Modified hybrid projection methods for finding common solutions to variational inequality problems

DOI 10.1007/s10589-016-9857-6Modified hybrid projection methods for finding common solutions to variational inequality problems Dang Van Hieu 1 · Pham Ky Anh 1 · Le Dung Muu 2 Received:

DOI 10.1007/s10589-016-9857-6

Modified hybrid projection methods for finding

common solutions to variational inequality problems

Dang Van Hieu 1 · Pham Ky Anh 1 · Le Dung Muu 2

Received: 29 February 2016

© Springer Science+Business Media New York 2016

Abstract In this paper we propose several modified hybrid projection methods for

solving common solutions to variational inequality problems involving monotone

and Lipschitz continuous operators Based on differently constructed half-spaces, theproposed methods reduce the number of projections onto feasible sets as well as thenumber of values of operators needed to be computed Strong convergence theoremsare established under standard assumptions imposed on the operators An extension ofthe proposed algorithm to a system of generalized equilibrium problems is consideredand numerical experiments are also presented

Keywords Variational inequality· Equilibrium problem · Generalized equilibriumproblem· Gradient method · Extragradient method

Mathematics Subject Classification 65Y05· 65K15 · 68W10 · 47H05 · 47H10

1 Department of Mathematics, Vietnam National University, Hanoi, 334 Nguyen Trai,

Thanh Xuan, Hanoi, Vietnam

2 Institute of Mathematics, VAST, Hanoi, 18 Hoang Quoc Viet, Hanoi, Vietnam

1 Introduction

Let{K i}N

i=1be nonempty closed convex subsets of a real Hilbert space H such that

i=1K i = ∅ We consider the following common solutions to variational

inequality problems (CSVIP) introduced in [11–13]

i=1: K i → H are given operators.

In what follows, we assume that each operator A i satisfies the following assumptions

G1 A i is monotone on K i

G2 A i is L - Lipschitz continuous on K i

G3 The solution set F of Problem1is nonempty

If N = 1 then CSVIP (1) becomes the classical variational inequality problem (VIP)[17,20,26,27]:

Find x∗∈ K such that:A(x∗), x − x∗≥ 0, ∀x ∈ K, (2)

where A : K → H is a monotone and L - Lipschitz continuous operator and K

is a nonempty closed convex subset of H Let us denote the solution set of VIP

(2) by V I (A, K ) Problem 1 is a generalization of many other problems ing: convex feasibility problems, common fixed point problems, common minimizerproblems, common saddle-point problems, hierarchical variational inequality prob-lems, variational inequality problems over the intersection of convex sets, etc., see[3 5,7,12,21,22]

includ-In this paper, we focus on projection methods, which together with regularizationones are fundamental methods for solving VIPs with monotone and Lipschitz contin-uous mappings The extragradient method was first introduced by Korpelevich [28] in

1976 for the saddle point problem and then was extended to VIPs It was proved that

in a finite dimensional space [28], the sequence{x n} defined by



y n = P K (x n − λA(x n )),

x n+1= P K (x n − λA(y n )), (3)

whereλ ∈ (0, 1

L ) converges to some point in V I (A, K ) However, in infinite

dimen-sional Hilbert spaces, the extragradient method only converges weakly In recentyears, the extragradient method has received a lot of attention, see, for example,[10,14,15,24,30,31] and the references therein Nadezhkina and Takahashi [32] intro-duced the following hybrid extragradient method

For solving CSVIP (1) with N > 1, Censor et al [12] proposed a strongly

con-vergent hybrid algorithm (CGRS’s method) where all operators A i , i = 1, , N are

multi-valued mappings from H to 2 H For the sake of simplicity, we recall this

algo-rithm when the mappings A i , i = 1, , N are single-valued, λ i

n = λ and γ i

n = 1 2(see, Algorithm 3.1 in [12] for more details) as follows:

large-In this paper, motivated and inspired by the results of Censor et al [12] and Malitskyand Semenov [31], we introduce the following hybrid algorithm for solving CSVIP(1)

Algorithm 1.2 (Modified hybrid projection method for CSVIPs)

Algorithm1.2needs only N projections onto the feasible sets K i and N values of A i

per each iteration Thus, based on slightly different half-spaces C n ias suggested in [31],

we can reduce a half of number of computations required in Algorithm1.1 Besides,Algorithm1.1requires the monotonicity and Lipschitz continuity of the operators A i

on the whole space H , while these properties are assumed to be satisfied only on the feasible sets K i in Algorithm1.2 Note that C n i and Q n are either closed half-

spaces or whole space Therefore, the projection x n+1= P C n ∩Q n (x0) can be found by

