# An Approximate Proximal Point Algorithm for Maximal Monotone Inclusion Problems

Authors: Lingling Huang; Sanyang Liu; Weifeng Gao
Journal: Mathematical Problems in Engineering
 (2011)

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## Abstract

This paper presents and analyzes a strongly convergent approximate proximal point algorithm for finding zeros of maximal monotone operators in Hilbert spaces. The proposed method combines the proximal subproblem with a more general correction step which takes advantage of more information on the existing iterations. As applications, convex programming problems and generalized variational inequalities are considered. Some preliminary computational results are reported.


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## Body

## 1. Introduction

LetH be a real Hilbert space with inner product 〈·,·〉 and induced norm ∥·∥=〈·,·〉. A set-valued operator T:H→2H is maximal monotone. A canonical problem associated with T is the maximal monotone inclusion problem, that is, to find a vector z∈H such that(1.1)0∈T(z).
The maximal monotone inclusion problem provides a powerful general framework for the study of many important optimization problems, such as convex programming problems and variational inequalities, see [1, 2], for example. Nowadays, it has received considerable attention. The interested readers may consult the monographs by Facchinei and Pang [3] and the survey papers [4–14].Many methods have been proposed to solve the maximal monotone inclusion problem. One of the classical methods is the proximal point algorithm (PPA) which was originally proposed by Martinet [5]. Let uk∈H be the current iteration, PPA generates the next iteration by solving the following proximal subproblem:(1.2)uk∈u+βkT(u).
If the sequence {βk} is chosen bounded from above zero, then the sequence {uk} generated by (1.2) converges weakly to a solution of (1.1). However, since (1.2) is a nonsmooth equation and an implicit scheme, solving (1.2) exactly is either impossible or as difficult as solving the original problem (1.1), see [6]. This makes straightforward applications of PPA impractical in many cases. To overcome this drawback, Rockafellar [7] generalized PPA and presented the following approximate proximal point algorithm:(1.3)uk+ek∈u+βkT(u),
where the error sequence {ek} satisfies a summable error criterion. Because of the relaxed accuracy requirement, the approximate proximal point algorithm is more practical than the exact one. Furthermore, Rockafellar posed an open question: the sequence generated by (1.3) converges strongly or not. In 1992, by exhibiting a proper closed convex function in a infinite-dimensional Hilbert space l2, Güler [8] showed that it does not converge strongly in general. Naturally, the question arises whether PPA can be modified preferably in a simple way, so that strong convergence is guaranteed.There is one point deserving to be paid attention to. The weak and strong convergences are only conceptions in the infinite dimensional spaces. Many real-world problems in economics and engineering are modeled in the infinite dimensional spaces, such as the optimal control and structural design problems. However, when we solve infinite-dimensional problems, numerical implementations of algorithms are certainly applied to finite dimensional approximations of these problems. Nevertheless, as is pointed out in [9], it is important to develop convergence theory for the infinite-dimensional case, because it guarantees the robustness and stability with respect to discretization schemes employed for obtaining finite dimensional approximations of infinite dimensional problems.Recently, a number of researchers have concentrated on the developments of the approximate proximal point algorithms on theoretical analysises, algorithm designs, and practical applications. To find the solution of the desired problem in a restricted areaC⊂H, many authors used an additional projection or extragradient step to correct the approximate solution and thus presented the prediction-correction proximal point algorithms. Recent developments on the approximate proximal point algorithms also focus on replacing the correction step with more general steps, see, for example, [10–13]. To mention a few, for any fixed vector u∈C, Zhou et al. [12] presented the following correction step:(1.4)uk+1=αku+(1-αk)PC(ũk-ek),
where ũk∈C is generated by (1.3). For suitably chosen parameter sequence {αk} and under certain assumptions on the error term ek, they proved the method converges strongly. Later, Qin et al. [13] developed a more general iterative scheme:(1.5)uk+1=αku+βkPC(ũk-ek)+γkPCfk,
where ũk∈C is generated by (1.3) and {fk} is a bounded sequence. For suitably chosen parameter sequences {αk}, {βk}, and {γk} and under certain assumptions on the error term ek, they got the strong convergence of the method too.In this paper, we propose a strongly convergent approximate proximal point algorithm for the maximal monotone inclusion problems by combining the proximal subproblem (1.3) with a more general correction step. Compared with methods [10–13], the proposed method takes advantage of more information on the existing iterations. For practical implementation, we give two applications of the proposed method to convex programming problems and generalized variational inequalities. Preliminary numerical experiments, reported in Section 5, demonstrate that the efficiency of the method is in practice.This paper is organized as follows. Section2 introduces some useful preliminaries. Section 3 describes the proposed method formally and presents the convergence analysis. Section 4 discusses some applications of the proposed method. Section 5 presents some numerical experiments, and some final conclusions are given in the last section.Throughout this paper, we assume that the solution set of (1.1), denoted by T-10, is nonempty.


