A Korpelevich-like algorithm for variational inequalities

Zhitao Wu; Yonghong Yao; Yeong-Cheng Liou; Hong-Jun Li

doi:10.1186/1029-242X-2013-76

Research
Open Access

A Korpelevich-like algorithm for variational inequalities

Zhitao Wu¹,
Yonghong Yao¹Email author,
Yeong-Cheng Liou² and
Hong-Jun Li¹

Journal of Inequalities and Applications20132013:76

https://doi.org/10.1186/1029-242X-2013-76

Received: 6 October 2012
Accepted: 4 February 2013
Published: 28 February 2013

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^{*} \in C

such that

〈 A x^{*}, x - x^{*} 〉 \geq 0, \forall x \in C,

(1.1)

where C is a nonempty closed convex subset of a real Hilbert space H and

A : C \to H

is a given mapping, is a fundamental problem in variational analysis and, in particular, in optimization theory. For related works, please see [1–20] and the references contained therein. Especially, Yao, Marino and Muglia [21] presented the following modified Korpelevich method for solving (1.1):

\begin{aligned} y_{n} = P_{C} [x_{n} - λ A x_{n} - α_{n} x_{n}], \\ x_{n + 1} = P_{C} [x_{n} - λ A y_{n} + μ (y_{n} - x_{n})], n \geq 0 . \end{aligned}

(1.2)

Recently, Aoyama, Iiduka and Takahashi [22] extended the variational inequality (1.1) to Banach spaces as follows:

Find x^{*} \in C such that 〈 A x^{*}, J (x - x^{*}) 〉 \geq 0, \forall x \in C,

(1.3)

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 (1.3). The generalized variational inequality (1.3) is connected with the fixed point problem for nonlinear mappings. For solving the above generalized variational inequality (1.3), Aoyama, Iiduka and Takahashi [22] introduced the iterative algorithm

x_{n + 1} = α_{n} x_{n} + (1 - α_{n}) Q_{C} [x_{n} - λ_{n} A x_{n}], n \geq 0,

(1.4)

where

Q_{C}

is a sunny nonexpansive retraction from E onto C and

{α_{n}} \subset (0, 1)

,

{λ_{n}} \subset (0, \infty)

are two real number sequences. Motivated by (1.4), Yao and Maruster [23] presented a modification of (1.4) as follows:

x_{n + 1} = β_{n} x_{n} + (1 - β_{n}) Q_{C} [(1 - α_{n}) (x_{n} - λ A x_{n})], n \geq 0 .

(1.5)

Motivated and inspired by the above algorithms (1.2), (1.4) and (1.5), in this paper, we suggest an extragradient-type method via the sunny nonexpansive retraction for solving the variational inequalities (1.3) in Banach spaces. It is shown that the presented algorithm converges strongly to a special solution of the variational inequality (1.3).

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) \in J (x - y)

such that

〈 A x - A y, j (x - y) 〉 \geq 0

for all

x, y \in C

. A mapping A of C into E is said to be α-strongly accretive if for

α > 0

,

〈 A x - A y, j (x - y) 〉 \geq α {∥ x - y ∥}^{2}

for all

x, y \in C

. A mapping A of C into E is said to be α-inverse-strongly accretive if for

α > 0

,

〈 A x - A y, j (x - y) 〉 \geq α {∥ A x - A y ∥}^{2}

for all $x, y \in C$ .

Let

U = {x \in E : ∥ x ∥ = 1}

. A Banach space E is said to uniformly convex if for each

ϵ \in (0, 2]

, there exists

δ > 0

such that for any

x, y \in U

,

∥ x - y ∥ \geq ϵ implies ∥ \frac{x + y}{2} ∥ \leq 1 - δ .

