖ ( 𝐹 ( 𝑢 𝑘 ) − 𝐹 ( 𝑢 ∗ ) ) + 𝛽 𝑘 𝑀 − 1 ( 𝑢 𝑘 − 𝑢 ∗ ) ‖ ‖ 2 𝑀 ‖ ‖ − 𝛾 ( 2 − 𝛾 ) 𝑒 ( 𝑢 𝑘 , 𝛽 𝑘 ) ‖ ‖ 2 𝑀 . ( 3 . 8 )

Now, we give the self-adaptive rule in choosing the parameter 𝛽 𝑘 . For the sake of balance, we hope that ‖ ‖ ( 𝐹 ( 𝑢 𝑘 + 1 ) − 𝐹 ( 𝑢 𝑘 ‖ ‖ ) ) 𝑀 ≈ ‖ ‖ 𝛽 𝑘 𝑀 − 1 ( 𝑢 𝑘 + 1 − 𝑢 𝑘 ) ‖ ‖ 𝑀 . ( 3 . 9 ) That is, for given constant 𝜏 > 0 , if ‖ ‖ ( 𝐹 ( 𝑢 𝑘 + 1 ) − 𝐹 ( 𝑢 𝑘 ‖ ‖ ) ) 𝑀 ‖ ‖ 𝛽 > ( 1 + 𝜏 ) 𝑘 𝑀 − 1 ( 𝑢 𝑘 + 1 − 𝑢 𝑘 ) ‖ ‖ 𝑀 , ( 3 . 1 0 ) we should increase 𝛽 𝑘 in the next iteration; on the other hand, we should decrease 𝛽 𝑘 when ‖ ‖ ( 𝐹 ( 𝑢 𝑘 + 1 ) − 𝐹 ( 𝑢 𝑘 ‖ ‖ ) ) 𝑀 < 1 ‖ ‖ 𝛽 ( 1 + 𝜏 ) 𝑘 𝑀 − 1 ( 𝑢 𝑘 + 1 − 𝑢 𝑘 ) ‖ ‖ 𝑀 . ( 3 . 1 1 ) Let 𝜔 𝑘 = ‖ ‖ ( 𝐹 ( 𝑢 𝑘 + 1 ) − 𝐹 ( 𝑢 𝑘 ‖ ‖ ) ) 𝑀 ‖ ‖ 𝛽 𝑘 𝑀 − 1 ( 𝑢 𝑘 + 1 − 𝑢 𝑘 ) ‖ ‖ 𝑀 . ( 3 . 1 2 ) Then we give 𝛽 𝑘 + 1 ⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩ ∶ = 1 + 𝜏 𝑘 𝛽 𝑘 , i f 𝜔 𝑘 > 1 ( 1 + 𝜏 ) , 1 + 𝜏 𝑘 𝛽 𝑘 , i f 𝜔 𝑘 < 1 , 𝛽 ( 1 + 𝜏 ) 𝑘 , o t h e r w i s e . ( 3 . 1 3 ) 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 𝛽 0 . 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 { 𝑢 𝑘 } generated by the proposed self-adaptive exact implicit method converges to a solution of V V I ( Ω , 𝐹 ) .

