R E S E A R C H
Open Access
On over-relaxed
(
A
,
η
,
m
)-proximal point
algorithm frameworks with errors and
applications to general variational inclusion
problems
Heng-you Lan
**Correspondence: [email protected] Department of Mathematics, Sichuan University, Chengdu, Sichuan 610064, P.R. China Department of Mathematics, Sichuan University of Science & Engineering, Zigong, Sichuan 643000, P.R. China
Abstract
The purpose of this paper is to provide some remarks for the main results of the paper Verma (Appl. Math. Lett. 21:142-147, 2008). Further, by using the generalized proximal operator technique associated with the (A,
η
,m)-monotone operators, we discuss the approximation solvability of general variational inclusion problem forms in Hilbert spaces and the convergence analysis of iterative sequences generated by the over-relaxed (A,η
,m)-proximal point algorithm frameworks with errors, which generalize the hybrid proximal point algorithm frameworks due to Verma. MSC: 47H05; 49J40Keywords: (A,
η
,m)-monotonicity; generalized proximal operator technique; over-relaxed (A,η
,m)-proximal point algorithm frameworks with errors; general variational inclusion problem; convergence analysis1 Introduction
In , Verma [] developed a general framework for a hybrid proximal point algorithm using the notion of (A,η)-monotonicity (also referred to as (A,η)-maximal monotonicity or (A,η,m)-monotonicity in literature) and explored convergence analysis for this algo-rithm in the context of solving the following variational inclusion problems along with some results on the resolvent operator corresponding to (A,η)-monotonicity: Find a so-lution to
∈M(x), (.)
whereM:H→His a set-valued mapping on a real Hilbert spaceH.
We remark that the problem (.) provides us a general and unified framework for study-ing a wide range of intereststudy-ing and important problems arisstudy-ing in mathematics, physics, engineering sciences, economics finance,etc.For more details, see [–] and the follow-ing example.
Example .[] LetV:Rn→Rbe a local Lipschitz continuous function, and letKbe a closed convex set inRn. Ifx∗∈Rnis a solution to the following problem:
min
x∈KV(x),
then
∈∂Vx∗+NK
x∗,
where∂V(x∗) denotes the subdifferential ofV atx∗, andNK(x∗) the normal cone ofK atx∗.
Very recently, Huang and Noor [] have pointed out ‘the question on whether the strong convergence holds or not for the over-relaxed proximal point algorithm is still open’. Verma [] also pointed out ‘the over-relaxed proximal point algorithm is of interest in the sense that it is quite application-oriented, but nontrivial in nature’. In [, ], we dis-cussed the convergence of iterative sequences generated by the hybrid proximal point al-gorithm frameworks associated with (A,η,m)-monotonicity when operatorAis strongly monotone and Lipschitz continuous.
Motivated and inspired by the recent works, in this paper, we correct the main result of the paper []. Further, by using the generalized proximal operator technique associated with the (A,η,m)-monotone operators, we discuss the approximation solvability of general variational inclusion problem forms in Hilbert spaces and the convergence analysis of iter-ative sequences generated by the over-relaxed (A,η,m)-proximal point algorithm frame-works with errors, which generalize the hybrid proximal point algorithm frameframe-works due to Verma [].
2 Preliminaries
In the sequel, letHbe a real Hilbert space with the norm · and the inner product·,· and Hdenote the family of all subsets ofH.
Definition . A single-valued operatorA:H→His said to be
(i) r-strongly monotone, if there exists a positive constantrsuch that
A(x) –A(y),x–y≥rx–y, ∀x,y∈H;
(ii) s-Lipschitz continuous, if there exists a constants> such that
A(x) –A(y)≤sx–y, ∀x,y∈H.
Ifs= , thenAis called nonexpansive.
Definition . LetA:H→Handη:H×H→Hbe two nonlinear (in general) opera-tors. A set-valued operatorM:H→His said to be
(i) maximal monotone if for any(y,v)∈Graph(M) ={(y,v)∈H×H|v∈M(y)},
(ii) r-stronglyη-monotone if there exists a positive constantrsuch that
u–v,η(x,y)≥rx–y, ∀(x,u), (y,v)∈Graph(M),
whereηis said to beτ-Lipschitz continuous if there exists a constantτ> such that
η(x,y)≤τx–y, ∀x,y∈H;
(iii) m-relaxedη-monotone if there exists a positive constantmsuch that
u–v,η(x,y)≥–mx–y, ∀(x,u), (y,v)∈Graph(M);
Similarly, ifη(x,y) =x–yfor allx,y∈H, we can obtain the definition of strong mono-tonicity and relaxed monomono-tonicity.