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 \to 0} \frac{∥ x + t y ∥ - ∥ x ∥}{t}

(2.1)

exists for all

x, y \in U

. It is also said to be uniformly smooth if the limit (2.1) is attained uniformly for

x, y \in U

. The norm of E is said to be Frechet differentiable if for each

x \in U

, the limit (2.1) is attained uniformly for

y \in U

. And we define a function

ρ : [0, \infty) \to [0, \infty)

called the modulus of smoothness of E as follows:

ρ (τ) = sup {\frac{1}{2} (∥ x + y ∥ + ∥ x - y ∥) - 1 : x, y \in X, ∥ x ∥ = 1, ∥ y ∥ = τ} .

It is known that E is uniformly smooth if and only if ${lim}_{τ \to 0} ρ (τ) / τ = 0$ . Let q be a fixed real number with $1 < q \leq 2$ . Then a Banach space E is said to be q-uniformly smooth if there exists a constant $c > 0$ such that $ρ (τ) \leq c τ^{q}$ for all $τ > 0$ .

We need the following lemmas for the proof of our main results.

Lemma 2.1 [24]

Let q be a given real number with

1 < q \leq 2

and let E be a q-uniformly smooth Banach space. Then

{∥ x + y ∥}^{q} \leq {∥ x ∥}^{q} + q 〈 y, J_{q} (x) 〉 + 2 {∥ K y ∥}^{q}

for all

x, y \in E

, where K is the q-uniformly smoothness constant of E and

J_{q}

is the generalized duality mapping from E into

2^{E^{*}}

defined by

J_{q} (x) = {f \in E^{*} : 〈 x, f 〉 = {∥ x ∥}^{q}, ∥ f ∥ = {∥ x ∥}^{q - 1}}, \forall x \in 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 (Q x + t (x - Q x)) = Q x,

whenever $Q x + t (x - Q x) \in C$ for $x \in C$ and $t \geq 0$ . A mapping Q of C into itself is called a retraction if $Q^{2} = Q$ . If a mapping Q of C into itself is a retraction, then $Q z = z$ for every $z \in R (Q)$ , 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 2.2 [25]

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 - Q u, j (y - Q u) 〉 \leq 0

for all $u \in C$ and $y \in D$ .

Lemma 2.3 [22]

Let C be a nonempty closed convex subset of a smooth Banach space X. Let

Q_{C}

be a sunny nonexpansive retraction from X onto C and let A be an accretive operator of C into X. Then for all

λ > 0

,

S (C, A) = F (Q_{C} (I - λ A)),

where $S (C, A) = {x^{*} \in C : 〈 A x^{*}, J (x - x^{*}) 〉 \geq 0, \forall x \in C}$ .

Lemma 2.4 [26]

Let C be a nonempty closed convex subset of a real 2-uniformly smooth Banach space X. Let the mapping

A : C \to X

be α-inverse-strongly accretive. Then we have

{∥ (I - λ A) x - (I - λ A) y ∥}^{2} \leq {∥ x - y ∥}^{2} + 2 λ (K^{2} λ - α) {∥ A x - A y ∥}^{2} .

In particular, if $0 \leq λ \leq \frac{α}{K^{2}}$ , then $I - λ A$ is nonexpansive.

Proof Indeed, for all

x, y \in C

, from Lemma 2.1, we have

\begin{array}{rcl} {∥ (I - λ A) x - (I - λ A) y ∥}^{2} & = & {∥ (x - y) - λ (A x - A y) ∥}^{2} \\ \leq & {∥ x - y ∥}^{2} - 2 λ 〈 A x - A y, j (x - y) 〉 + 2 K^{2} λ^{2} {∥ A x - A y ∥}^{2} \\ \leq & {∥ x - y ∥}^{2} - 2 λ α {∥ A x - A y ∥}^{2} + 2 K^{2} λ^{2} {∥ A x - A y ∥}^{2} \\ = & {∥ x - y ∥}^{2} + 2 λ (K^{2} λ - α) {∥ A x - A y ∥}^{2} . \end{array}

It is clear that if $0 \leq λ \leq \frac{α}{K^{2}}$ , then $I - λ A$ is nonexpansive. □

Lemma 2.5 [27]

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 ${x_{n}}$ is a sequence of C such that $x_{n} \to x$ weakly and $x_{n} - T x_{n} \to 0$ strongly, then x is a fixed point of T.

