A Korpelevich-like algorithm for variational inequalities | SpringerLink

Wu et al. Journal of Inequalities and Applications 2013, 2013:76 http://www.journalofinequalitiesandapplications.com/content/2013/1/76

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A Korpelevich-like algorithm for variational inequalities Zhitao Wu1 , Yonghong Yao1* , Yeong-Cheng Liou2 and Hong-Jun Li1 *

Correspondence: [email protected] 1 Department 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 spaces

1 Introduction Now it is well-known that the variational inequality of finding x* ∈ C such that   * Ax , x – x* ≥ ,

∀x ∈ C,

(.)

where C is a nonempty closed convex subset of a real Hilbert space H and A : C → H is a given mapping, is a fundamental problem in variational analysis and, in particular, in optimization 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 – αn xn ],   xn+ = PC xn – λAyn + μ(yn – xn ) ,

(.)

n ≥ .

Recently, Aoyama, Iiduka and Takahashi [] extended the variational inequality (.) to Banach spaces as follows:    Find x* ∈ C such that Ax* , J x – x* ≥ ,

∀x ∈ C,

(.)

where C is a nonempty closed convex subset of a real Banach space E. We use S(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+ = αn xn + ( – αn )QC [xn – λn Axn ],

n ≥ ,

(.)

© 2013 Wu et al.; licensee Springer. This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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where QC is a sunny nonexpansive retraction from E onto C and {αn } ⊂ (, ), {λn } ⊂ (, ∞) are two real number sequences. Motivated by (.), Yao and Maruster [] presented a modification of (.) as follows:   xn+ = βn xn + ( – 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 Let C be a nonempty closed convex subset of a real Banach space E. Recall that a mapping A of C into E is said to be accretive if there exists j(x – y) ∈ J(x – y) such that   Ax – Ay, j(x – y) ≥  for all x, y ∈ C. A mapping A of C into E is said to be α-strongly accretive if for α > ,   Ax – Ay, j(x – y) ≥ αx – y for all x, y ∈ C. A mapping A of C into E is said to be α-inverse-strongly accretive if for α > ,   Ax – Ay, j(x – y) ≥ αAx – Ay for all x, y ∈ C. Let U = {x ∈ E : x = }. A Banach space E is said to uniformly convex if for each  ∈ (, ], there exists δ >  such that for any x, y ∈ U, x – y ≥ 

  x + y  ≤  – δ.  implies   

It is known that a uniformly convex Banach space is reflexive and strictly convex. A Banach space E is said to be smooth if the limit lim

t→

x + ty – x t

(.)

exists for all x, y ∈ U. It is also said to be uniformly smooth if the limit (.) is attained uniformly for x, y ∈ U. The norm of E is said to be Frechet differentiable if for each x ∈ U, the limit (.) is attained uniformly for y ∈ U. And we define a function ρ : [, ∞) → [, ∞) called the modulus of smoothness of E as follows:

  ρ(τ ) = sup x + y + x – y –  : x, y ∈ X, x = , y = τ .  It is known that E is uniformly smooth if and only if limτ → ρ(τ )/τ = . Let q be a fixed real number with  < q ≤ . Then a Banach space E is said to be q-uniformly smooth if there exists a constant c >  such that ρ(τ ) ≤ cτ q for all τ > . We need the following lemmas for the proof of our main results.

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Lemma . [] Let q be a given real number with  < q ≤  and let E be a q-uniformly smooth Banach space. Then   x + yq ≤ xq + q y, Jq (x) + Kyq for all x, y ∈ E, where K is the q-uniformly smoothness constant of E and Jq is the generalized * duality mapping from E into E defined by  Jq (x) = f ∈ E* : x, f  = xq , f  = xq– ,

∀x ∈ E.

Let D be a subset of C and let Q be a mapping of C into D. Then Q is said to be sunny if   Q Qx + t(x – Qx) = Qx, whenever Qx + t(x – Qx) ∈ C for x ∈ C and t ≥ . A mapping Q of C into itself is called a retraction if Q = Q. If a mapping Q of C into itself is a retraction, then Qz = z for every z ∈ R(Q), where R(Q) is the range of Q. A subset D of C is called a sunny nonexpansive retract of C if there exists a sunny nonexpansive retraction from C onto D. 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 nonexpansive 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 QC be a sunny nonexpansive retraction from X onto C and let A be an accretive operator of C into X. Then for all λ > ,   S(C, A) = F QC (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 be α-inverse-strongly accretive. Then we have     (I – λA)x – (I – λA)y ≤ x – y + λ K  λ – α Ax – Ay . In particular, if  ≤ λ ≤

α , K

then I – λA is nonexpansive.