Definition . LetA:H→Hber-strongly monotone. The operatorM:H→His said to beA-maximal monotone if
(i) Mism-relaxed monotone; (ii) R(A+ρM) =Hforρ> .
Definition . LetA:H→Hber-stronglyη-monotone. ThenM:H→His said to be (A,η,m)-monotone if
(i) Mism-relaxedη-monotone; (ii) R(A+ρM) =Hforρ> .
Lemma .[] LetHbe a real Hilbert space,A:H→Hbe r-strongly monotone,and M:H→Hbe A-maximal monotone.Then the resolvent operator associated with M and defined by
JρM,A(x) = (A+ρM)–(x), ∀x∈H,
isr–ρm-Lipschitz continuous.
Lemma .[] LetHbe a real Hilbert space,A:H→Hbe r-stronglyη-monotone,M:
H→Hbe(A,η,m)-maximal monotone,andη:H×H→Hbeτ-Lipschitz continuous.
Then the generalized resolvent operator associated with M and defined by
JρM,,Aη(x) = (A+ρM)–(x), ∀x∈H,
isr–τρm-Lipschitz continuous.
3 Remarks and algorithm frameworks
Lemma .[] LetHbe a real Hilbert space,A:H→Hbe r-stronglyη-monotone,and M:H→Hbe(A,η,m)-maximal monotone.Then the following statements are mutually equivalent:
(i) An elementx∈His a solution to(.). (ii) For anx∈H,we have
x=JρM,,Aη
A(x),
whereJρM,,Aη= (A+ρM)–.
Lemma .[] LetHbe a real Hilbert space,A:H→Hbe r-strongly monotone,and M:H→Hbe A-maximal monotone.Then the following statements are mutually equiv-alent:
(i) An elementx∈His a solution to(.). (ii) For anx∈H,we have
x=JρM,A
A(x),
whereJM
ρ,A= (A+ρM)–.
In [], by using Lemmas ., ., ., and ., the author obtained the following main results on the convergence rate (or convergence), which hold only whenr–ρm> :
Theorem V(See [, p., Theorem .]) LetHbe a real Hilbert space,let A:H→H
be r-strongly monotone and s-Lipschitz continuous,and let M:H→H be A-maximal monotone.For an arbitrarily chosen initial point x,suppose that the sequence{xn}is
gen-erated by an iterative procedure
xn+= ( –αn)xn+αnyn, ∀n≥,
and ynsatisfies
yn–JρMn,A
A(xn)≤δnyn–xn,
where JM
ρn,A= (A+ρnM)–and{δn},{αn},{ρn} ⊂[,∞)are scalar sequences such that ∞
n=
δn<∞, δn→, α=lim sup
n→∞ αk< , ρn↑ρ≤+∞.
Then the sequence{xn}converges linearly to a solution of(.)with the convergence rate
– α – ( –α) s
r–ρm–
α
s r–ρm
–
α
< ,
for c=r–ρm,
c<( –α)s–
( –α)s+ ( –α)αs
–α , ( –α)s>
and for
c>( –α)s+
( –α)s+ ( –α)αs –α .
Theorem V(See [, p., Theorem .]) LetHbe a real Hilbert space,let A:H→Hbe r-stronglyη-monotone and s-Lipschitz continuous,and let M:H→Hbe(A,η)-maximal monotone.Letη:H×H→Hbeτ-Lipschitz continuous.For an arbitrarily chosen initial point x,suppose that the sequence{xn}is generated by an iterative procedure
xn+= ( –αn)xn+αnyn, ∀n≥,
and ynsatisfies
yn–JρM,n,Aη
A(xn)≤n–yn–xn,
where JM
ρn,A = (A+ρnM)– and {αn},{ρn} ⊂[,∞) are scalar sequences such that α =
lim supn→∞αk< ,ρn↑ρ≤+∞.Then the sequence{xn}converges linearly to a solution
of(.)for
r<( –α)sτ–
( –α)sτ+ ( –α)αsτ –α ,
( –α)sτ>( –α)sτ+ ( –α)αsτ,
and for
r>( –α)sτ+
( –α)sτ+ ( –α)αsτ –α .
In the sequel, we give the following remarks to show that the main proof of Theorems . and . of [] is worth correcting.
Remark . By ther-strongly monotonicity ands-Lipschitz continuity of the underlying operatorA, it follows that for allx,y∈H, ifx=y,
rx–y≤A(x) –A(y),x–y≤A(x) –A(y)· x–y ≤sx–y,
showing thatr≤s.
Remark . From Remark ., it is easy to prove that the convergence rateθn> in p. of [] forn≥. Therefore, the strong convergence of [, Theorem .], is not true.