## 2. Preliminaries

This section summarizes some fundamental concepts and lemmas that are useful in the consequent analysis.Definition 2.1.
LetHbe a real Hilbert space and let T:H→2H be a set-valued operator. Then(i)
the effective domain ofT, denoted by D(T), is
(2.1)D(T)={u:∃v∈H,(u,v)∈T};(ii)
the range or image ofT, denoted by R(T), is
(2.2)R(T)={v:u∈H,(u,v)∈T};(iii)
the graph ofT, denoted by G(T), is
(2.3)G(T)={(u,v)∈H×H:u∈D(T),v∈R(T)};(iv)
the inverse ofT, denoted by T-1, is
(2.4)T-1={(v,u):(u,v)∈T}.Definition 2.2.
LetH be a real Hilbert space and let T:H→2H be a set-valued operator. Then, T is called monotone on H if
(2.5)〈u-u′,v-v′〉≥0,∀(u,v),(u′,v′)∈T.
Furthermore, a monotone operator T is called maximal monotone if its graph G(T) is not properly contained in the graph of any other monotone operator on H.Definition 2.3.
LetH be a real Hilbert space, let C be a nonempty closed convex subset of H, and let  T be an operator from C into C. Then,(i)
T is called firmly nonexpansive if
(2.6)‖v′-v‖2≤〈u′-u,v′-v〉,∀(u,v),(u′,v′)∈T;(ii)
T is called nonexpansive if
(2.7)‖v′-v‖≤‖u′-u‖,∀(u,v),(u′,v′)∈T.Definition 2.4.
LetHbe a real Hilbert space and let C be a nonempty closed convex subset of H. Then, the orthogonal projection from H onto C, denoted by PC(·), is
(2.8)PC(u)=argmin⁡⁡{‖u-v‖:v∈C},∀u∈H.It is easy to verify that the orthogonal projection operator is nonexpansive.Definition 2.5.
Given any positive scalarβ and operator T, define the resolvent of T by
(2.9)Jβ=(I+βT)-1,
where I denotes the identity operator on H. Also, define the Yosida approximation Tβ by
(2.10)Tβ=1β(I-Jβ).We know thatTβx∈TJβx for all x∈H, ∥Tβx∥≤|Tx| for all x∈D(T), where |Tx|=inf{∥y∥:y∈Tx} and T-10=F(Jβ) for all β>0.In the following, we list some useful lemmas.Lemma 2.6 (see [14]).
Letβ be any positive scalar. An operator  T is monotone if and only if its resolvent  Jβ is firmly nonexpansive. Furthermore,  T is maximal monotone if and only if  Jβ is firmly nonexpansive and  D(Jβ)=H.Lemma 2.7 (see [15]).
LetT be a maximal monotone operator and let  β be a positive scalar, then
(2.11)u∈Jβ-1(ũ)⟺ũ=Jβ(u).Lemma 2.8 (see [16]).
For allu∈H,  limt→∞Jtu exists and it is the point of  T-10 nearest to  u.Lemma 2.9 (see [17]).
Let{ak},  {bk} and  {ck} be sequences of positive numbers satisfying:
(2.12)ak+1≤(1-tk)ak+bk+ck,k≥0,
where {tk} is a sequence in [0,1]. Assume that the following conditions are satisfied(i)
tk→0 as k→∞ and ∑k=0∞tk=∞;(ii)
bk=o(tk);(iii)
∑k=0∞ck<∞.Then limk→∞ak=0.