Proof Indeed, for all x, y ∈ C, from Lemma ., we have     (I – λA)x – (I – λA)y = (x – y) – λ(Ax – Ay)   ≤ x – y – λ Ax – Ay, j(x – y) + K  λ Ax – Ay

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≤ x – y – λαAx – Ay + K  λ Ax – Ay   = x – y + λ K  λ – α Ax – Ay . It is clear that if  ≤ λ ≤

α , K

then I – λA is 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 | < ∞. Then limn→∞ 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) E is a uniformly convex and -uniformly smooth Banach space with a weakly sequentially continuous duality mapping; (A) C is a nonempty closed convex subset of E; (A) A : C → E is an α-strongly accretive and L-Lipschitz continuous mapping with S(C, A) = ∅; (A) QC is a sunny nonexpansive retraction from E onto C. 3.2 Parameters restrictions (P) λ, μ and γ are three positive constants satisfying: (i) γ ∈ (, ), λ ∈ [a, b] for some a, b with  < a < b < K αL ; (ii) μλ < K αL where K is the smooth constant of E.

(P) {αn } is a sequence in (, ) such that limn→∞ αn =  and ∞ n= αn = ∞. Algorithm . For given x ∈ C, define a sequence {xn } iteratively by ⎧ ⎨y = Q [( – α )x – λAx ], n C n n n ⎩xn+ = ( – γ )xn + γ QC [xn – λAyn + μ(yn – xn )],

n ≥ .

(.)

Theorem . The sequence {xn } generated by (.) converges strongly to Q (), where Q is a sunny nonexpansive retraction of E onto S(C, A). Proof Let p ∈ S(C, A). First, from Lemma ., we have p = QC [p – δAp] for all δ > . In λ Ap)] for all n ≥ . particular, p = QC [p – λAp] = QC [αn p + ( – αn )(p – –α n

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Since A : C → E is α-strongly accretive and L-Lipschitzian, it must be

α -inverseL

strongly accretive mapping. Thus, by Lemma ., we have     (I – λA)x – (I – λA)y ≤ x – y + λ K  λ – α Ax – Ay . L Since αn →  and λ ∈ [a, b] ⊂ (, K αL ), we get αn <  –

K  L λ α

for enough large n. Without

loss of generality, we may assume that for all n ∈ N, αn <  – Hence, I –

λ A –αn

K  L λ , α

i.e.,

λ –αn

∈ (, K αL ).

is nonexpansive.

From (.), we have        yn – p = QC ( – αn )xn – λAxn – QC αn p + ( – αn ) p –

  λ Ap    – αn        λ λ  α – p – (–p) + ( – α ) x – Ax Ap ≤ n n n n    – αn  – αn        λ λ  ≤ αn p + ( – αn )  I –  – α A xn – I –  – α A p n n

≤ αn p + ( – αn )xn – p.

(.)

By (.) and (.), we have     xn+ – p ≤ ( – γ )xn – p + γ QC xn – λAyn + μ(yn – xn ) – p      λ  = ( – γ )xn – p + γ QC ( – μ)xn + μ yn – Ayn μ     λ – QC ( – μ)p + μ p – Ap   μ ≤ ( – γ )xn – p        λ λ  + γ ( – μ)(xn – p) + μ yn – Ayn – p – Ap   μ μ ≤ ( – γ )xn – p + ( – μ)γ xn – p       λ λ  + μγ  y – p – Ay Ap –  n μ n  μ ≤ ( – μγ )xn – p + μγ yn – p ≤ ( – μγ )xn – p + μγ αn p + μγ ( – αn )xn – p = ( – μγ αn )xn – p + μγ αn p  ≤ max xn – p, p .. .  ≤ max x – p, p . Hence, {xn } is bounded.

(.)

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Set zn = QC [xn – λAyn + μ(yn – xn )]. From (.), we have xn+ = ( – γ )xn + γ zn for all n ≥ . Then we have      yn – yn–  = QC ( – αn )xn – λAxn – QC ( – αn– )xn– – λAxn–         λ λ  Axn – ( – αn– ) xn– – Axn–  ≤ ( – αn ) xn –   – αn  – αn–       λ λ  ≤ ( – αn )  xn –  – α Axn – xn– –  – α Axn–  n n + |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– )        λ λ  Ay Ay y – – – y ≤ ( – μ)xn – xn–  + μ n n n– n–   μ μ ≤ ( – μ)xn – xn–  + μyn – yn–  ≤ ( – μαn )xn – xn–  + |αn – αn– |xn– . It follows that   lim sup zn – zn–  – xn – xn–  ≤ . n→∞

This together with Lemma . implies that lim xn+ – xn  = .

n→∞

From (.), we have     yn – p ≤ αn (–p) + ( – αn ) xn – 

    λ λ Axn – p – Ap    – αn  – αn       λ λ  ≤ αn p + ( – αn )  xn –  – α Axn – p –  – α Ap  n n    α K λ   Axn – Ap . ≤ αn p + ( – αn )xn – p + λ –  – α n L

From (.), (.) and (.), we obtain xn+ – p

       λ λ  Ay Ap ≤ ( – γ )xn – p + γ  ( – μ)(x – p) + μ y – – p – n n n   μ μ       λ λ  Ay Ap ≤ ( – γ )xn – p + γ ( – μ)xn – p + γ μ y – – p –   n μ n μ

(.)