Proof. Let 𝜉 𝑘 ∶ = 2 𝜏 𝑘 + 𝜏 𝑘 2 . Then from the assumption that ∑ ∞ 𝑘 = 0 𝜏 𝑘 < + ∞ , we have that ∑ ∞ 𝑘 = 0 𝜉 𝑘 < + ∞ , which means that ∏ ∞ 𝑘 = 0 ( 1 + 𝜉 𝑘 ) < + ∞ . Denote 𝐶 𝑠 ∶ = ∞ 𝑖 = 0 𝜉 𝑖 , 𝐶 𝑝 ∶ = ∞ 𝑖 = 0 1 + 𝜉 𝑖 . ( 3 . 1 4 ) From (3.8), for any 𝑢 ∗ ∈ Ω ∗ , that is, an arbitrary solution of V V I ( Ω , 𝐹 ) , we have ‖ ‖ ( 𝐹 ( 𝑢 𝑘 + 1 ) − 𝐹 ( 𝑢 ∗ ) ) + 𝛽 𝑘 + 1 𝑀 − 1 ( 𝑢 𝑘 + 1 − 𝑢 ∗ ) ‖ ‖ 2 𝑀 ≤ 1 + 𝜉 𝑘 ‖ ‖ ( 𝐹 ( 𝑢 𝑘 ) − 𝐹 ( 𝑢 ∗ ) ) + 𝛽 𝑘 𝑀 − 1 ( 𝑢 𝑘 − 𝑢 ∗ ) ‖ ‖ 2 𝑀 ≤ 𝑘 𝑖 = 0 1 + 𝜉 𝑖 ‖ ‖ ( 𝐹 ( 𝑢 0 ) − 𝐹 ( 𝑢 ∗ ) ) + 𝛽 0 𝑀 − 1 ( 𝑢 0 − 𝑢 ∗ ) ‖ ‖ 2 𝑀 ( 3 . 1 5 ) ≤ 𝐶 𝑝 ‖ ‖ ( 𝐹 ( 𝑢 0 ) − 𝐹 ( 𝑢 ∗ ) ) + 𝛽 0 𝑀 − 1 ( 𝑢 0 − 𝑢 ∗ ) ‖ ‖ 2 𝑀 < + ∞ . ( 3 . 1 6 ) This, together with the monotonicity of the mapping 𝐹 , means that the generated sequence { 𝑢 𝑘 } is bounded.
Also from (3.8), we have ‖ ‖ 𝛾 ( 2 − 𝛾 ) 𝑒 ( 𝑢 𝑘 , 𝛽 𝑘 ) ‖ ‖ 2 𝑀 ≤ ( 1 + 𝜏 𝑘 ) 2 ‖ ‖ ( 𝐹 ( 𝑢 𝑘 ) − 𝐹 ( 𝑢 ∗ ) ) + 𝛽 𝑘 𝑀 − 1 ( 𝑢 𝑘 − 𝑢 ∗ ) ‖ ‖ 2 𝑀 − ‖ ‖ ( 𝐹 ( 𝑢 𝑘 + 1 ) − 𝐹 ( 𝑢 ∗ ) ) + 𝛽 𝑘 + 1 𝑀 − 1 ( 𝑢 𝑘 + 1 − 𝑢 ∗ ) ‖ ‖ 2 𝑀 = ‖ ‖ ( 𝐹 ( 𝑢 𝑘 ) − 𝐹 ( 𝑢 ∗ ) ) + 𝛽 𝑘 𝑀 − 1 ( 𝑢 𝑘 − 𝑢 ∗ ) ‖ ‖ 2 𝑀 − ‖ ‖ ( 𝐹 ( 𝑢 𝑘 + 1 ) − 𝐹 ( 𝑢 ∗ ) ) + 𝛽 𝑘 + 1 𝑀 − 1 ( 𝑢 𝑘 + 1 − 𝑢 ∗ ) ‖ ‖ 2 𝑀 + 𝜉 𝑘 ‖ ‖ ( 𝐹 ( 𝑢 𝑘 ) − 𝐹 ( 𝑢 ∗ ) ) + 𝛽 𝑘 𝑀 − 1 ( 𝑢 𝑘 − 𝑢 ∗ ) ‖ ‖ 2 𝑀 . ( 3 . 1 7 ) Adding both sides of the above inequality, we obtain 𝛾 ( 2 − 𝛾 ) ∞ 𝑘 = 𝑘 0 ‖ ‖ 𝑒 ( 𝑢 𝑘 , 𝛽 𝑘 ) ‖ ‖ 2 𝑀 ≤ ‖ ‖ ( 𝐹 ( 𝑢 0 ) − 𝐹 ( 𝑢 ∗ ) ) + 𝛽 0 𝑀 − 1 ( 𝑢 0 − 𝑢 ∗ ) ‖ ‖ 2 𝑀 + ∞ 𝑘 = 0 𝜉 𝑘 ‖ ‖ ( 𝐹 ( 𝑢 𝑘 ) − 𝐹 ( 𝑢 ∗ ) ) + 𝛽 𝑘 𝑀 − 1 ( 𝑢 𝑘 − 