## 3. The Algorithm and Convergence Analysis

In this section, we analyze the proposed approximate proximal point algorithm formally.Algorithm 3.1.
Givenu0∈C,u∈Cand{βk}⊂(0,∞)withβk→∞ask→∞.Findũk∈Csatisfying
(3.1)uk+ek∈ũk+βkT(ũk),
under the inexact error criterion:
(3.2)‖ek‖≤ηk‖uk-ũk‖,sup⁡k≥0ηk=v<1.
Generate the new iteration by
(3.3)uk+1=λku+δkPC[uk-αk(uk-ũk+ek)]+γkũk,
where the stepsize αk is defined by
(3.4)αk=〈uk-ũk,uk-ũk+ek〉‖uk-ũk+ek‖2,
and {λk},{δk}, and {γk} are real sequences in [0,1] satisfying(i)
λk+δk+γk=1;(ii)
limk→∞λk=0 and ∑k=0∞λk=∞;(iii)
∑k=0∞δk<∞.The following remark gives the relationships between Algorithm3.1 and some existing algorithms.Remark 3.2.
Whenλk=0,δk=1,γk=0,αk=1, Algorithm 3.1 reduces to the method proposed by He et al. [10]; when λk=0,δk=1,γk=0 without considering the error term ek in (3.3), Algorithm 3.1 reduces to the method proposed by Yang and He [11]; when γk=0,αk=1, Algorithm 3.1 reduces to the method proposed by Zhou et al. [12]; when αk=1, Algorithm 3.1 reduces to the method proposed by Qin et al. [13] with PCfk=ũk.In the following, we give the convergence analysis of Algorithm3.1, beginning with some lemmas. For convenience, we use the notation(3.5)dk=uk-ũk+ek.Lemma 3.3 (see [6]).
LetH be a real Hilbert space and let  C be a nonempty closed convex subset of  H. For given  uk∈H,βk>0 and  ek∈H, there exists  ũk∈C conforming to the set-valued equation (3.1) Furthermore, for any  p∈T-10, one has
(3.6)〈uk-ũk,dk〉≤〈uk-p,dk〉.Lemma 3.4.
Letαk be defined by (3.4), then
(3.7)αk>12.Proof.
By the notation ofdk, it follows from (3.2) that
(3.8)〈uk-ũk,dk〉=‖uk-ũk‖2+〈xk-ũk,ek〉>12‖uk-ũk‖2+〈uk-ũk,ek〉+12‖ek‖2=12‖dk‖2.
By the selection of αk, the proof is complete.Lemma 3.5.
Letp be any zero of   T in C, then
(3.9)‖PC(uk-αkdk)-p‖≤‖uk-p‖.Proof.
Sincep∈C and PC(·) is nonexpansive, by Lemma 3.3, we have
(3.10)‖PC(uk-αkdk)-p‖2≤‖uk-αkdk-p‖2=‖uk-p‖2-2αk〈uk-p,dk〉+αk2‖dk‖2≤‖uk-p‖2-2αk〈uk-ũk,dk〉+αk2‖dk‖2.
Considering the last two terms of the above equality, by the definition of αk and dk, we have