Lemma 2.6 [28]

Assume

{a_{n}}

is a sequence of nonnegative real numbers such that

a_{n + 1} \leq (1 - γ_{n}) a_{n} + δ_{n}, n \geq 0,

where

{γ_{n}}

is a sequence in

(0, 1)

and

{δ_{n}}

is a sequence in R such that

(a)

$\sum_{n = 0}^{\infty} γ_{n} = \infty$ ;
(b)

${lim sup}_{n \to \infty} δ_{n} / γ_{n} \leq 0$ or $\sum_{n = 0}^{\infty} | δ_{n} | < \infty$ .

Then ${lim}_{n \to \infty} a_{n} = 0$ .

3 Main results

In this section, we present our Korpelevich-like algorithm and consequently we will show its strong convergence.

3.1 Conditions assumptions

(A1) E is a uniformly convex and 2-uniformly smooth Banach space with a weakly sequentially continuous duality mapping;

(A2) C is a nonempty closed convex subset of E;

(A3) $A : C \to E$ is an α-strongly accretive and L-Lipschitz continuous mapping with $S (C, A) \neq \emptyset$ ;

(A4) $Q_{C}$ is a sunny nonexpansive retraction from E onto C.

3.2 Parameters restrictions

(P1) λ, μ and γ are three positive constants satisfying:

(i)

$γ \in (0, 1)$ , $λ \in [a, b]$ for some a, b with $0 < a < b < \frac{α}{K^{2} L^{2}}$ ;
(ii)

$\frac{λ}{μ} < \frac{α}{K^{2} L^{2}}$ where K is the smooth constant of E.

(P2) ${α_{n}}$ is a sequence in $(0, 1)$ such that ${lim}_{n \to \infty} α_{n} = 0$ and $\sum_{n = 1}^{\infty} α_{n} = \infty$ .

Algorithm 3.1 For given

x_{0} \in C

, define a sequence

{x_{n}}

iteratively by

{\begin{matrix} y_{n} = Q_{C} [(1 - α_{n}) x_{n} - λ A x_{n}], \\ x_{n + 1} = (1 - γ) x_{n} + γ Q_{C} [x_{n} - λ A y_{n} + μ (y_{n} - x_{n})], n \geq 0 . \end{matrix}

(3.1)

Theorem 3.2 The sequence ${x_{n}}$ generated by (3.1) converges strongly to $Q^{'} (0)$ , where $Q^{'}$ is a sunny nonexpansive retraction of E onto $S (C, A)$ .

Proof Let $p \in S (C, A)$ . First, from Lemma 2.2, we have $p = Q_{C} [p - δ A p]$ for all $δ > 0$ . In particular, $p = Q_{C} [p - λ A p] = Q_{C} [α_{n} p + (1 - α_{n}) (p - \frac{λ}{1 - α_{n}} A p)]$ for all $n \geq 0$ .

Since

A : C \to E

is α-strongly accretive and L-Lipschitzian, it must be

\frac{α}{L^{2}}

-inverse-strongly accretive mapping. Thus, by Lemma 2.4, we have

{∥ (I - λ A) x - (I - λ A) y ∥}^{2} \leq {∥ x - y ∥}^{2} + 2 λ (K^{2} λ - \frac{α}{L^{2}}) {∥ A x - A y ∥}^{2} .

Since $α_{n} \to 0$ and $λ \in [a, b] \subset (0, \frac{α}{K^{2} L^{2}})$ , we get $α_{n} < 1 - \frac{K^{2} L^{2} λ}{α}$ for enough large n. Without loss of generality, we may assume that for all $n \in N$ , $α_{n} < 1 - \frac{K^{2} L^{2} λ}{α}$ , i.e., $\frac{λ}{1 - α_{n}} \in (0, \frac{α}{K^{2} L^{2}})$ . Hence, $I - \frac{λ}{1 - α_{n}} A$ is nonexpansive.