Haugazeau’s method [8, Corollary 29.8] or any available method of convex quadraticprogramming [9, Chapter 8] Some numerical experiments, comparing Algorithm1.2

with Algorithm1.1, are performed Figures 1and2 illustrate the iterative steps ofAlgorithms1.1and1.2for the case N = 2, respectively

This paper is organized as follows: In Sect.2we recall some definitions and liminary results used in the paper Section3deals with the convergence analysis ofAlgorithm1.2and its modification Section4presents an extension of Algorithm1.2togeneralized equilibrium problems In Sect.5, we perform some numerical experiments

pre-to illustrate the proposed algorithms in comparison with Algorithm1.1

2 Preliminaries

Let C be a nonempty closed convex subset of a real Hilbert space H We begin with

some concepts of the monotonicity of an operator, see, [1,25] An operator A : C → H

is said to be

i strongly monotone on C if there exists a constant η > 0 such that

A(x) − A(y), x − y ≥ η||x − y||2, ∀x, y ∈ C;

ii monotone on C if

A(x) − A(y), x − y ≥ 0, ∀x, y ∈ C;

iii α - inverse strongly monotone on C if there exists a positive constant α such that

A(x) − A(y), x − y ≥ α||A(x) − A(y)||2, ∀x, y ∈ C;

iv maximal monotone if it is monotone and its graph G (A) := {(x, A(x)) : x ∈ C}

is not a proper subset of the graph of any other monotone mapping;

v L - Lipschitz continuous on C if there exists a positive constant L such that

||A(x) − A(y)|| ≤ L||x − y||, ∀x, y ∈ C.

We have the following result

Lemma 2.1 [37] Let C be a nonempty, closed convex subset of a Hilbert space H

and A be a monotone, hemicontinuous mapping of C into H Then

V I (A, C) = {u ∈ C : v − u, A(v) ≥ 0, ∀v ∈ C}

Remark 2.1 Lemma2.1ensures that the solution set of VIP (2) is closed and convex

For every x ∈ H, the metric projection P C x of x onto C is defined by

P C x

Since C is a nonempty closed and convex subset of H , P C x exists and is unique It is

well-known that the metric projection P C : H → C has the following

characteriza-tions

Lemma 2.2 [1,18] Let P C : H → C be the metric projection from H onto a nonempty

closed convex subset C of H Then

i P C is 1 - inverse strongly monotone on H , i.e., for all x , y ∈ H,

The following lemmas will be used for proving convergence theorems in Sect.3

Lemma 2.3 [36] Let C be a nonempty closed convex subset of a Hilbert space H and

let A be a monotone and hemi-continuous mapping of C into H with D(A) = C Let

Then Q is maximal monotone and Q−10= V I (A, C).

Lemma 2.4 [31] Let {a n }, {b n }, {c n } be nonnegative real sequences , α, β ∈  and

for all n ≥ 0 the following inequality holds

a n ≤ b n + βc n − αc n+1.

If∞

n=0b n < +∞ and α > β ≥ 0 then lim n→∞a n = 0.

Proof We divide the proof of Theorem3.1into four steps.

Claim 1 The following estimate holds

From the relations (10) and (11), we conclude

and Lemma2.1, we obtain



+ 2λ A i (y i

n−1), y i

n+1− y i n

Claim 2 The sets F , C n , Q n are closed and convex, F ⊂ C n ∩ Q n for all n≥ 0, and

From the definitions, C n i , Q n are closed half-spaces or the whole space H , hence they

are closed and convex subsets for all n ≥ 0 Claim 1 and the definition of C i

nensure

that F ⊂ C i

n for all n ≥ 0 and i = 1, , N It is clear that F ⊂ C0 ∩ Q0 Assume that

F ⊂ C n ∩ Q n for some n ≥ 0 From x n+1= P C n ∩Q n (x0) and Lemma2.2.iii, we seethatz − x n+1, x0− x n+1 ≤ 0 for all z ∈ Cn ∩ Q n In particular, this is also true for

z ∈ F ⊂ C n ∩ Q n By the definition of Q n+1, F ⊂ Q n+1 Hence, F ⊂ C n+1∩ Q n+1

Thus, by induction, F ⊂ C n ∩ Q n for all n≥ 0

From the definition of Q nand Lemma2.2.iii, x n = P Q n (x0) It follows from Lemma

2.2.ii that

||z − x n||2≤ ||z − x0||2− ||x n − x0||2, ∀z ∈ Q n (16)

Substituting z = x†:= P F (x0) ∈ Q ninto inequality (16), one has

||x†− x0||2− ||x n − x0||2≥ ||x†− x n||2≥ 0, (17)

which implies that the sequence{||x n − x0||}, and therefore {x n}, are bounded

Sub-stituting z = x n+1∈ Q ninto inequality (16), one also gets

Moreover, by (23), the sequence

y i n is bounded because of the boundedness of{x n}

Claim 3 If p is any weak cluster point of {x n } then p ∈ F.

