Regularization and Iterative Methods for Monotone Variational Inequalities
© X. Xu and H.-K. Xu. 2010
Received: 16 September 2009
Accepted: 23 November 2009
Published: 7 December 2009
We provide a general regularization method for monotone variational inequalities, where the regularizer is a Lipschitz continuous and strongly monotone operator. We also introduce an iterative method as discretization of the regularization method. We prove that both regularization and iterative methods converge in norm.
A typical example of monotone operators is the subdifferential of a proper convex lower semicontinuous function.
The dual VIP of (1.1) is the following VIP:
The following equivalence between the dual VIP (1.6) and the primal VIP (1.1) plays a useful role in our regularization in Section 2.
Lemma 1.1 (cf. ).
However, if fails to be Lipschitz continuous or strongly monotone, then the result of the above theorem is false in general. We will assume that is Lipschitz continuous, but do not assume strong monotonicity of . Thus, VIP (1.1) is ill-posed and regularization is needed; moreover, a solution is often sought through iteration methods.
In the special case where is of the form , with being a nonexpansive mapping, regularization and iterative methods for VIP (1.1) have been investigated in literature; see, for example, [5–19]; work related to variational inequalities of monotone operators can be found in [20–25], and work related to iterative methods for nonexpansive mappings can be found in [26–33].
The aim of this paper is to provide a regularization and its induced iteration method for VIP (1.1) in the general case. The paper is structured as follows. In the next section we present a general regularization method for VI (1.1) with the regularizer being a Lipschitz continuous and strongly monotone operator. In Section 3, by discretizing the implicit method of the regularization obtained in Section 2, we introduce an iteration process and prove its strong convergence. In the final section, Section 4, we apply the results obtained in Sections 2 and 3 to a convex minimization problem.
Since VIP (1.1) is usually ill-posed, regularization is necessary, towards which we let be a Lipschitz continuous, everywhere defined, strongly monotone, and single-valued operator. Consider the following regularized variational inequality problem:
To analyze more details of VI (2.1) (or its equivalent fixed point equation (2.2)), we need to impose more assumptions on the operators and . Assume that and are Lipschitz continuous with Lipschiz constants , respectively. We also assume that is -strongly monotone; namely, there is a constant satisfying the property
Below we always assume that satisfies (2.6) so that is a -contraction from into itself. Therefore, for such a choice of , has a unique fixed point in which is denoted as whose asymptotic behavior when is given in the following result.
Now (2.10) follows immediately from (2.15).
Proof of Theorem 2.2.
In Theorem 2.2, we have proved that if the solution set of VIP (1.1) is nonempty, then the net of the solutions of the regularized VIPs (2.1) is bounded (and hence converges in norm). The converse is indeed also true; that is, the boundedness of the net implies that the solution set of VIP (1.1) is nonempty. As a matter of fact, suppose that is bounded and is a constant such that for all .
By Lemma 1.1, we have
3. Iterative Method
From the fixed point equation (2.2), it is natural to consider the following iteration method that generates a sequence according to the recursion:
Assume in addition that
To prove Theorem 3.1, we need a lemma below.
Lemma 3.2 (cf. ).
Proof of Theorem 3.1.
Let , where . By assumption (i) and Lemma 2.1, is a contraction and has a unique fixed point which is denoted by . Moreover, by Theorem 2.2, converges in norm to the unique solution of VI (2.9). Therefore, it suffices to prove that as .
satisfy the assumptions (i)–(iv) of Theorem 3.1.
Consider the constrained convex minimization problem:
It is known that the minimization (4.1) is equivalent to the variational inequality problem:
Therefore, applying Theorems 2.2 and 3.1, we get the following result.
Assume that (4.1) has a solution. Assume in addition that
The authors are grateful to the anonymous referees for their comments and suggestions which improved the presentation of this paper. This paper is dedicated to Professor William Art Kirk for his significant contributions to fixed point theory. The first author was supported in part by a fund (Grant no. 2008ZG052) from Zhejiang Administration of Foreign Experts Affairs. The second author was supported in part by NSC 97-2628-M-110-003-MY3, and by DGES MTM2006-13997-C02-01.
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