From (3.1), we have

\begin{array}{rcl} ∥ y_{n} - p ∥ & = & ∥ Q_{C} [(1 - α_{n}) x_{n} - λ A x_{n}] - Q_{C} [α_{n} p + (1 - α_{n}) (p - \frac{λ}{1 - α_{n}} A p)] ∥ \\ \leq & ∥ α_{n} (- p) + (1 - α_{n}) [(x_{n} - \frac{λ}{1 - α_{n}} A x_{n}) - (p - \frac{λ}{1 - α_{n}} A p)] ∥ \\ \leq & α_{n} ∥ p ∥ + (1 - α_{n}) ∥ (I - \frac{λ}{1 - α_{n}} A) x_{n} - (I - \frac{λ}{1 - α_{n}} A) p ∥ \\ \leq & α_{n} ∥ p ∥ + (1 - α_{n}) ∥ x_{n} - p ∥ . \end{array}

(3.2)

By (3.1) and (3.2), we have

\begin{array}{rcl} ∥ x_{n + 1} - p ∥ & \leq & (1 - γ) ∥ x_{n} - p ∥ + γ ∥ Q_{C} [x_{n} - λ A y_{n} + μ (y_{n} - x_{n})] - p ∥ \\ = & (1 - γ) ∥ x_{n} - p ∥ + γ ∥ Q_{C} [(1 - μ) x_{n} + μ (y_{n} - \frac{λ}{μ} A y_{n})] \\ - Q_{C} [(1 - μ) p + μ (p - \frac{λ}{μ} A p)] ∥ \\ \leq & (1 - γ) ∥ x_{n} - p ∥ \\ + γ ∥ (1 - μ) (x_{n} - p) + μ [(y_{n} - \frac{λ}{μ} A y_{n}) - (p - \frac{λ}{μ} A p)] ∥ \\ \leq & (1 - γ) ∥ x_{n} - p ∥ + (1 - μ) γ ∥ x_{n} - p ∥ \\ + μ γ ∥ (y_{n} - \frac{λ}{μ} A y_{n}) - (p - \frac{λ}{μ} A p) ∥ \\ \leq & (1 - μ γ) ∥ x_{n} - p ∥ + μ γ ∥ y_{n} - p ∥ \\ \leq & (1 - μ γ) ∥ x_{n} - p ∥ + μ γ α_{n} ∥ p ∥ + μ γ (1 - α_{n}) ∥ x_{n} - p ∥ \\ = & (1 - μ γ α_{n}) ∥ x_{n} - p ∥ + μ γ α_{n} ∥ p ∥ \\ \leq & max {∥ x_{n} - p ∥, ∥ p ∥} \\ ⋮ \\ \leq & max {∥ x_{0} - p ∥, ∥ p ∥} . \end{array}

(3.3)

Hence, ${x_{n}}$ is bounded.

Set

z_{n} = Q_{C} [x_{n} - λ A y_{n} + μ (y_{n} - x_{n})]

. From (3.1), we have

x_{n + 1} = (1 - γ) x_{n} + γ z_{n}

for all

n \geq 0

. Then we have

\begin{array}{rcl} ∥ y_{n} - y_{n - 1} ∥ & = & ∥ Q_{C} [(1 - α_{n}) x_{n} - λ A x_{n}] - Q_{C} [(1 - α_{n - 1}) x_{n - 1} - λ A x_{n - 1}] ∥ \\ \leq & ∥ (1 - α_{n}) (x_{n} - \frac{λ}{1 - α_{n}} A x_{n}) - (1 - α_{n - 1}) (x_{n - 1} - \frac{λ}{1 - α_{n - 1}} A x_{n - 1}) ∥ \\ \leq & (1 - α_{n}) ∥ (x_{n} - \frac{λ}{1 - α_{n}} A x_{n}) - (x_{n - 1} - \frac{λ}{1 - α_{n}} A x_{n - 1}) ∥ \\ + | α_{n} - α_{n - 1} | ∥ x_{n - 1} ∥ \\ \leq & (1 - α_{n}) ∥ x_{n} - x_{n - 1} ∥ + | α_{n} - α_{n - 1} | ∥ x_{n - 1} ∥, \end{array}

