R E S E A R C H
Open Access
A Korpelevich-like algorithm for variational
inequalities
Zhitao Wu
1, Yonghong Yao
1*, Yeong-Cheng Liou
2and Hong-Jun Li
1*Correspondence:
1Department of Mathematics,
Tianjin Polytechnic University, Tianjin, 300387, China Full list of author information is available at the end of the article
Abstract
A Korpelevich-like algorithm has been introduced for solving a generalized variational inequality. It is shown that the presented algorithm converges strongly to a special solution of the generalized variational inequality.
MSC: 47H05; 47J25
Keywords: Korpelevich-like algorithm; sunny nonexpansive retraction; generalized variational inequalities;
α
-inverse-strongly accretive mappings; Banach spaces1 Introduction
Now it is well-known that the variational inequality of findingx*∈Csuch that
Ax*,x–x*≥, ∀x∈C, (.)
whereCis a nonempty closed convex subset of a real Hilbert spaceHandA:C→His a given mapping, is a fundamental problem in variational analysis and, in particular, in opti-mization theory. For related works, please see [–] and the references contained therein. Especially, Yao, Marino and Muglia [] presented the following modified Korpelevich method for solving (.):
yn=PC[xn–λAxn–αnxn],
xn+=PC
xn–λAyn+μ(yn–xn)
, n≥.
(.)
Recently, Aoyama, Iiduka and Takahashi [] extended the variational inequality (.) to Banach spaces as follows:
Findx*∈Csuch thatAx*,Jx–x*≥, ∀x∈C, (.)
whereCis a nonempty closed convex subset of a real Banach spaceE. We useS(C,A) to denote the solution set of (.). The generalized variational inequality (.) is connected with the fixed point problem for nonlinear mappings. For solving the above generalized variational inequality (.), Aoyama, Iiduka and Takahashi [] introduced the iterative algorithm
xn+=αnxn+ ( –αn)QC[xn–λnAxn], n≥, (.)
where QC is a sunny nonexpansive retraction from E ontoC and{αn} ⊂(, ),{λn} ⊂
(,∞) are two real number sequences. Motivated by (.), Yao and Maruster [] pre-sented a modification of (.) as follows:
xn+=βnxn+ ( –βn)QC
( –αn)(xn–λAxn)
, n≥. (.)
Motivated and inspired by the above algorithms (.), (.) and (.), in this paper, we suggest an extragradient-type method via the sunny nonexpansive retraction for solving the variational inequalities (.) in Banach spaces. It is shown that the presented algorithm converges strongly to a special solution of the variational inequality (.).
2 Preliminaries
LetCbe a nonempty closed convex subset of a real Banach spaceE. Recall that a mapping AofCintoEis said to beaccretiveif there existsj(x–y)∈J(x–y) such that
Ax–Ay,j(x–y)≥
for allx,y∈C. A mappingAofCintoEis said to beα-strongly accretiveif forα> ,
Ax–Ay,j(x–y)≥αx–y
for all x,y∈C. A mappingAofCintoE is said to beα-inverse-strongly accretiveif for
α> ,
Ax–Ay,j(x–y)≥αAx–Ay for allx,y∈C.
LetU={x∈E:x= }. A Banach spaceEis said touniformly convexif for each∈
(, ], there existsδ> such that for anyx,y∈U,
x–y ≥ implies x+y
≤ –δ.
It is known that a uniformly convex Banach space is reflexive and strictly convex. A Banach spaceEis said to besmoothif the limit
lim
t→
x+ty–x
t (.)
exists for allx,y∈U. It is also said to beuniformly smoothif the limit (.) is attained uniformly forx,y∈U. The norm ofEis said to be Frechet differentiable if for eachx∈U, the limit (.) is attained uniformly fory∈U. And we define a functionρ: [,∞)→[,∞) called themodulus of smoothnessofEas follows:
ρ(τ) =sup
x+y+x–y– :x,y∈X,x= ,y=τ
.
It is known thatEis uniformly smooth if and only iflimτ→ρ(τ)/τ= . Letqbe a fixed real
number with <q≤. Then a Banach spaceEis said to beq-uniformly smooth if there exists a constantc> such thatρ(τ)≤cτqfor allτ> .
Lemma .[] Let q be a given real number with <q≤and let E be a q-uniformly smooth Banach space.Then
x+yq≤ xq+qy,Jq(x)
+ Kyq
for all x,y∈E,where K is the q-uniformly smoothness constant of E and Jqis the generalized
duality mapping from E intoE*defined by
Jq(x) =
f ∈E*:x,f=xq,f=xq–, ∀x∈E.