𝑢 ∗ ) ‖ ‖ 2 𝑀 ≤ ‖ ‖ ( 𝐹 ( 𝑢 0 ) − 𝐹 ( 𝑢 ∗ ) ) + 𝛽 0 𝑀 − 1 ( 𝑢 0 − 𝑢 ∗ ) ‖ ‖ 2 𝑀 + ∞ 𝑘 = 0 𝜉 𝑘 𝐶 𝑝 ‖ ‖ ( 𝐹 ( 𝑢 0 ) − 𝐹 ( 𝑢 ∗ ) ) + 𝛽 0 𝑀 − 1 ( 𝑢 0 − 𝑢 ∗ ) ‖ ‖ 2 𝑀 = 1 + 𝐶 𝑠 𝐶 𝑝 ‖ ‖ ( 𝐹 ( 𝑢 0 ) − 𝐹 ( 𝑢 ∗ ) ) + 𝛽 0 𝑀 − 1 ( 𝑢 0 − 𝑢 ∗ ) ‖ ‖ 2 𝑀 < + ∞ , ( 3 . 1 8 ) where the second inequality follows from (3.15). Thus, we have l i m 𝑘 → ∞ ‖ ‖ 𝑒 ( 𝑢 𝑘 , 𝛽 𝑘 ) ‖ ‖ 𝑀 = 0 , ( 3 . 1 9 ) which, from Lemma 2.3, means that l i m 𝑘 → ∞ ‖ ‖ 𝑒 ( 𝑢 𝑘 , 𝛽 𝐿 ) ‖ ‖ 𝑀 ≤ l i m 𝑘 → ∞ ‖ ‖ 𝑒 ( 𝑢 𝑘 , 𝛽 𝑘 ) ‖ ‖ 𝑀 = 0 . ( 3 . 2 0 )
Since { 𝑢 𝑘 } is bounded, it has at least one cluster point. Let 𝑢 be a cluster point of { 𝑢 𝑘 } and let { 𝑢 𝑘 𝑗 } be the subsequence converging to 𝑢 . Since 𝑒 ( 𝑢 , 𝛽 𝐿 ) is continuous, taking limit in (3.20) along the subsequence, we get ‖ ‖ 𝑒 ( 𝑢 , 𝛽 𝐿 ) ‖ ‖ 𝑀 = l i m 𝑗 → ∞ ‖ ‖ 𝑒 ( 𝑢 𝑘 𝑗 , 𝛽 𝐿 ) ‖ ‖ 𝑀 = 0 . ( 3 . 2 1 ) Thus, from Lemma 2.1, 𝑢 is a solution of V V I ( Ω , 𝐹 ) .
In the following we prove that the sequence { 𝑢 𝑘 } has exactly one cluster point. Assume that ̂ 𝑢 is another cluster point of { 𝑢 𝑘 } , which is different from 𝑢 . Because 𝑢 is a cluster point of the sequence { 𝑢 𝑘 } and 𝐹 is monotone, there is a 𝑘 0 > 0 such that ‖ ‖ 𝐹 ( 𝑢 𝑘 0 ) − 𝐹 ( 𝑢 ) + 𝛽 𝑘 0 𝑀 − 1 ( 𝑢 𝑘 0 − ‖ ‖ 𝑢 ) 𝑀 ≤ 𝛿 2 𝐶 𝑝 , ( 3 . 2 2 ) where ‖ ‖ 𝛿 ∶ = ( 𝐹 ( ̂ 𝑢 ) − 𝐹 𝑢 ) + 𝛽 𝑘 0 𝑀 − 1 ( ̂ 𝑢 − ‖ ‖ 𝑢 ) M . ( 3 . 