# Super-Relaxed ( )-Proximal Point Algorithms, Relaxed ( )-Proximal Point Algorithms, Linear Convergence Analysis, and Nonlinear Variational Inclusions

- Ravi P. Agarwal
^{1, 2}Email author and - Ram U. Verma
^{1, 3}

**2009**:957407

**DOI: **10.1155/2009/957407

© R. P. Agarwal and R. U. Verma. 2009

**Received: **26 June 2009

**Accepted: **30 August 2009

**Published: **27 September 2009

## Abstract

We glance at recent advances to the general theory of maximal (set-valued) monotone mappings and their role demonstrated to examine the convex programming and closely related field of nonlinear variational inequalities. We focus mostly on applications of the super-relaxed ( )-proximal point algorithm to the context of solving a class of nonlinear variational inclusion problems, based on the notion of maximal ( )-monotonicity. Investigations highlighted in this communication are greatly influenced by the celebrated work of Rockafellar (1976), while others have played a significant part as well in generalizing the proximal point algorithm considered by Rockafellar (1976) to the case of the relaxed proximal point algorithm by Eckstein and Bertsekas (1992). Even for the linear convergence analysis for the overrelaxed (or super-relaxed) ( )-proximal point algorithm, the fundamental model for Rockafellar's case does the job. Furthermore, we attempt to explore possibilities of generalizing the Yosida regularization/approximation in light of maximal ( )-monotonicity, and then applying to first-order evolution equations/inclusions.

## 1. Introduction and Preliminaries

where is a set-valued mapping on .

As a matter of fact, Rockafellar did demonstrate the weak convergence and strong convergence separately in two theorems, but for the strong convergence a further imposition of the Lipschitz continuity of at 0 plays the crucial part. Let us recall these results.

Theorem 1.1 (see [1]).

where , , and is bounded away from zero. Suppose that the sequence is bounded in the sense that there exists at least one solution to .

Remark 1.2.

The situation changes when if the convex function attains its minimum nonuniquely.

Next we look, unlike Theorem 1.1, at [1, Theorem 2] in which Rockafellar achieved a linear convergence of the sequence by considering the Lipschitz continuity of at 0 instead.

Theorem 1.3 (see [1]).

As a specialization, we have

That means, the proximal point algorithm for is a minimizing method for .

There is an abundance of literature on proximal point algorithms with applications mostly followed by the work of Rockafellar [1], but we focus greatly on the work of Eckstein and Bertsekas [2], where they have relaxed the proximal point algorithm in the following form and applied to the Douglas-Rachford splitting method. Now let us have a look at the relaxed proximal point algorithm introduced and studied in [2].

Algorithm 1.4.

are scalar sequences.

As a matter of fact, Eckstein and Bertsekas [2] applied Algorithm 1.4 to approximate a weak solution to (1.1). In other words, they established Theorem 1.1 using the relaxed proximal point algorithm instead.

Theorem 1.5 (see [2, Theorem 3]).

then the sequence converges weakly to a zero of .

takes care of the Lipschitz continuity issue.

As we look back into the literature, general maximal monotonicity has played a greater role to studying convex programming as well as variational inequalities/inclusions. Later it turned out that one of the most fundamental algorithms applied to solve these problems was the proximal point algorithm. In [2], Eckstein and Bertsekas have shown that much of the theory of the relaxed proximal point algorithm and related algorithms can be passed along to the Douglas-Rachford splitting method and its specializations, for instance, the alternating direction method of multipliers.

Just recently, Verma [3] generalized the relaxed proximal point algorithm and applied to the approximation solvability of variational inclusion problems of the form (1.1). Recently, a great deal of research on the solvability of inclusion problems is carried out using resolvent operator techniques, that have applications to other problems such as equilibria problems in economics, optimization and control theory, operations research, and mathematical programming.