LetDbe a subset ofCand letQbe a mapping ofCintoD. ThenQis said to besunnyif
QQx+t(x–Qx)=Qx,
wheneverQx+t(x–Qx)∈Cforx∈Candt≥. A mappingQofCinto itself is called a retractionifQ=Q. If a mappingQofCinto itself is a retraction, thenQz=zfor every
z∈R(Q), whereR(Q) is the range ofQ. A subsetDofCis called asunny nonexpansive retract ofCif there exists a sunny nonexpansive retraction fromContoD. We know the following lemma concerning sunny nonexpansive retraction.
Lemma .[] Let C be a closed convex subset of a smooth Banach space E,let D be a nonempty subset of C and Q be a retraction from C onto D.Then Q is sunny and nonex-pansive if and only if
u–Qu,j(y–Qu)≤ for all u∈C and y∈D.
Lemma .[] Let C be a nonempty closed convex subset of a smooth Banach space X. Let QCbe a sunny nonexpansive retraction from X onto C and let A be an accretive operator
of C into X.Then for allλ> ,
S(C,A) =FQC(I–λA)
,
where S(C,A) ={x*∈C:Ax*,J(x–x*) ≥,∀x∈C}.
Lemma .[] Let C be a nonempty closed convex subset of a real-uniformly smooth Banach space X.Let the mapping A:C→X beα-inverse-strongly accretive.Then we have
(I–λA)x– (I–λA)y≤ x–y+ λKλ–αAx–Ay.
In particular,if≤λ≤Kα,then I–λA is nonexpansive.
Proof Indeed, for allx,y∈C, from Lemma ., we have
(I–λA)x– (I–λA)y=(x–y) –λ(Ax–Ay)
≤ x–y– λαAx–Ay+ KλAx–Ay
=x–y+ λKλ–αAx–Ay.
It is clear that if ≤λ≤Kα, thenI–λAis nonexpansive.
Lemma .[] Let C be a nonempty bounded closed convex subset of a uniformly convex Banach space E and let T be a nonexpansive mapping of C into itself.If{xn}is a sequence
of C such that xn→x weakly and xn–Txn→strongly,then x is a fixed point of T.
Lemma .[] Assume{an}is a sequence of nonnegative real numbers such that
an+≤( –γn)an+δn, n≥,
where{γn}is a sequence in(, )and{δn}is a sequence in R such that
(a) ∞n=γn=∞;
(b) lim supn→∞δn/γn≤or
∞
n=|δn|<∞.
Thenlimn→∞an= .
3 Main results
In this section, we present our Korpelevich-like algorithm and consequently we will show its strong convergence.
3.1 Conditions assumptions
(A) Eis a uniformly convex and -uniformly smooth Banach space with a weakly
sequentially continuous duality mapping; (A) Cis a nonempty closed convex subset ofE;
(A) A:C→Eis anα-strongly accretive andL-Lipschitz continuous mapping with
S(C,A)=∅;
(A) QCis a sunny nonexpansive retraction fromEontoC.
3.2 Parameters restrictions
(P) λ,μandγ are three positive constants satisfying: (i) γ∈(, ),λ∈[a,b]for somea,bwith <a<b< α
KL; (ii) μλ<KαL whereKis the smooth constant ofE. (P) {αn}is a sequence in(, )such thatlimn→∞αn= and
∞
n=αn=∞.
Algorithm . For givenx∈C, define a sequence{xn}iteratively by
⎧ ⎨ ⎩
yn=QC[( –αn)xn–λAxn],
xn+= ( –γ)xn+γQC[xn–λAyn+μ(yn–xn)], n≥.
(.)
Theorem . The sequence{xn}generated by(.)converges strongly to Q(),where Qis
a sunny nonexpansive retraction of E onto S(C,A).
Proof Letp∈S(C,A). First, from Lemma ., we havep=QC[p–δAp] for allδ> . In
Since A:C →E is α-strongly accretive and L-Lipschitzian, it must be Lα -inverse-strongly accretive mapping. Thus, by Lemma ., we have
(I–λA)x– (I–λA)y≤ x–y+ λ
Kλ– α L
Ax–Ay.
Sinceαn→ andλ∈[a,b]⊂(,KαL), we getαn< –K
Lλ
α for enough largen. Without loss of generality, we may assume that for alln∈N,αn< – K
Lλ α ,i.e.,
λ
–αn ∈(,
α
KL). Hence,I––λα
nAis nonexpansive.