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    λ K  λ α   –  Ayn – Ap ≤ ( – γ μ)xn – p + γ μ yn – p + μ μ L      K λ α    ≤ γ μ αn p + ( – αn )xn – p + λ – Axn – Ap  – α n L    K λ α –  Ayn – Ap + ( – γ μ)xn – p + γ λ μ L    K λ α = αn γ μp + ( – γ μαn )xn – p + γ λμ –  Axn – Ap  – αn L    K λ α –  Ayn – Ap . + γ λμ μ L 

Therefore, we have       α K λ K λ α  ≤ –γ λμ –  Ayn – Ap –  Axn – Ap – γ λμ  – αn L μ L ≤ αn γ μp + xn – p – xn+ – p    = αn γ μp + xn – p + xn+ – p xn – p – xn+ – p   ≤ αn γ μp + xn – p + xn+ – p xn – xn+ . Since αn →  and xn – xn+  →  , we obtain lim Axn – Ap = lim Ayn – Ap = .

n→∞

n→∞

It follows that lim Ayn – Axn  = .

n→∞

Since A is α-strongly accretive, we deduce Ayn – Axn  ≥ αyn – xn , which implies that lim yn – xn  = ,

n→∞

that is,     lim QC ( – αn )xn – λAxn – xn  = .

n→∞

It follows that   lim QC [xn – λAxn ] – xn  = .

n→∞

(.)

Next, we show that    lim sup Q (), j xn – Q () ≥ . n→∞

(.)

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To show (.), since {xn } is bounded, we can choose a sequence {xni } of {xn } converging weakly to z such that       lim sup Q (), j xn – Q () = lim sup Q (), j xni – Q () . n→∞

(.)

i→∞

We first prove z ∈ S(C, A). It follows that   lim QC (I – λA)xni – xni  = .

(.)

i→∞

By Lemma . and (.), we have z ∈ F(QC (I – λA)), it follows from Lemma . that z ∈ S(C, A). Now, from (.) and Lemma ., we have       lim sup Q (), j xn – Q () = lim sup Q (), j xni – Q () n→∞

i→∞

   = Q (), j z – Q () ≥ .

Noticing that xn – yn  → , we deduce that    lim sup Q (), j yn – Q () ≥ . n→∞

λ λ Since yn = QC [(–αn )(xn – –α Axn )] and Q () = QC [αn Q ()+(–αn )(Q ()– –α AQ ())] n n for all n ≥ , we can deduce from Lemma . that



  QC ( – αn ) xn –

λ Axn  – αn

   – ( – αn ) xn –

λ Axn  – αn



   , j yn – Q () ≤ 

and   αn Q () + ( – αn ) Q () –

λ AQ ()  – αn

  – QC αn Q () + ( – αn ) Q () –



λ AQ ()  – αn

    , j yn – Q () ≤ .

Therefore, we have   yn – Q ()     ( – α = Q ) xn – C n 

 λ Axn  – αn   – QC αn Q () + ( – αn ) Q () –

λ AQ ()  – αn       λ ≤ αn –Q () + ( – αn ) xn – Axn  – αn      λ – Q () – AQ () , j yn – Q ()  – αn

   

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       ≤ –αn Q (), j yn – Q () + ( – αn )  xn –  – Q () –

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λ Axn  – αn



   λ yn – Q () AQ ()    – αn        ≤ –αn Q (), j yn – Q () + ( – αn )xn – Q ()yn – Q ()         – αn  xn – Q () + yn – Q () , ≤ –αn Q (), j yn – Q () +  which implies that        yn – Q () ≤ ( – αn )xn – Q () + αn –Q (), j yn – Q () .

(.)

Finally, we will prove that the sequence xn → Q (). As a matter of fact, from (.) and (.), we have   xn+ – Q ()   ≤ ( – γ )xn – Q ()          λ λ     Ay AQ +γ ( – μ) x – Q () + μ y – () – () – Q n n n   μ μ         λ λ    Ay AQ ≤ ( – γ μ)xn – Q () + γ μ y – () – () – Q  n μ n  μ     ≤ ( – γ μ)xn – Q () + γ μyn – Q ()      ≤ ( – γ μαn )xn – Q () + γ μαn –Q (), j yn – Q () . Applying Lemma . to the last inequality, we conclude that xn converges strongly to Q (). 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 1 Department of Mathematics, Tianjin Polytechnic University, Tianjin, 300387, China. 2 Department 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. Optimization 42, 309-321 (1997) 3. Iusem, AN, Lucambio Pe´rez, LR: An extragradient-type algorithm for non-smooth variational inequalities. Optimization 48, 309-332 (2000) 4. Solodov, MV, Tseng, P: Modified projection-type methods for monotone variational inequalities. SIAM J. Control Optim. 34, 1814-1830 (1996)

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doi:10.1186/1029-242X-2013-76 Cite this article as: Wu et al.: A Korpelevich-like algorithm for variational inequalities. Journal of Inequalities and Applications 2013 2013:76.

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