For each i = 1, , N, set

which implies that

n→∞||A i (y i

n+1) − A i (y i

Assume that there exists a subsequence of{x n } converging weakly to p Without loss

of generality, we can write x n  p as n → ∞ Since ||x n − y i

n+1|| → 0, y i

n  p as

n → ∞ Passing to the limit in (27) as n→ ∞ and employing relation (28) and theboundedness of

y n i , we obtainx − p, y ≥ 0 for all (x, y) ∈ G(Q i ) Thus, from

the maximal monotonicity of Q iand Lemma2.3, one has p ∈ Q−1i (0) = V I (A i , K i )

for all 1≤ i ≤ N Hence, p ∈ F.

Claim 4 The sequences{x n} and y n i converge strongly to x†:= P F (x0).

From (17) we obtain||x n − x0|| ≤ ||x†− x0|| for all n ≥ 0 This together with the

weak lower semicontinuity of the norm||.|| implies that

||p − x0|| ≤ lim inf

n→∞||x n − x0|| ≤ lim sup

n→∞||x n − x0|| ≤ ||x†− x0||.

By the definition of x†, p = x†and limn→∞||x n − x0|| = ||x†− x0|| Finally, since

x n − x0  x†− x0 , the Kadec-Klee property of H ensures that x n − x0 → x†− x0

or x n → x†= P F (x0) as n → ∞ By the uniqueness of x†, the whole sequence{x n}

converges strongly to x† From Claim 2, we can conclude that

Remark 3.1 By induction, we can show that C n is the intersection of finitely many

closed half-spaces Actually, the number of half-spaces increases precisely by N after

each iterative step However, for our tested problems, Algorithm3.3converges morequickly than Algorithms1.1and1.2due to the shrinking property of the sequence

From x n = P C n (x0) and Lemma2.2.ii, we have

||u − x n||2+ ||x n − x0||2≤ ||u − x0||2, ∀u ∈ C n (30)

Substituting u = x†:= P F (x0) ∈ C ninto inequality (30), we obtain

||x n − x0||2 is creasing Hence, there exists the limit of the sequence

nonde-||x n − x0||2 From relation(31) we have

which implies that

∞



n=1

||x n+1− x n||2< +∞.

The rest of the proof of Theorem3.2is similar to that of Theorem3.1 

4 An extension to finitely many generalized equilibrium problems

Let K i , i = 1, , N be nonempty closed convex subsets of a real Hilbert space H

such that K := ∩N

i=1K i = ∅ Let f i : K i ×K i →  be bifunctions such that f i (x, x) =

0 for all x ∈ K i and A i : K i → H be operators In this section, we consider the

following problem of finding common solutions to generalized equilibrium problems(CSGEP) [23,33,34]

Problem 2 Find x∗∈ K such that

f i (x∗, y) +A i (x∗), y − x∗≥ 0, ∀y ∈ K i , i = 1, , N.

If N = 1 then Problem2 becomes the following generalized equilibrium problem[16,38]:

Find x∗∈ K such that f (x∗, y) +A(x∗), y − x∗≥ 0, ∀y ∈ K, (32)

where f : K × K →  is a bifunction and A : K → H is an operator Let us denote

the solution set of (32) by G E P ( f, A) Some methods for CSGEPs can be found in

[23,33–35] Almost existing methods require a strict assumption on the strong (or

inverse-strong) monotonicity of A i In this section, we assume that A i is monotone

and Lipschitz continuous We recall that a bifunction f : K × K →  is called

i monotone if f (x, y) + f (y, x) ≤ 0 for all x, y ∈ K ;

ii n - cyclically monotone (see, [2]) if for each cycle x1 , x2, , x n , x n+1= x1 ∈ K ,

n



i=1

An example for a bifunction f : × →  which satisfies the n - cyclic monotonicity

as f (x, y) = x(y − x) Some other cyclically monotone operators can be found in

[6]

For solving Problem2, we assume that the operators A isatisfy the conditions G1−G2

and the bifunctions f i satisfy the following conditions

A1 f i (x, x) = 0 for all x ∈ K i;