In this survey, we first discuss in detail the history of proximal point algorithms with their applications to general nonlinear variational inclusion problems, and then we recall some significant developments, especially the relaxation of proximal point algorithms with applications to the Douglas-Rachford splitting method. At the second stage, we turn our attention to over-relaxed proximal point algorithms and their contribution to the linear convergence. We start with some introductory materials to the over-relaxed -proximal point algorithm based on the notion of maximal -monotonicity, and recall some investigations on approximation solvability of a general class of nonlinear inclusion problems involving maximal -monotone mappings in a Hilbert space setting. As a matter fact, we examine the convergence analysis of the over-relaxed -proximal point algorithm for solving a class of nonlinear inclusions. Also, several results on the generalized firm nonexpansiveness and generalized resolvent mapping are given. Furthermore, we explore the real impact of recently obtained results on the celebrated work of Rockafellar, most importantly in the case of over-relaxed (or super-relaxed) proximal point algorithms. For more details, we refer the reader [1–55].

where is the Yosida regularization of , while there is an equivalent form , that is characterized as the Yosida approximation of with parameter . It seems in certain ways that it is easier to solve the Yosida inclusion than (1.1). In other words, provides better solvability conditions under right choice for than itself. To prove this assertion, let us recall the following existence theorem.

Theorem 1.6.

Let be a set-valued maximal monotone mapping on . Then the following statements are equivalent.

(i)An element is a solution to .

(ii) .

where the Lipschitz constant is .

Proof.

This completes the proof.

Indeed, the Yosida approximation and its equivalent form are related to this identity. Let us consider

On the other hand, we have the inverse resolvent identity that lays the foundation of the Yosida approximation.

Lemma 1.7 (see [26, Lemma 12.14]).

Proof.

which is the required assertion.

are single valued, in fact maximal monotone and nonexpansive.

The contents for the paper are organized as follows. Section 1 deals with a general historical development of the relaxed proximal point algorithm and its variants in conjunction with maximal -monotonicity, and with the approximation solvability of a class of nonlinear inclusion problems using the convergence analysis for the proximal point algorithm as well as for the relaxed proximal point algorithm. Section 2 introduces and derives some results on unifying maximal -monotonicity and generalized firm nonexpansiveness of the generalized resolvent operator. In Section 3, the role of the over-relaxed -proximal point algorithm is examined in detail in terms of its applications to approximating the solution of the inclusion problem (1.1). Finally, Section 4 deals with some important specializations that connect the results on general maximal monotonicity, especially to several aspects of the linear convergence.

## 2. General Maximal *η*-Monotonicity

Definition 2.1.

Let be a multivalued mapping on . The map is said to be

Definition 2.2.

Let be a mapping on . The map is said to be

In light of Definitions 2.1(vii) and 2.2(ii), notions of cocoerciveness and firm nonexpansiveness coincide, but differ in applications much depending on the context.

Definition 2.3.

A map is said to be

Definition 2.4.

Let be a multivalued mapping on , and let be another mapping. The map is said to be

Definition 2.5.

A map is said to be maximal -monotone if

(1) is -monotone,

(2) for .

Proposition 2.6.

Let be a -strongly monotone mapping, and let be a maximal -monotone mapping. Then is maximal -monotone for , where is the identity mapping.

Proof.

The proof follows on applying Definition 2.5.

Proposition 2.7 (see [4]).

Let be -strongly monotone, and let be maximal -monotone. Then generalized resolvent operator is single valued, where is the identity mapping.

Proof.

Since is -strongly monotone, it implies . Thus, is single valued.

Definition 2.8.

Proposition 2.9 (see [4]).

Proof.

Proposition 2.10 (see [4]).

Let be a real Hilbert space, let be maximal -monotone, and let be -strongly monotone.

Proof.

When and in Proposition 2.10, we have the following.

Proposition 2.11.

Let be a real Hilbert space, let be maximal -monotone, and let be -strongly monotone.

For and in Proposition 2.10, we find a result of interest as follows.

Proposition 2.12.

Let be a real Hilbert space, let be maximal -monotone, and let be strongly monotone.

For in Proposition 2.10, we have the following result.

Proposition 2.13.

Let be a real Hilbert space, let be maximal -monotone, and let be strongly monotone.

## 3. The Over-Relaxed (*η*)-Proximal Point Algorithm

This section deals with the over-relaxed -proximal point algorithm and its application to approximation solvability of the inclusion problem (1.1) based on the maximal -monotonicity. Furthermore, some results connecting the -monotonicity and corresponding resolvent operator are established, that generalize the results on the firm nonexpansiveness [2], while the auxiliary results on maximal -monotonicity and general maximal monotonicity are obtained.

Theorem 3.1.

Let be a real Hilbert space, and let be maximal -monotone. Then the following statements are mutually equivalent.

(i)An element is a solution to (1.1).

Proof.

It follows from the definition of the generalized resolvent operator corresponding to .

Note that Theorem 3.1 generalizes [2, Lemma 2] to the case of a maximal -monotone mapping.