(3.11)-2αk〈uk-ũk,dk〉+αk2‖dk‖2=-αk〈uk-ũk,dk〉≤-αk[‖uk-ũk‖2-‖uk-ũk‖‖ek‖].
Taking into account that ∥ek∥≤ηk∥uk-ũk∥, by Lemma 3.4, we further obtain
(3.12)‖PC(uk-αkdk)-p‖2≤‖uk-p‖2-12(1-ηk)‖uk-ũk‖2.
Since supk≥0ηk=v<1, the assertion follows from (3.12) immediately.We now prove the strong convergence of Algorithm3.1Theorem 3.6.
Let{uk} be generated by Algorithm 3.1. Suppose that  ∑k=0∞∥ek∥<∞. Then, the sequence  {uk} converges strongly to a zero point  z of  T, where  z=limt→∞Jtu.Proof.
We divide the proof into three parts.Claim 1.
Show that the sequence{uk} is bounded.
For anyp∈T-10. Set M=max{∥u-p∥,∥u0-p∥,supk≥0∥ũk-p∥}. We want to prove that
(3.13)‖uk-p‖≤M,∀k≥0.
It is easy to see that (3.13) is true for k=0. Now, assume that (3.13) holds for some k≥0. We prove that (3.13) holds for k+1. By the definition of uk+1, it follows from λk+δk+γk=1 that
(3.14)‖uk+1-p‖=‖λku+δkPC(uk-αkdk)+γkũk-p‖≤λk‖u-p‖+δk‖PC(uk-αkdk)-p‖+γk‖ũk-p‖≤λk‖u-p‖+δk‖uk-p‖+γk‖ũk-p‖≤M,
where the second inequality follows from Lemma 3.5. Hence, the sequence {uk} is bounded.Claim 2.
Show thatlim supk→∞〈u-z,uk+1-z〉≤0, where z=limt→∞Jtu. Note that the existence of limt→∞Jtu is guaranteed by Lemma 2.8.
SinceT is maximal monotone, Ttu∈TJtu and Tβkuk∈TJβkuk, we have(3.15)〈u-Jtu,Jβkuk-Jtu〉=-t〈Ttu,Jtu-Jβkuk〉=-t〈Ttu-Tβkuk,Jtu-Jβkuk〉-t〈Tβkuk,Jtu-Jβkuk〉≤-tβk〈uk-Jβkuk,Jtu-Jβkuk〉.
Since βk→∞ as k→∞, for any t>0, we obtain
(3.16)limsup⁡k→∞〈u-Jtu,Jβkuk-Jtu〉≤0.
By the nonexpansivity of Jβk, we have
(3.17)‖Jβk(uk+ek)-Jβkuk‖≤‖uk+ek-uk‖=‖ek‖.
Since ∥ek∥→0ask→∞, we obtain
(3.18)lim⁡k→∞⁡‖Jβk(uk+ek)-Jβkuk‖=0,
which combines with (3.16) yielding that
(3.19)limsup⁡k→∞⁡〈u-Jtu,Jβk(uk+ek)-Jtu〉≤0.
By Lemma 2.7, we get
(3.20)‖PC(ũk-ek)-Jβk(uk+ek)‖≤‖(ũk-ek)-Jβk(uk+ek)‖=‖ek‖.
Thus,
(3.21)lim⁡k→∞⁡‖PC(ũk-ek)-Jβk(uk+ek)‖=0.
Furthermore, we have
(3.22)limsup⁡k→∞⁡〈u-Jtu,PC(ũk-ek)-Jtu〉≤0.
Adopting the notation u̅k=(δk/(δk+γk))PC(uk-αkdk)+(γk/(δk+γk))ũk. Since λk+δk+γk=1, then
(3.23)uk+1=λku+(1-λk)u̅k.
Since PC(·) is nonexpansive, by the notation of dk, we obtain