and thus

\begin{array}{rcl} ∥ z_{n} - z_{n - 1} ∥ & = & ∥ Q_{C} [x_{n} - λ A y_{n} + μ (y_{n} - x_{n})] - Q_{C} [x_{n - 1} - λ A y_{n - 1} + μ (y_{n - 1} - x_{n - 1})] ∥ \\ \leq & (1 - μ) ∥ x_{n} - x_{n - 1} ∥ + μ ∥ (y_{n} - \frac{λ}{μ} A y_{n}) - (y_{n - 1} - \frac{λ}{μ} A y_{n - 1}) ∥ \\ \leq & (1 - μ) ∥ x_{n} - x_{n - 1} ∥ + μ ∥ y_{n} - y_{n - 1} ∥ \\ \leq & (1 - μ α_{n}) ∥ x_{n} - x_{n - 1} ∥ + | α_{n} - α_{n - 1} | ∥ x_{n - 1} ∥ . \end{array}

It follows that

\underset{n \to \infty}{lim sup} (∥ z_{n} - z_{n - 1} ∥ - ∥ x_{n} - x_{n - 1} ∥) \leq 0 .

This together with Lemma 2.6 implies that

lim_{n \to \infty} ∥ x_{n + 1} - x_{n} ∥ = 0 .

From (3.2), we have

\begin{array}{rcl} {∥ y_{n} - p ∥}^{2} & \leq & {∥ α_{n} (- p) + (1 - α_{n}) [(x_{n} - \frac{λ}{1 - α_{n}} A x_{n}) - (p - \frac{λ}{1 - α_{n}} A p)] ∥}^{2} \\ \leq & α_{n} {∥ p ∥}^{2} + (1 - α_{n}) {∥ (x_{n} - \frac{λ}{1 - α_{n}} A x_{n}) - (p - \frac{λ}{1 - α_{n}} A p) ∥}^{2} \\ \leq & α_{n} {∥ p ∥}^{2} + (1 - α_{n}) {∥ x_{n} - p ∥}^{2} + 2 λ (\frac{K^{2} λ}{1 - α_{n}} - \frac{α}{L^{2}}) {∥ A x_{n} - A p ∥}^{2} . \end{array}

(3.4)

From (3.1), (3.3) and (3.4), we obtain

Therefore, we have

\begin{array}{rcl} 0 & \leq & - 2 γ λ μ (\frac{K^{2} λ}{1 - α_{n}} - \frac{α}{L^{2}}) {∥ A x_{n} - A p ∥}^{2} - 2 γ λ μ (\frac{K^{2} λ}{μ} - \frac{α}{L^{2}}) {∥ A y_{n} - A p ∥}^{2} \\ \leq & α_{n} γ μ {∥ p ∥}^{2} + {∥ x_{n} - p ∥}^{2} - {∥ x_{n + 1} - p ∥}^{2} \\ = & α_{n} γ μ {∥ p ∥}^{2} + (∥ x_{n} - p ∥ + ∥ x_{n + 1} - p ∥) (∥ x_{n} - p ∥ - ∥ x_{n + 1} - p ∥) \\ \leq & α_{n} γ μ {∥ p ∥}^{2} + (∥ x_{n} - p ∥ + ∥ x_{n + 1} - p ∥) ∥ x_{n} - x_{n + 1} ∥ . \end{array}

Since

α_{n} \to 0

and

∥ x_{n} - x_{n + 1} ∥ \to 0

, we obtain

lim_{n \to \infty} ∥ A x_{n} - A p ∥ = lim_{n \to \infty} ∥ A y_{n} - A p ∥ = 0 .

It follows that

lim_{n \to \infty} ∥ A y_{n} - A x_{n} ∥ = 0 .

