Regularization and Iterative Methods for Monotone Variational Inequalities

Abstract

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.

1. Introduction

Variational inequalities (VIs) have widely been studied (see the monographs [1–3]). A monotone variational inequality problem (VIP) is stated as finding a point with the following property:

(1.1)

where is a nonempty closed convex subset of a real Hilbert space with inner product and norm , respectively, and is a monotone operator in with domain .

Recall that is monotone if

(1.2)

A typical example of monotone operators is the subdifferential of a proper convex lower semicontinuous function.

Variational inequality problems are equivalent to fixed point problems. As a matter of fact, solves VIP (1.1) if and only if solves the following fixed point problem (FPP), for any ,

(1.3)

where is the metric (or nearest point) projection from onto ; namely, for each , is the unique point in with the property

(1.4)

The equivalence between VIP (1.1) and FPP (1.3) is an immediate consequence of the following characterization of :

(1.5)

The dual VIP of (1.1) is the following VIP:

(1.6)

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. [4]).

Assume that is monotone and weakly continuous along segments (i.e., weakly as for ), then the dual VIP (1.6) is equivalent to the primal VIP (1.1).

To guarantee the existence and uniqueness of a solution of VIP (1.1), one has to impose conditions on the operator . The following existence and uniqueness result is well known.

Theorem 1.2.

If is Lipschitz continuous and strongly monotone, then there exists one and only one solution to VIP (1.1).

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.

2. Regularization

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:

(2.1)

Since is strongly monotone, VI (2.1) has a unique solution which is denoted by . Indeed, VI (2.1) is equivalent to the fixed point equation

(2.2)

where , with .

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

(2.3)

Lemma 2.1.

If is chosen in such a way that

(2.4)

then is a contraction with contraction coefficient

(2.5)

Moreover, if

(2.6)

then

(2.7)

hence, is a -contraction.

Proof.

Noticing that is -Lipschitzian and -strongly monotone, we deduce that, for ,

(2.8)

It turns out that if satisfies (2.4), then is a contraction with coefficient given by the left side of (2.5).

Finally, it is straightforward that (2.7) holds provided that satisfies (2.6).

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.

Theorem 2.2.

Assume that

(a) is monotone on and weakly continuous along segments in (i.e., weakly as for ),

(b) is -monotone on ,

(c)the solution set of VI (1.1) is nonempty.

For , let be the unique solution of the regularized VIP (2.1). Then, as , converges in norm to a point in which is the unique solution of the VIP

(2.9)

Therefore, if one takes to be the identity operator, then the regularized solution of the corresponding regularized VIP (2.1) converges in norm to the minimal norm point of the solution set .

To prove Theorem 2.2, we first prove the boundedness of the net .

Lemma 2.3.

Assume that is monotone on . Assume conditions (b) and (c) in Theorem 2.2. Then is bounded; indeed, for any ,

(2.10)

Proof.

We have (2.1) holds for all . In particular, for , we have

(2.11)

It turns out that

(2.12)

Since is monotone and is -strongly monotone, we have

(2.13)

Substituting them into (2.12) we obtain

(2.14)

However, since , . We therefore get from (2.14) that

(2.15)

Now (2.10) follows immediately from (2.15).

Proof of Theorem 2.2.

Since is bounded by Lemma 2.3, the set of weak limit points as of the net , , is nonempty. Pick a and let be a null sequence in the interval such that weakly as . We first show that . To see this we use the equivalent dual VI of (2.1):

(2.16)

Thus, we have, for all and ,

(2.17)

Taking the limit as yields that

(2.18)

It turns out that .

We next prove that the sequence actually converges to strongly. Replacing in (2.15) with gives

(2.19)

Now it is straightforward from (2.19) that the weak convergence to of implies strong convergence to of .

The relation (2.15) particularly implies that, for ,

(2.20)

which in turns implies that every point solves the VIP

(2.21)

or equivalently, the VIP

(2.22)

However, since is strongly monotone, the solution to VIP (2.22) is unique. This has shown that the unique solution of VIP (2.22) is the strong limit of the net .

Finally, if is the identity operator, then VIP (2.22) is reduced to

(2.23)

This is equivalent to

(2.24)

which immediately implies that for all and hence is the minimal norm of .

Remark 2.4.

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

(2.25)

Since is bounded, we can easily see that every weak cluster point of the net solves the VIP

(2.26)

This is the dual VI to the primal VI (2.1); hence is a solution of VI (2.1) by Lemma 1.1.

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:

(3.1)

where the initial guess is selected arbitrarily, and and are two sequences of positive numbers in . Put in another way, is the unique solution in of the following VIP:

(3.2)

Theorem 3.1.

Assume that

(a) is -Lipschitz continuous and monotone on ,

(b) is -Lipschitz continuous and -monotone on

(c)the solution set of VI (1.1) is nonempty.

Assume in addition that

(i)

(ii) as ,

(iii),

(iv),

then the sequence generated by the algorithm (3.1) converges in norm to the unique solution of VI (2.9).

To prove Theorem 3.1, we need a lemma below.

Lemma 3.2 (cf. [20]).

Assume that is a sequence of nonnegative real numbers such that

(3.3)

where and are real sequences such that

(i) for all , and ;

(ii)

then .

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 .

To see this, observing that is a -contraction, we obtain

(3.4)

However, we have

(3.5)

Since is bounded, it turns out that, for an appropriate constant ,

(3.6)

Substituting (3.6) into (3.4) and setting , we get

(3.7)

where

(3.8)

with . Assumptions (iii) and (iv) assure that and as , respectively. Therefore, we can apply lemma to (3.7) to conclude that ; hence, in norm.

Remark 3.3.

Assume satisfy , then it is not hard to see that for an appropriate constant ,

(3.9)

satisfy the assumptions (i)–(iv) of Theorem 3.1.

4. Application

Consider the constrained convex minimization problem:

(4.1)

where is a closed convex subset of a real Hilbert space and is a real-valued convex function. Assume that is continuously differentiable with a Lipschitz continuous gradient:

(4.2)

where is a constant.

It is known that the minimization (4.1) is equivalent to the variational inequality problem:

(4.3)

Therefore, applying Theorems 2.2 and 3.1, we get the following result.

Theorem 4.1.

Assume the Lipschitz continuity (4.2) for the gradient .

• (a) For , let be the unique solution of the regularized VIP

(4.4)

Equivalently, is the unique solution in of the regularized minimization problem:

(4.5)

Then, as , remains bounded if and only if (4.1) has a solution, and in this case, converges in norm to the minimal norm solution of (4.1).

• (b) Assume that (4.1) has a solution. Assume in addition that

(i)

(ii) as ,

(iii),

(iv)

Starting , one defines by the iterative algorithm

(4.6)

Then converges in norm to the minimum-norm solution of the constrained minimization problem (4.1).

Proof.

Apply Theorems 2.2 and 3.1 to the case where and is the identity operator to get the conclusions in (a) and (b).