(3.24)‖u̅k-PC(ũk-ek)‖=‖δkδk+γkPC(uk-αkdk)+γkδk+γkũk-PC(ũk-ek)‖≤δkδk+γk‖PC(uk-αkdk)-PC(ũk-ek)‖+γkδk+γk‖ũk-PC(ũk-ek)‖≤δkδk+γk‖uk-αkdk-ũk+ek‖+γkδk+γk‖ek‖=|1-αk|δkδk+γk‖dk‖+γkδk+γk‖ek‖.
Since ∥ek∥→0 as k→∞ and ∑k=0∞δk<∞, we get
(3.25)lim⁡k→∞⁡‖u̅k-PC(ũk-ek)‖=0.
By using (3.23), we get
(3.26)‖uk+1-PC(ũk-ek)‖=‖λku+(1-λk)u̅k-PC(ũk-ek)‖(3.27)≤λk‖u-PC(ũk-ek)‖+(1-λk)‖u̅k-PC(ũk-ek)‖.
Since limk→∞λk=0, it follows from (3.25) and (3.27) that
(3.28)lim⁡k→∞⁡‖uk+1-PC(ũk-ek)‖=0.
Combining (3.28) with (3.22), we obtain
(3.29)limsup⁡k→∞⁡〈u-Jtu,uk+1-Jtu〉≤0.
Note that z=limt→∞Jtu, we get
(3.30)limsup⁡k→∞⁡〈u-z,uk+1-z〉≤0.Claim 3.
Show thatuk→z as k→∞.
By Lemma2.7 and by the nonexpansivity of Jβk, we have(3.31)‖ũk-z‖=‖Jβk(uk+ek)-z‖(3.32)≤‖uk-z‖+‖ek‖.
By the definition of uk+1, we have
(3.33)uk+1-z=λku+δkPC(uk-αkdk)+γkũk-z=λk(u-z)+δk[PC(uk-αkdk)-ũk]+(1-λk)(ũk-z).
Since PC(·) is nonexpansive, we obtain
(3.34)‖uk+1-z‖2=〈λk(u-z)+δk[PC(uk-αkdk)-ũk]+(1-λk)(ũk-z),uk+1-z〉=λk〈u-z,uk+1-z〉+δk〈PC(uk-αkdk)-ũk,uk+1-z〉+(1-λk)〈ũk-z,uk+1-z〉≤λk〈u-z,uk+1-z〉+δk‖PC(uk-αkdk)-ũk‖‖uk+1-z‖+(1-λk)‖ũk-z‖‖uk+1-z‖≤λk〈u-z,uk+1-z〉+δk‖uk-αkdk-ũk‖‖uk+1-z‖+(1-λk)‖ũk-z‖‖uk+1-z‖.
Now, we consider the last two terms on the right-hand side of the above equality. Since λk∈[0,1], it follows from (3.32) that
(3.35)δk∥uk-αkdk-ũk∥∥uk+1-z∥+(1-λk)∥ũk-z∥∥uk+1-z∥≤δk‖uk-αkdk-ũk‖‖uk+1-z‖+(1-λk)(‖uk-z‖+‖ek‖)‖uk+1-z‖≤1-λk2(‖uk-z‖2+‖uk+1-z‖2)+[δk‖uk-αkdk-ũk‖+(1-λk)‖ek‖]‖uk+1-z‖≤1-λk2‖uk-z‖2+12‖uk+1-z‖2+[δk‖uk-αkdk-ũk‖+(1-λk)‖ek‖]‖uk+1-z‖.
Consequently, we get
(3.36)‖uk+1-z‖2≤λk〈u-z,uk+1-z〉+1-λk2‖uk-z‖2+12‖uk+1-z‖2+[δk‖uk-αkdk-ũk‖+(1-λk)‖ek‖]‖uk+1-z‖.
Furthermore, we obtain
(3.37)‖uk+1-z‖2≤(1-λk)‖uk-z‖2+2λk〈u-z,uk+1-z〉+2[δk‖uk-αkdk-ũk‖+(1-λk)‖ek‖]‖uk+1-z‖.
Since ∑k=0∞δk<∞ and ∑k=0∞∥ek∥<∞, we have
(3.38)∑k=0∞2[δk‖uk-αkdk-ũk‖+(1-λk)‖ek‖]‖uk+1-z‖<∞.
Set σk=max{〈u-z,uk+1-z〉,0}, we have σk→0 as k→∞. Denote ak=∥uk-z∥2, bk=2λkσk and ck=2[δk∥uk-αkdk-ũk∥+(1-λk)∥ek∥]∥uk+1-z∥, by Lemma 2.9, we conclude that ak→0 as k→∞, that is, uk→z as k→∞. The proof is complete.