A2 f iis 3 - cyclically monotone;

A3 for all x , y, z ∈ K i,

lim

t→0sup f i (tz + (1 − t)x, y) ≤ f i (x, y);

A4 for each x ∈ K i , the function f i (x, ) is convex and lower semicontinuos;

Note that Assumptions A1 and A2 imply that f i is monotone on K i Indeed, from A2

we have

f i (x, y) + f i (y, z) + f i (z, x) ≤ 0, ∀x, y, z ∈ K i

Particularly, by substituting z = x into the last inequality and using assumption A1, one has f i (x, y) + f i (y, x) ≤ 0, ∀x, y ∈ K i Thus f i is monotone on K i

The following results concern with the bifunction f : C × C → .

Lemma 4.5 [16] Let C be a nonempty closed and convex subset of a Hilbert space H,

f be a bifunction from C × C to  satisfying the conditions A1 − A4 and let r > 0,

x ∈ H Then, there exists z ∈ C such that

f (z, y) +1

r y − z, z − x ≥ 0, ∀y ∈ C.

Lemma 4.6 [16] Let C be a nonempty closed and convex subset of a Hilbert space

H , f be a bifunction from C ×C to  satisfying the conditions A1−A4 For all r > 0

and x ∈ H, define the mapping

(C4) E P ( f, C) is closed and convex.

Next, we extend Algorithm1.2to the following algorithm for solving CSGEPs

Algorithm 4.4 (An extension of the modified hybrid projection method for CSGEPs) Initialization: Choose x0 = x1 ∈ H, y i

We have the following result.

Theorem 4.3 Let K i , i = 1, , N be closed and convex subsets of a real Hilbert space H such that K = ∩N

By arguing similarly to the proofs of (10)–(12), we get



+ 2r A i (y i

n−1), y i

n+1− y i n

From the definitions of y n i and T f i

Claim 7 {x n } then p ∈ F.

Without loss of generality, we assume that x n  p Since ||y i

n − x n || → 0, y i

n  p.

From the fact

y n i ⊂ K i and the weak closedness of the convex set K i, we can

conclude that p ∈ K i It follows from (41) and the triangle inequality that

Due to the monotonicity of f iwe find

For each t ∈ (0, 1] and y ∈ K i , set y t = ty + (1 − t)p It follows from the convexity

of K i that y t ∈ K i Thus, the monotonicity of A i and relation (44) yield

Remark 4.1 Algorithm3.3can be extended to CSGEPs in the same manner as rithm4.4.

nite), q i is a vector inm The feasible set K i = K ⊂  m is a closed convex subset

defined by K = {x ∈  m : Ax ≤ b}, where A is an l × m matrix and b is a tive vector It is clear that A i is monotone and Lipschitz continuous with the constant

nonnega-L = max {||M i || : i = 1, , N} The initial data is listed in Table1

We see that K is a polyhedral convex set The sets C n i , Q nin Algorithms1.1and

1.2are either the half-spaces or the whole spacem , thus C n ∩ Q nis also a polyhedral

convex set which, in general, is the intersection of N + 1 half-spaces The set C ninAlgorithm3.3is the intersection of n N half-spaces.

All projections on half-spaces are explicitly defined All projections onto polyhedralconvex sets are effectively performed by Haugazeau’s method [8, Corollary 29.8] with

error TOL In this example, we choose q i = 0 Thus, the solution set F = {0} We

compare the execution time (CPU in second) and the number of iterations (Iter.) forAlgorithms1.1,1.2and3.3 The numerical results are showed in Table2

Example 2 Let H be the functional space L2[0, 1] and K i be the unit ball B [0, 1] ⊂ H.

In this example, we consider the operators A i : K i → H defined by

Table 1 The initial data

The starting points x0, x1 x0= x 1= (1, 1, , 1) T∈  m The starting points yi0, yi

1 = P Ki(x0− λAi(yi

0))

4L, k = 3

The size l× m of matrix A l= 20 and m = 2, 5, 10, 20

The matrixes A,b,B i, Ci, Di Generated randomly

0 for all x ∈ K i and A i : K i → H be operators In this section, we consider the

following problem of finding common solutions to generalized... example for a bifunction f : × →  which satisfies the n - cyclic monotonicity

as f (x, y) = x(y − x) Some other cyclically monotone operators can be found in

[6]

For. .. extend Algorithm1. 2to the following algorithm for solving CSGEPs

Algorithm 4.4 (An extension of the modified hybrid projection method for CSGEPs) Initialization: Choose x0 = x1