From (.), we have
yn–p=
QC
( –αn)xn–λAxn
–QC
αnp+ ( –αn)
p– λ
–αn
Ap
≤αn(–p) + ( –αn)
xn– λ
–αn
Axn
–
p– λ
–αn
Ap
≤αnp+ ( –αn)
I– λ –αn
A
xn–
I– λ
–αn
A
p
≤αnp+ ( –αn)xn–p. (.)
By (.) and (.), we have
xn+–p ≤( –γ)xn–p+γQC
xn–λAyn+μ(yn–xn)
–p = ( –γ)xn–p+γ
QC
( –μ)xn+μ
yn– λ μAyn
–QC
( –μ)p+μ
p– λ
μAp
≤( –γ)xn–p
+γ( –μ)(xn–p) +μ
yn– λ μAyn
–
p–λ
μAp
≤( –γ)xn–p+ ( –μ)γxn–p
+μγ
yn– λ μAyn
–
p–λ
μAp
≤( –μγ)xn–p+μγyn–p
≤( –μγ)xn–p+μγ αnp+μγ( –αn)xn–p
= ( –μγ αn)xn–p+μγ αnp
≤maxxn–p,p
.. .
≤maxx–p,p
. (.)
Setzn=QC[xn–λAyn+μ(yn–xn)]. From (.), we havexn+= ( –γ)xn+γznfor all
n≥. Then we have
yn–yn–=QC
( –αn)xn–λAxn
–QC
( –αn–)xn––λAxn– ≤( –αn)
xn– λ
–αn
Axn
– ( –αn–)
xn–– λ
–αn–
Axn–
≤( –αn)
xn–
λ
–αn
Axn
–
xn–– λ
–αn
Axn–
+|αn–αn–|xn–
≤( –αn)xn–xn–+|αn–αn–|xn–,
and thus
zn–zn–=QC
xn–λAyn+μ(yn–xn)
–QC
xn––λAyn–+μ(yn––xn–) ≤( –μ)xn–xn–+μ
yn–
λ μAyn
–
yn–– λ μAyn–
≤( –μ)xn–xn–+μyn–yn– ≤( –μαn)xn–xn–+|αn–αn–|xn–.
It follows that
lim sup
n→∞
zn–zn––xn–xn–
≤.
This together with Lemma . implies that
lim
n→∞xn+–xn= .
From (.), we have
yn–p≤
αn(–p) + ( –αn)
xn– λ
–αn
Axn
–
p– λ
–αn
Ap
≤αnp+ ( –αn)
xn–
λ
–αn
Axn
–
p– λ
–αn
Ap
≤αnp+ ( –αn)xn–p+ λ
Kλ
–αn
– α
L
Axn–Ap. (.)
From (.), (.) and (.), we obtain
xn+–p
≤( –γ)xn–p+γ
( –μ)(xn–p) +μ
yn– λ μAyn
–
p–λ
μAp
≤( –γ)xn–p+γ( –μ)xn–p+γ μ
yn–
λ μAyn
–
p– λ
μAp
≤( –γ μ)xn–p+γ μ
yn–p+
λ μ
Kλ μ –
α
L
Ayn–Ap
≤γ μ
αnp+ ( –αn)xn–p+ λ
Kλ
–αn
– α
L
Axn–Ap
+ ( –γ μ)xn–p+ γ λ
Kλ μ –
α
L
Ayn–Ap
=αnγ μp+ ( –γ μαn)xn–p+ γ λμ
Kλ
–αn
– α
L
Axn–Ap
+ γ λμ
Kλ μ –
α
L
Ayn–Ap.
Therefore, we have
≤–γ λμ
Kλ
–αn
– α
L
Axn–Ap– γ λμ
Kλ μ –
α
L
Ayn–Ap
≤αnγ μp+xn–p–xn+–p
=αnγ μp+
xn–p+xn+–p
xn–p–xn+–p
≤αnγ μp+
xn–p+xn+–p
xn–xn+.
Sinceαn→ andxn–xn+ → , we obtain
lim
n→∞Axn–Ap=nlim→∞Ayn–Ap= .
It follows that
lim
n→∞Ayn–Axn= .
SinceAisα-strongly accretive, we deduce
Ayn–Axn ≥αyn–xn,
which implies that
lim
n→∞yn–xn= ,
that is,
lim
n→∞QC
( –αn)xn–λAxn
–xn= .