## 4. Applications

This section considers two interesting applications, convex programming problems and generalized variational inequalities.LetH be a real Hilbert space, let C be a nonempty closed convex subset of H, and let f:C→(-∞,+∞] be a proper closed convex function. Consider the following convex programming problem:(4.1)min⁡x∈C⁡f(x).
In 1965, Moreau [18] indicated that if f is a proper closed convex function, ∂f is a maximal monotone operator, where ∂f(z) represents the subdifferential of f, that is,(4.2)∂f(u)={y∈H:f(z)≥f(u)+〈z-u,y〉,z∈H},∀u∈H.
Since z is a minimizer of f if and only if 0∈∂f(z). Thus, the problem (4.1) can be directly transformed to the maximal monotone inclusion problem (1.1). Note that(4.3)uk+ek∈ũk+βk∂f(ũk)
is equivalent to the following equation:(4.4)ũk=argminu∈H⁡{f(u)+12βk‖u-uk-ek‖2}.
Hence, when dealing with the convex programming problems, we use (4.4) instead of (3.1), see [12, 13], for example. Specifically, Theorem 3.6 could be detailed as follows.Theorem 4.1.
LetH be a real Hilbert space, let  C be a nonempty closed convex subset of  H, and let  f:H→(-∞,∞] be a proper closed convex function such that  ∂f(0)∩C≠∅. Let  {βk}⊂(0,∞) with  βk→∞ as  k→∞,  {ek} be a sequence in  H satisfying  ∥ek∥≤ηk∥uk-ũk∥ with  supk≥0ηk=v<1 and  ∑k=0∞∥ek∥<∞. Given  u0∈C and  u∈C, let  {uk} be generated by
(4.5)ũk=argminu∈H⁡{f(u)+12βk‖u-uk-ek‖2},uk+1=λku+δkPC[uk-αk(uk-ũk+ek)]+γkũk,
where the stepsize αk is defined by
(4.6)αk=〈uk-ũk,uk-ũk+ek〉‖uk-ũk+ek‖2,
and {λk},{δk}, and {γk} are real sequences in [0,1] satisfying(i)
λk+δk+γk=1;(ii)
limk→∞λk=0 and ∑k=0∞λk=∞;(iii)
∑k=0∞δk<∞.Then, the sequence {uk} converges strongly to a minimizer of f nearest to u.We now turn to another application of the proposed method. In recent years, the approximate proximal point algorithms are a family important methods to solve monotone variational inequalities, see [19–21], for example. Let H be a real Hilbert space, let C be a nonempty closed convex subset of H, and let F:C→2H be a monotone set-valued mapping. Consider the following generalized variational inequality (GVI(F,C)): find a vector u*∈C and ω*∈F(u*) such that(4.7)〈u-u*,ω*〉≥0,∀u∈C.
When F is single-valued, GVI(F,C) reduces to the classical variational inequality VI(F,C). Let T(u)=ξ(u)+NC(u), whereξ(u)∈F(u) is single-valued and NC(·) denotes the normal cone operator with respect to C, namely,(4.8)NC(u):={{y:〈y,v-u〉≤0,∀v∈C},ifu∈C;∅,otherwise.
In this way, GVI(F,C) can be transformed to the maximal monotone inclusion problem (1.1) easily. In particular, for given uk and βk>0, using the proximal subproblem (3.1) to solve the problem (4.7) is equivalent to find ũk∈C and ξk∈F(ũk) such that(4.9)ũk=PC[uk-βkξk+ek].
Specifically, when dealing with GVI(F,C), Theorem 3.6 could be detailed as follows.Theorem 4.2.
LetH be a real Hilbert space,let  C be a nonempty closed convex subset of  H, and let  F:C→2H be a monotone set-valued mapping. Suppose that the solution set of variational inequality problem (4.7) is nonempty. Let  {βk}⊂(0,∞) with  βk→∞ as  k→∞,  {ek} be a sequence in  H such that  ∥ek∥≤ηk∥uk-ũk∥ with  supk≥0ηk=v<1 and  ∑k=0∞∥ek∥<∞. Given  u0∈C and  u∈C, let  {uk} be generated by
(4.10)ũk=PC[uk-βkξk+ek],ξk∈F(ũk),uk+1=λku+δkPC[uk-αk(uk-ũk+ek)]+γkũk,
where the stepsize αk is defined by
(4.11)αk=〈uk-ũk,uk-ũk+ek〉‖uk-ũk+ek‖,
and {λk},{δk}, and {γk} are real sequences in [0,1] satisfying(i)
λk+δk+γk=1;(ii)
limk→∞λk=0 and ∑k=0∞λk=∞;(iii)
∑k=0∞δk<∞.Then, the sequence {uk} converges strongly to a solution of GVI(F,C) nearest to u.


## 5. Preliminary Computational Results

In this section, we give some numerical experiments and present comparisons between Algorithm3.1 and the algorithm presented in Qin et al. [13]. All the codes are written in MATLAB 7.0 and run on the computer with an Intel Core2 1.86 GHz CPU, and Windows XP system. Throughout the computational experiments, the parameters are chosen as v=0.9,βk=k+1,λk=1/10(k+1),δk=1/2k. The stopping criterion is ∥min{uk,F(uk)}∥∞≤10-6.Example 5.1.
Consider the following generalized variational inequality problem which is tested in [22] by four variables. Let C={u∈R+4:∑i=14ui=1} and F:C→2R4 be defined by
(5.1)F(u)={(t,t+2u2,t+3u3,t+4u4):t∈[0,1]}.
Then, (1,0,0,0) is a solution of this problem.We solve this problem with different starting points. The numbers of iterations (It. num.) and the computation times (CPU(s)) are summarized in Table1. From Table 1, we can see that Algorithm 3.1 performs better. In addition, for the considered problem, the iterative numbers and the computational times are not very sensitive to the starting point.Table 1
Numerical results for different starting point.
Starting pointAlgorithm3.1Algorithm in [13]It. num.CPU(s)It. num.CPU(s)(0,0,0,1)300.2310370.3470(0,0,1,0)320.2310370.3470(0,0.5,0.5,0)330.2310380.3790(0.5,0,0.5,0)330.2470380.3780


## 6. Conclusion

This paper suggests an approximate proximal point algorithm for the maximal monotone inclusion problems by adopting a more general correction step. Under suitable and standard assumptions on the algorithm parameters, we get the strong convergence of the algorithm. Note that Algorithm3.1 here includes some existing methods as special cases. Therefore, the proposed algorithm is expected to be widely applicable.



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*Source: 262073-2011-12-29.xml*