2 3 ) On the other hand, since 𝑢 ∈ Ω ∗ and 𝑢 ∗ is an arbitrary solution, by setting 𝑢 ∗ ∶ = 𝑢 in (3.15), we have for all 𝑘 ≥ 𝑘 0 , ‖ ‖ ( 𝐹 ( 𝑢 𝑘 ) − 𝐹 ( 𝑢 ) ) + 𝛽 𝑘 𝑀 − 1 ( 𝑢 𝑘 − ‖ ‖ 𝑢 ) 2 𝑀 ≤ 𝑘 𝑖 = 𝑘 0 1 + 𝜉 𝑖 ‖ ‖ ( 𝐹 ( 𝑢 𝑖 ) − 𝐹 ( 𝑢 ) ) + 𝛽 𝑖 𝑀 − 1 ( 𝑢 𝑖 − ‖ ‖ 𝑢 ) 2 𝑀 ≤ 𝐶 𝑝 ‖ ‖ ( 𝐹 ( 𝑢 𝑘 0 ) − 𝐹 ( 𝑢 ) ) + 𝛽 𝑘 0 𝑀 − 1 ( 𝑢 𝑘 0 − ‖ ‖ 𝑢 ) 2 𝑀 , ( 3 . 2 4 ) that is, ‖ ‖ ( 𝐹 ( 𝑢 𝑘 ) − 𝐹 ( 𝑢 ) ) + 𝛽 𝑘 𝑀 − 1 ( 𝑢 𝑘 − ‖ ‖ 𝑢 ) 𝑀 ≤ 𝐶 𝑝 1 / 2 ‖ ‖ ( 𝐹 ( 𝑢 𝑘 0 ) − 𝐹 ( 𝑢 ) ) + 𝛽 𝑘 0 𝑀 − 1 ( 𝑢 𝑘 0 − ‖ ‖ 𝑢 ) 𝑀 ≤ 𝛿 2 𝐶 𝑝 1 / 2 . ( 3 . 2 5 ) Then, ‖ ‖ ( 𝐹 ( 𝑢 𝑘 ) − 𝐹 ( ̂ 𝑢 ) ) + 𝛽 𝑘 𝑀 − 1 ( 𝑢 𝑘 ‖ ‖ − ̂ 𝑢 ) 𝑀 ≥ ‖ ‖ ( 𝐹 ( ̂ 𝑢 ) − 𝐹 𝑢 ) + 𝛽 𝑘 𝑀 − 1 ( ̂ 𝑢 − ‖ ‖ 𝑢 ) 𝑀 − ‖ ‖ ( 𝐹 ( 𝑢 𝑘 ) − 𝐹 ( 𝑢 ) ) + 𝛽 𝑘 𝑀 − 1 ( 𝑢 𝑘 − ‖ ‖ 𝑢 ) 𝑀 . ( 3 . 2 6 ) Using the monotonicity of 𝐹 and the choosing rule of 𝛽 𝑘 , we have ‖ ‖ ( 𝐹 ( 𝑢 ) − 𝐹 ( ̂ 𝑢 ) ) + 𝛽 𝑘 𝑀 − 1 ( ‖ ‖ 𝑢 − ̂ 𝑢 ) 2 𝑀 = ‖ ‖ ( 𝐹 ( 𝑢 ) − 𝐹 ( ̂ 𝑢 ) ) + 𝛽 𝑘 − 1 𝑀 − 1 ( 𝑢 − ̂ 𝑢 ) + ( 𝛽 𝑘 − 𝛽 𝑘 − 1 ) 𝑀 − 1 ( ‖ ‖ 𝑢 − ̂ 𝑢 ) 2 𝑀 = ‖ ‖ ( 𝐹 ( 𝑢 ) − 𝐹 ( ̂ 𝑢 ) ) + 𝛽 𝑘 − 1 𝑀 − 1 ( ‖ ‖ 𝑢 − ̂ 𝑢 ) 2 𝑀 + ‖ ‖ ( 𝛽 𝑘 − 𝛽 𝑘 − 1 ) 𝑀 − 1 ( ‖ ‖ 𝑢 − ̂ 𝑢 ) 2 𝑀 𝛽 + 2 𝑘 − 𝛽 𝑘 − 1 𝑢 − ̂ 𝑢 𝑇 𝐹 𝑢 − 𝐹 ( ̂ 𝑢 ) + 𝛽 𝑘 − 1 𝑀 − 1 ≥ 1 𝑢 − ̂ 𝑢 ( 1 + 𝜏 𝑘 − 1 ) 2 ‖ ‖ ( 𝐹 ( 𝑢 ) − 𝐹 ( ̂ 𝑢 ) ) + 𝛽 𝑘 − 1 𝑀 − 1 ( ‖ ‖ 𝑢 − ̂ 𝑢 ) 2 𝑀 ≥ 1 𝐶 𝑝 ‖ ‖ ( 𝐹 ( 𝑢 ) − 𝐹 ( ̂ 𝑢 ) ) + 𝛽 𝑘 0 𝑀 − 1 ( ‖ ‖ 𝑢 − ̂ 𝑢 ) 2 𝑀 . ( 3 . 2 7 ) Combing (3.25)–(3.27), we have that for any 𝑘 ≥ 𝑘 0 , ‖ ‖ ( 𝐹 ( 𝑢 𝑘 ) − 𝐹 ( ̂ 𝑢 ) ) + 𝛽 𝑘 𝑀 − 1 ( 𝑢 𝑘 ‖ ‖ − ̂ 𝑢 ) 𝑀 ≥ 𝛿 𝐶 𝑝 1 / 2 − 𝛿 2 𝐶 𝑝 1 / 2 = 𝛿 2 𝐶 𝑝 1 / 2 > 0 , ( 3 . 2 8 ) which means that ̂ 𝑢 cannot be a cluster point of { 𝑢 𝑘 } . Thus, { 𝑢 𝑘 } has just one cluster point.