It follows that
lim
n→∞QC[xn–λAxn] –xn= . (.)
Next, we show that
lim sup
n→∞
Q(),jxn–Q()
To show (.), since{xn}is bounded, we can choose a sequence{xni}of{xn}converging
weakly tozsuch that
lim sup
n→∞
Q(),jxn–Q()
=lim sup
i→∞
Q(),jxni–Q
(). (.)
We first provez∈S(C,A). It follows that
lim
i→∞QC(I–λA)xni–xni= . (.)
By Lemma . and (.), we havez∈F(QC(I–λA)), it follows from Lemma . thatz∈
S(C,A).
Now, from (.) and Lemma ., we have
lim sup
n→∞
Q(),jxn–Q()
=lim sup
i→∞
Q(),jxni–Q
()
=Q(),jz–Q()
≥.
Noticing thatxn–yn →, we deduce that
lim sup
n→∞
Q(),jyn–Q()
≥.
Sinceyn=QC[(–αn)(xn––λαnAxn)] andQ
() =Q
C[αnQ()+(–αn)(Q()––λαnAQ
())]
for alln≥, we can deduce from Lemma . that
QC
( –αn)
xn– λ
–αn
Axn
–
( –αn)
xn– λ
–αn
Axn
,jyn–Q()
≤
and
αnQ() + ( –αn)
Q() – λ –αn
AQ()
–QC
αnQ() + ( –αn)
Q() – λ –αn
AQ()
,jyn–Q()
≤.
Therefore, we have
yn–Q()
=QC
( –αn)
xn– λ
–αn
Axn
–QC
αnQ() + ( –αn)
Q() – λ –αn
AQ()
≤
αn
–Q()+ ( –αn)
xn– λ
–αn
Axn
–
Q() – λ –αn
AQ()
,jyn–Q()
≤–αnQ(),jyn–Q()
+ ( –αn)
xn– λ
–αn
Axn
–
Q() – λ –αnAQ
()y
n–Q() ≤–αn
Q(),jyn–Q()
+ ( –αn)xn–Q()yn–Q() ≤–αn
Q(),jyn–Q()
+ –αn
xn–Q
()
+yn–Q()
,
which implies that
yn–Q()
≤( –αn)xn–Q()
+ αn
–Q(),jyn–Q()
. (.)
Finally, we will prove that the sequencexn→Q(). As a matter of fact, from (.) and
(.), we have
xn+–Q()
≤( –γ)xn–Q()
+γ( –μ)xn–Q()
+μ
yn– λ μAyn
–
Q() – λ
μAQ
()
≤( –γ μ)xn–Q()
+γ μ
yn– λ μAyn
–
Q() – λ
μAQ
()
≤( –γ μ)xn–Q()
+γ μyn–Q()
≤( –γ μαn)xn–Q()
+ γ μαn
–Q(),jyn–Q()
.
Applying Lemma . to the last inequality, we conclude thatxnconverges strongly toQ().
This completes the proof.
Competing interests
The authors declare that they have no competing interests.
Authors’ contributions
All authors contributed equally and significantly in writing this paper. All authors read and approved the final manuscript.
Author details
1Department of Mathematics, Tianjin Polytechnic University, Tianjin, 300387, China.2Department of Information
Management, Cheng Shiu University, Kaohsiung, 833, Taiwan.
Acknowledgements
Yonghong Yao was supported in part by NSFC 11071279 and NSFC 71161001-G0105. Yeong-Cheng Liou was supported in part by NSC 101-2628-E-230-001-MY3 and NSC 101-2622-E-230-005-CC3.