In fact, from Remark . and the definition of the convergence rate in line , p. of [], we have the following estimate:
s≥r≥r–ρnm> , i.e.,
s r–ρnm
and
θn= – αn – ( –αn)
s r–ρnm
– αn
s r–ρnm
–
αn
= – αn+ αn( –αn)
s r–ρnm
+αn
s r–ρnm
+αn
= ( –αn)+ ( –αn) sαn
r–ρnm +
sαn
r–ρnm
= ( –αn) + sαn
r–ρnm
= –αn
– s
r–ρnm
> , (.)
it is becauseαn> for alln≥.
Remark . Similarly, we can show that the conditions for the convergence of [, Theo-rem .] must be revised.
Indeed, from ≤α< and the assumption, it follows that the conditions for the con-vergence of a sequence{xn}generated by the iterative algorithm are equivalent to
( –α)c– ( –α)sτc–αsτ> , c=r–ρm,
that is,
– ( –α) sτ
r–ρm–
α
sτ
r–ρm
–
α> ,
which should be revised because it follows from the assumption, (.), and Remark . that
sτ <r–ρm≤r≤s, i.e., τ< .
Thus, ifτ≥, then the conditions for the convergence are not true.
Next, in order to illustrate the main results in [], we construct the following over-relaxed proximal point algorithm frameworks with errors based on Lemmas . and ..
Algorithm . Step . Choose an arbitrary initial pointu∈H.
Step . Choose sequences{αn},{δn}, and{ρn}such that forn≥,{αn},{δn}, and{ρn}are three sequences in [,∞) satisfying
∞
n=
δn<∞, ρn↑ρ.
Step . Let{xn} ⊂Hbe generated by the following iterative procedure:
where{en}is an error sequence inHto take into account a possible inexact computation of the operator point, which satisfies∞n=en<∞, andynsatisfies
yn–JM, η ρn,A
A(xn)≤δnyn–xn,
wheren≥,JρM,n,Aη = (A+ρnM)–andρn> .
Step . Ifxnandynsatisfy (.) to sufficient accuracy, stop; otherwise, setn:=n+ and return to Step .
Remark . Ifen≡,δn=n, andαn< forn≥, then Algorithm . is reduced to the iterative algorithm in Theorem . of [].
Algorithm . Step . Choose an arbitrary initial pointx∈H.
Step . Choose sequences{αn},{δn}, and{ρn}such that forn≥,{αn},{δn}, and{ρn}are three sequences in [,∞) satisfying
∞
n=
δn<∞, ρn↑ρ.
Step . Let{xn} ⊂Hbe generated by the following iterative procedure:
xn+= ( –αn)xn+αnyn+en, ∀n≥, (.)
where{en}is an error sequence inHto take into account a possible inexact computation of the operator point, which satisfies∞n=en<∞, andynsatisfies
yn–JρMn,A
A(xn)≤δnyn–xn,
wheren≥,JρMn,A= (A+ρnM)
–andρ n> .
Step . Ifxnandynsatisfy (.) to sufficient accuracy, stop; otherwise, setn:=n+ and return to Step .
Remark . Ifen≡ andαn< forn≥, then Algorithm . is reduced to the iterative algorithm in Theorem . of [].
4 Convergence analysis
In this section, we apply the over-relaxed proximal point Algorithms . and . to ap-proximate the solution of (.), and as a result, we end up showing linear convergence.
Theorem . LetHbe a real Hilbert space,A:H→Hbe r-strongly monotone and s-Lipschitz continuous, η:H×H→Hbeτ-Lipschitz continuous, and M:H→H be
(A,η,m)-maximal monotone.If forγ >,
A(xn) –Ax∗,JρM,n,Aη
A(xn)–JρM,n,Aη
Ax∗≥γJρM,n,Aη
A(xn)–JρM,n,Aη
Ax∗,
and there exists a constantρ∈(,mr)such that
α+skα– γ(α– )< , k= τ
then the sequence{xn}generated by Algorithm.converges linearly to a solution x∗of the
problem(.)with the convergence rate
ϑ=
–α
–γ
sτ
r–ρm
–α – (γ – )
sτ
r–ρm
< ,
whereα=lim supn→∞αn> andρn↑ρ.
Proof Letx∗be a solution of the problem (.). Then it follows from Lemma . that
x∗= ( –αn)x∗+αnJρM,n,Aη
Ax∗. (.)
Let
zn+= ( –αn)xn+αnJM, η ρn,A
A(xn)
, ∀n≥.