Next, we present a generalization to the relaxed proximal point algorithm [3] based on the maximal -monotonicity.

Algorithm 3.2 (see [4]).

are scalar sequences such that .

Algorithm 3.3.

are scalar sequences such that .

For in Algorithm 3.2, we have the following.

Algorithm 3.4.

are scalar sequences.

In the following result [4], we observe that Theorems 1.1 and 1.3 are unified and are generalized to the case of the -maximal monotonicity and super-relaxed proximal point algorithm. Also, we notice that this result in certain respects demonstrates the importance of the firm nonexpansiveness rather than of the nonexpansiveness.

Theorem 3.5 (see [4]).

where , , , , and .

Suppose that the sequence is bounded in the sense that there exists at least one solution to .

are scalar sequences such that and .

where , , and sequences and satisfy , , , and .

Proof.

Therefore, . Then, in light of Theorem 3.1, any solution to (1.1) is a fixed point of , and hence a zero of .

Now we begin verifying the boundedness of the sequence leading to .

where .

Thus, the sequence is bounded.

where .

that is, .

where .

where .

for and .

for setting .

Theorem 3.6.

satisfy , , , and .

Then the sequence converges weakly to a solution of (1.1).

Proof.

The proof is similar to that of the first part of Theorem 3.5 on applying the generalized representation lemma.

Theorem 3.7.

where , , , , and .

Suppose that the sequence is bounded in the sense that there exists at least one solution to .

are scalar sequences such that and .

where , , and sequences and satisfy , , and .

Proof.

The proof is similar to that of Theorem 3.5.

## 4. Some Specializations

Finally, we examine some significant specializations of Theorem 3.5 in this section. Let us start with and and applying Proposition 2.11.

Theorem 4.1.

where , , , , and .

Suppose that the sequence is bounded in the sense that there exists at least one solution to .

are scalar sequences such that and .

where , , and sequences and satisfy , , and .

Proof.

Therefore, . Then, in light of Theorem 3.1, any solution to (1.1) is a fixed point of , and hence a zero of .

Now we begin verifying the boundedness of the sequence leading to .

where .

Thus, the sequence is bounded.

where .

that is, .

where .

where .

for and .

for setting .

Second we examine Theorem 3.5 when and , but in this case there is no need to include the proof.

Theorem 4.2.

where , , , , and .

Suppose that the sequence is bounded in the sense that there exists at least one solution to .

are scalar sequences such that and .

where , , and sequences and satisfy , , and .

Finally, we consider the case when in Theorem 3.5, especially using Proposition 2.13. In this situation, the inclusion of the complete proof seems to be appropriate.

Theorem 4.3.

where , , , , and .

Suppose that the sequence is bounded in the sense that there exists at least one solution to .

are scalar sequences such that and .

where , , and sequences and satisfy , , , and .

Proof.

Therefore, . Then, in light of Theorem 3.1, any solution to (1.1) is a fixed point of , and hence a zero of .

Now we begin examining the boundedness of the sequence leading to .

Thus, the sequence is bounded.

where .

Thus, the sequence is bounded.

where .

that is, .

where .

for and .

for setting .

Note that if we set in Theorem 4.3, we get a result connecting [2] to the case of a linear convergence setting, but the algorithm remains overrelaxed (or superrelaxed). In this context, we state the following results before we start examining Theorem 4.7, the main result on linear convergence in the maximal monotone setting. Note that based on Proposition 4.6, notions of cocoercivity and firm nonexpansiveness coincide, though it is well known that they may differ in usage much depending on the context.

Theorem 4.4.

Let be a real Hilbert space, and let be maximal monotone. Then the following statements are mutually equivalent.

(i)An element is a solution to (1.1).

Proof.

It follows from the definition of the generalized resolvent operator corresponding to .

Next, we present the super-relaxed Proximal point algorithm based on the maximal monotonicity.

Algorithm 4.5.

are scalar sequences such that .

Proposition 4.6.

Theorem 4.7.

where , , , , , and .

Suppose that the sequence is bounded in the sense that there exists at least one solution to .

are scalar sequences such that and .

where , , and sequences and satisfy , , , and .

Proof.

Therefore, . Then, in light of Theorem 4.4, any solution to (1.1) is a fixed point of , and hence a zero of .

Now we begin examining the boundedness of the sequence leading to .

Therefore, the sequence is bounded.

where .

Thus, the sequence is bounded.

where .

that is, .

where .

for and .

for setting .

## Authors’ Affiliations

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