Received: 6 October 2012 Accepted: 4 February 2013 Published: 28 February 2013 References
1. Korpelevich, GM: An extragradient method for finding saddle points and for other problems. Ekon. Mat. Metod.12, 747-756 (1976)
2. Iusem, AN, Svaiter, BF: A variant of Korpelevich’s method for variational inequalities with a new search strategy. Optimization42, 309-321 (1997)
3. Iusem, AN, Lucambio Pe´rez, LR: An extragradient-type algorithm for non-smooth variational inequalities. Optimization48, 309-332 (2000)
5. Lions, JL, Stampacchia, G: Variational inequalities. Commun. Pure Appl. Math.20, 493-517 (1967) 6. He, BS, Yang, ZH, Yuan, XM: An approximate proximal-extragradient type method for monotone variational
inequalities. J. Math. Anal. Appl.300, 362-374 (2004)
7. Bello Cruz, JY, Iusem, AN: A strongly convergent direct method for monotone variational inequalities in Hilbert space. Numer. Funct. Anal. Optim.30(1-2), 23-36 (2009)
8. Glowinski, R: Numerical Methods for Nonlinear Variational Problems. Springer, New York (1984)
9. Yao, Y, Noor, MA, Liou, YC: Strong convergence of a modified extra-gradient method to the minimum-norm solution of variational inequalities. Abstr. Appl. Anal.2012, Article ID 817436 (2012). doi:10.1155/2012/817436
10. Xu, HK, Kim, TH: Convergence of hybrid steepest-descent methods for variational inequalities. J. Optim. Theory Appl. 119, 185-201 (2003)
11. Yamada, I: The hybrid steepest descent for the variational inequality problems over the intersection of fixed points sets of nonexpansive mappings. In: Butnariu, D, Censor, Y, Reich, S (eds.) Inherently Parallel Algorithms in Feasibility and Optimization and Their Applications, pp. 473-504. Elsevier, New York (2001)
12. Yao, Y, Liou, YC, Kang, SM: Two-step projection methods for a system of variational inequality problems in Banach spaces. J. Glob. Optim.55, 801-811 (2013). doi:10.1007/s10898-011-9804-0
13. Yao, Y, Chen, R, Liou, YC: A unified implicit algorithm for solving the triple-hierarchical constrained optimization problem. Math. Comput. Model.55, 1506-1515 (2012)
14. Yao, Y, Noor, MA, Noor, KI, Liou, YC, Yaqoob, H: Modified extragradient method for a system of variational inequalities in Banach spaces. Acta Appl. Math.110, 1211-1224 (2010)
15. Cho, YJ, Yao, Y, Zhou, H: Strong convergence of an iterative algorithm for accretive operators in Banach spaces. J. Comput. Anal. Appl.10, 113-125 (2008)
16. Yao, Y, Cho, YJ, Liou, YC: Algorithms of common solutions for variational inclusions, mixed equilibrium problems and fixed point problems. Eur. J. Oper. Res.212, 242-250 (2011)
17. Cho, YJ, Kang, SM, Qin, X: On systems of generalized nonlinear variational inequalities in Banach spaces. Appl. Math. Comput.206, 214-220 (2008)
18. Cho, YJ, Qin, X: Systems of generalized nonlinear variational inequalities and its projection methods. Nonlinear Anal. 69, 4443-4451 (2008)
19. Ceng, LC, Ansari, QH, Yao, JC: Mann type steepest-descent and modified hybrid steepest-descent methods for variational inequalities in Banach spaces. Numer. Funct. Anal. Optim.29, 987-1033 (2008)
20. Sahu, DR, Wong, NC, Yao, JC: A unified hybrid iterative method for solving variational inequalities involving generalized pseudo-contractive mappings. SIAM J. Control Optim.50, 2335-2354 (2012)
21. Yao, Y, Marino, G, Muglia, L: A modified Korpelevich’s method convergent to the minimum norm solution of a variational inequality. Optimization (in press). doi:10.1080/02331934.2013.764522
22. Aoyama, K, Iiduka, H, Takahashi, W: Weak convergence of an iterative sequence for accretive operators in Banach spaces. Fixed Point Theory Appl.2006, Article ID 35390 (2006). doi:10.1155/FPTA/2006/35390
23. Yao, Y, Maruster, S: Strong convergence of an iterative algorithm for variational inequalities in Banach spaces. Math. Comput. Model.54, 325-329 (2011)
24. Xu, HK: Inequalities in Banach spaces with applications. Nonlinear Anal.16, 1127-1138 (1991) 25. Bruck, RE Jr.: Nonexpansive retracts of Banach spaces. Bull. Am. Math. Soc.76, 384-386 (1970)
26. Takahashi, W, Toyoda, M: Weak convergence theorems for nonexpansive mappings and monotone mappings. J. Optim. Theory Appl.118, 417-428 (2003)
27. Browder, FE: Nonlinear operators and nonlinear equations of evolution in Banach spaces. In: Nonlinear Functional Analysis (Proc. Sympos. Pure Math., vol. XVIII, Part 2, Chicago, Ill, 1968), pp. 1-308. Am. Math. Soc., Rhode Island (1976) 28. Xu, HK: Iterative algorithms for nonlinear operators. J. Lond. Math. Soc.66, 240-256 (2002)
doi:10.1186/1029-242X-2013-76