Thus, by the assumptions of the theorem, Lemma ., and (.), now we find the estimate
zn+–x∗
=( –αn)xn–x∗
+αnJρM,n,AηA(xn)–JρM,n,AηAx∗
=αnJρM,n,AηA(xn)–JρM,n,AηAx∗+( –αn)xn–x∗ + αnJρM,n,AηA(xn)–JρM,n,AηAx∗, ( –αn)xn–x∗
≤( –αn)xn–x∗+
αn+ γ αn( –αn)JρM,n,AηA(xn)–JρM,n,AηAx∗ ≤
( –αn)+αn+ γ αn( –αn) τ
(r–ρnm)
·sxn–x∗
=ϑnxn–x∗
,
where
ϑn=
–αn
–γ
sτ
r–ρnm
–αn – (γ – )
sτ
r–ρnm
.
Thus, we have
xn+–x∗≤ϑnxn–x∗.
Sincexn+= ( –αn)xn+αnyn+en,xn+–xn=αn(yn–xn) +en, it follows that
xn+–zn+
=( –αn)xn+αnyn+en– ( –αn)xn–αnJρM,n,Aη(xn)
=αn
yn–JρM,n,Aη(xn)+en
≤δnαn(yn–xn)+en
Using the above arguments, we estimate that
xn+–x∗
≤ xn+–zn++zn+–x∗
≤δnxn+–xn+ϑnxn–x∗+en
≤δnxn+–x∗+δnxn–x∗+ϑnxn–x∗+en.
This implies that
xn+–x∗≤ ϑn+δn
–δn
xn–x∗+ –δn
en. (.)
SinceAisr-strongly monotone (and hence, A(u) –A(v) ≥ru–v,∀u,v∈H), this implies from (.) that the{xn}converges linearly to a solutionx∗for
ϑn=
–αn
–γ
sτ
r–ρnm
–αn – (γ – )
sτ
r–ρnm
.
Hence, we have
lim sup
n→∞
ϑn+δn –δn
=lim sup
n→∞ ϑn
=
–α
–γ
sτ
r–ρm
–α – (γ– )
sτ
r–ρm
,
whereα=lim supn→∞αn,ρn↑ρ. This completes the proof.
Remark . The conditions (.) in Theorem . hold for some suitable values of con-stants, for example, α = .,γ = .,τ = ., r= ., ρ = .,m= ., s= . and the convergence rateθ= . < .
From Theorem ., we have the following result.
Theorem . LetHbe a real Hilbert space,A:H→Hbe r-strongly monotone with r>
and s-Lipschitz continuous,and M:H→Hbe A-maximal monotone.If forγ >,
A(xn) –Ax∗,JρMn,A
A(xn)–JρMn,A
Ax∗
≥γJM ρn,A
A(xn)
–JM ρn,A
Ax∗,
and there exists a constantρ∈(,r–
m )such that
α+α– γ(α– ) s
r–ρm
then the sequence{xn}generated by Algorithm.converges linearly to a solution x∗of the
problem(.)with the convergence rate
θ=
–α
–γ
s r–ρm
–α – (γ – )
s r–ρm
< ,
whereα=lim supn→∞αn> andρn↑ρ.
Remark . In Theorems . and ., if we apply thec-Lipschitz continuity ofM– in-stead, it seems that the strong convergence could be achieved (see, for example, [–]).
Remark . For an arbitrarily chosen initial pointx, let the iterative sequence{xn} gen-erate the following over-relaxed proximal point algorithm:
A(xn+) = ( –αn)A(xn) +αnyn,
andynsatisfy
yn–A
GA(xn)≤δnyn–A(xn),
wheren≥,G=JρM,n,Aη = (A+ρnM)
–, the resolvent operator associated withA-maximal
monotoneM, orG=JρMn,A= (A+ρnM)–, the resolvent operator associated with (A,η,m )-maximal monotoneM, and scalar sequences{αn},{ρn},{δn} ⊂[,∞). Then we can obtain the corresponding results by using the same method as in Theorem . (see, for example, [, ]). Therefore, the results presented in this paper improve, generalize, and unify the corresponding results of recent works.
Competing interests
The author declares that they have no competing interests. Author’s contributions
HYL conceived of the study and participated in its design and coordination, the proof of convergence of the theorems and gave some examples to show the main results. The author read and approved the final manuscript.
Acknowledgements
This work was supported by the Scientific Research Fund of Sichuan Provincial Education Department (10ZA136), Sichuan Province Youth Fund project (2011JTD0031) and the Cultivation Project of Sichuan University of Science and Engineering (2011PY01), and Artificial Intelligence of Key Laboratory of Sichuan Province (2012RYY04).
Received: 7 July 2012 Accepted: 6 January 2013 Published: 12 March 2013 References
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doi:10.1186/1029-242X-2013-97