4. Inexact Implicit Method and Convergence Analysis

The main task at each iteration of the implicit exact algorithm in the last section is to solve a system of nonlinear equations. To solve it exactly per iteration is time consuming, and there is little justification to solve it exactly, especially when the iterative point is far away from the solution set. Thus, in this section, we propose to solve the subproblem approximately. That is, for a given 𝑢 𝑘 , instead of finding the exact solution of (3.1), we would accept 𝑢 𝑘 + 1 as the new iterate if it satisfies ‖ ‖ 𝐹 ( 𝑢 𝑘 + 1 ) − 𝐹 ( 𝑢 𝑘 ) + 𝛽 𝑘 𝑀 − 1 ( 𝑢 𝑘 + 1 − 𝑢 𝑘 ) + 𝛾 𝑒 ( 𝑢 𝑘 , 𝛽 𝑘 ) ‖ ‖ 𝑀 ≤ 𝜂 𝑘 ‖ ‖ 𝑒 ( 𝑢 𝑘 , 𝛽 𝑘 ) ‖ ‖ 𝑀 , ( 4 . 1 ) where { 𝜂 𝑘 } is a nonnegative sequence with ∑ ∞ 𝑘 = 0 𝜂 𝑘 2 < + ∞ . 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. Let { 𝑢 𝑘 } be the sequence generated by the inexact implicit method. Then there exists a 𝑘 0 > 0 such that for any 𝑘 ≥ 𝑘 0 and 𝑢 ∗ ∈ Ω ∗ , ‖ ‖ ( 𝐹 ( 𝑢 𝑘 + 1 ) − 𝐹 ( 𝑢 ∗ ) ) + 𝛽 𝑘 𝑀 − 1 ( 𝑢 𝑘 + 1 − 𝑢 ∗ ) ‖ ‖ 2 𝑀 ≤ 1 + 4 𝜂 𝑘 2 𝛾 ‖ ‖ ( 2 − 𝛾 ) ( 𝐹 ( 𝑢 𝑘 ) − 𝐹 ( 𝑢 ∗ ) ) + 𝛽 𝑘 𝑀 − 1 ( 𝑢 𝑘 − 𝑢 ∗ ) ‖ ‖ 2 𝑀 − 1 2 ‖ ‖ 𝛾 ( 2 − 𝛾 ) 𝑒 ( 𝑢 𝑘 , 𝛽 𝑘 ) ‖ ‖ 2 𝑀 . ( 4 . 2 )

Proof. Denote 𝜃 𝑘 𝑢 ( 𝑢 ) ∶ = 𝐹 ( 𝑢 ) − 𝐹 𝑘 + 𝛽 𝑘 𝑀 − 1 𝑢 − 𝑢 𝑘 𝑢 + 𝛾 𝑒 𝑘 , 𝛽 𝑘 . ( 4 . 3 ) Then (4.1) can be rewritten as ‖ ‖ 𝜃 𝑘 ( 𝑢 𝑘 + 1 ) ‖ ‖ 𝑀 ≤ 𝜂 𝑘 ‖ ‖ 𝑒 ( 𝑢 𝑘 , 𝛽 𝑘 ) ‖ ‖ 𝑀 . ( 4 . 4 ) According to (4.3) and (2.7), ‖ ‖ 𝐹 ( 𝑢