Since A is α-strongly accretive, we deduce

∥ A y_{n} - A x_{n} ∥ \geq α ∥ y_{n} - x_{n} ∥,

which implies that

lim_{n \to \infty} ∥ y_{n} - x_{n} ∥ = 0,

that is,

lim_{n \to \infty} ∥ Q_{C} [(1 - α_{n}) x_{n} - λ A x_{n}] - x_{n} ∥ = 0 .

It follows that

lim_{n \to \infty} ∥ Q_{C} [x_{n} - λ A x_{n}] - x_{n} ∥ = 0 .

(3.5)

Next, we show that

\underset{n \to \infty}{lim sup} 〈 Q^{'} (0), j (x_{n} - Q^{'} (0)) 〉 \geq 0 .

(3.6)

To show (3.6), since

{x_{n}}

is bounded, we can choose a sequence

{x_{n_{i}}}

of

{x_{n}}

converging weakly to z such that

\underset{n \to \infty}{lim sup} 〈 Q^{'} (0), j (x_{n} - Q^{'} (0)) 〉 = \underset{i \to \infty}{lim sup} 〈 Q^{'} (0), j (x_{n_{i}} - Q^{'} (0)) 〉 .

(3.7)

We first prove

z \in S (C, A)

. It follows that

lim_{i \to \infty} ∥ Q_{C} (I - λ A) x_{n_{i}} - x_{n_{i}} ∥ = 0 .

(3.8)

By Lemma 2.5 and (3.8), we have $z \in F (Q_{C} (I - λ A))$ , it follows from Lemma 2.3 that $z \in S (C, A)$ .

Now, from (3.7) and Lemma 2.2, we have

\begin{array}{rcl} \underset{n \to \infty}{lim sup} 〈 Q^{'} (0), j (x_{n} - Q^{'} (0)) 〉 & = & \underset{i \to \infty}{lim sup} 〈 Q^{'} (0), j (x_{n_{i}} - Q^{'} (0)) 〉 \\ = & 〈 Q^{'} (0), j (z - Q^{'} (0)) 〉 \\ \geq & 0 . \end{array}

Noticing that

∥ x_{n} - y_{n} ∥ \to 0

, we deduce that

\underset{n \to \infty}{lim sup} 〈 Q^{'} (0), j (y_{n} - Q^{'} (0)) 〉 \geq 0 .

Since

y_{n} = Q_{C} [(1 - α_{n}) (x_{n} - \frac{λ}{1 - α_{n}} A x_{n})]

and

Q^{'} (0) = Q_{C} [α_{n} Q^{'} (0) + (1 - α_{n}) (Q^{'} (0) - \frac{λ}{1 - α_{n}} A Q^{'} (0))]

for all

n \geq 0

, we can deduce from Lemma 2.2 that

〈 Q_{C} [(1 - α_{n}) (x_{n} - \frac{λ}{1 - α_{n}} A x_{n})] - [(1 - α_{n}) (x_{n} - \frac{λ}{1 - α_{n}} A x_{n})], j (y_{n} - Q^{'} (0)) 〉 \leq 0

and

Therefore, we have

which implies that

{∥ y_{n} - Q^{'} (0) ∥}^{2} \leq (1 - α_{n}) {∥ x_{n} - Q^{'} (0) ∥}^{2} + 2 α_{n} 〈 - Q^{'} (0), j (y_{n} - Q^{'} (0)) 〉 .

(3.9)

Finally, we will prove that the sequence

x_{n} \to Q^{'} (0)

. As a matter of fact, from (3.1) and (3.9), we have

Applying Lemma 2.6 to the last inequality, we conclude that $x_{n}$ converges strongly to $Q^{'} (0)$ . This completes the proof. □

Declarations

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.

Authors’ Affiliations

(1)

Department of Mathematics, Tianjin Polytechnic University, Tianjin, 300387, China

(2)

Department of Information Management, Cheng Shiu University, Kaohsiung, 833, Taiwan

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