Parallel hybrid extragradient methods for

Numerical Algorithms manuscript No. (will be inserted by the editor)

Parallel hybrid extragradient methods for pseudomonotone equilibrium problems and nonexpansive mappings Dang Van Hieu · Le Dung Muu · Pham Ky Anh

Received: date / Accepted: date

Abstract In this paper we propose and analyze three parallel hybrid extragradient methods for finding a common element of the set of solutions of equilibrium problems involving pseudomonotone bifunctions {fi (x, y)}N i=1 and the in a real Hilbert space. set of fixed points of nonexpansive mappings {Sj }M j=1 Based on parallel computation we can reduce the overall computational effort under widely used conditions on the bifunctions fi (x, y) and the mappings Sj . A simple numerical example is given to illustrate the proposed parallel algorithms. Keywords Equilibrium problem · Pseudomonotone bifuction · Lipschitz-type continuity · Nonexpansive mapping · Hybrid method · Parallel computation

1 Introduction Let C be a nonempty closed convex subset of a real Hilbert space H. The equilibrium problem for a bifunction f : C × C → < ∪ {+∞}, satisfying condition f (x, x) = 0 for every x ∈ C, is stated as follows: Find x∗ ∈ C such that f (x∗ , y) ≥ 0

∀y ∈ C.

Dang Van Hieu Department of Mathematics, Vietnam National University, 334 Nguyen Trai, Thanh Xuan, Hanoi, Vietnam E-mail: [email protected] Le Dung Muu Institute of Mathematics, VAST, Hanoi, 18 Hoang Quoc Viet, Hanoi, Vietnam E-mail: [email protected] Pham Ky Anh Department of Mathematics, Vietnam National University, Hanoi, 334 Nguyen Trai, Thanh Xuan, Hanoi, Vietnam E-mail: [email protected]

(1)

2

D.V. Hieu, L.D. Muu, and P.K. Anh

The set of solutions of (1) is denoted by EP (f ). Problem (1) includes, as special cases, many mathematical models, such as, optimization problems, saddle point problems, Nash equilirium point problems, fixed point problems, convex differentiable optimization problems, variational inequalities, complementarity problems, etc., see [5, 14]. In recent years, many methods have been proposed for solving equilibrium problems, for instance, see [11, 18, 19, 21] and the references therein. A mapping T : C → C is said to be nonexpansive if ||T (x) − T (y)|| ≤ ||x − y|| for all x, y ∈ C. The set of fixed points of T is denoted by F (T ). For finding a common element of the set of solutions of monotone equilibrium problem (1) and the set of fixed points of a nonexpansive mapping T in Hilbert spaces, Tada and Takahashi [20] proposed the following hybrid method:  x0 ∈ C0 = Q0 = C,     z such that f (zn , y) + λ1n hy − zn , zn − xn i ≥ 0, ∀y ∈ C,  n ∈C   wn = αn xn + (1 − αn )T (zn ), C  n = {v ∈ C : ||wn − v|| ≤ ||xn − v||},    Q = {v ∈ C : hx0 − xn , v − xn i ≤ 0},    n xn+1 = PCn ∩Qn (x0 ). According to the above algorithm, at each step for determining the intermediate approximation zn we need to solve a strongly monotone regularized equilibrium problem Find zn ∈ C, such that f (zn , y) +

1 hy − zn , zn − xn i ≥ 0, ∀y ∈ C. λn

(2)

If the bifunction f is only pseudomonotone, then subproblem (2) is not necessarily strongly monotone, even not pseudomonotone, hence the existing algorithms using the monotoncity of the subproblem, cannot be applied. To overcome this difficulty, Anh [1] proposed the following hybrid extragradient method for finding a common element of the set of fixed points of a nonexpansive mapping T and the set of solutions of an equilibrium problem involving a pseudomonotone bifunction f :    x0 ∈ C, C0 = Q0 = C,  1 2    yn = arg min{λn f (xn , y) + 12 ||xn − y||2 : y ∈ C},    tn = arg min{λn f (yn , y) + 2 ||xn − y|| : y ∈ C}, zn = αn xn + (1 − αn )T (tn ),   C  n = {v ∈ C : ||zn − v|| ≤ ||xn − v||},    Q  n = {v ∈ C : hx0 − xn , v − xn i ≤ 0},   xn+1 = PCn ∩Qn (x0 ). Under certain assumptions, the strong convergence of the sequences {xn } , {yn } , {zn } to x† := PEP (f )∩F (T ) x0 has been established. Very recently, Anh and Chung [2] have proposed the following parallel hybrid method for finding a common fixed point of a finite family of relatively

Parallel hybrid extragradient methods

3

N

nonexpansive mappings {Ti }i=1 .  x0 ∈ C, C0 = Q0 = C,    i  y n )Ti (xn ), i =  n = αn xn + (1 − α 1, . . . , N, i   i = arg max i

y − x , y¯n := ynn , n 1≤i≤N n n C = {v ∈ C : ||v − y ¯ || ≤ ||v − x n n ||} ,  n    Q = {v ∈ C : hJx − Jx , x − vi ≥ 0} , 0 n n  n   x T = P x , n ≥ 0. n+1 Cn Qn 0

(3)

This algorithm was extended, modified and generelized by Anh and Hieu [3] for a finite family of asymptotically quasi φ-nonexpansive mappings in Banach spaces. According to algorithm (3), the intermediate approximations yni can be found in parallel. Then the farthest element from xn among all yni , i = 1, . . . , N, denoted by y¯n , is chosen. Using the element y¯n , the authors constructed two convex closed subsets Cn and Qn containing the set of common fixed points F and seperating the initial approximation x0 from F . The next approximation T xn+1 is defined as the projection of x0 onto the intersection Cn Qn . The purpose of this paper is to propose three parallel hybrid extragradient algorithms for finding a common element of the set of solutions of a finite family N of equilibrium problems for pseudomonotone bifunctions {fi }i=1 and the set M of fixed points of a finite family of nonexpansive mappings {Sj }j=1 in Hilbert spaces. We combine the extragradient method for dealing with pseudomonotone equilibrium problems (see, [1, 17]), and Mann’s or Halpern’s iterative algorithms for finding fixed points of nonexpansive mappings [12, 10], with parallel splitting-up techniques [2, 3], as well as hybrid methods (see, [1–3, 11, 16, 18, 19]) to obtain the strong convergence of iterative processes. The paper is organized as follows: In Section 2, we recall some definitions and preliminary results. Section 3 deals with novel parallel hybrid algorithms and their convergence analysis. Finally, in Section 4, we illustrate the propesed parallel hybrid methods by considering a simple numerical experiment. 2 Preliminaries In this section, we recall some definitions and results that will be used in the sequel. Let C be a nonempty closed convex of a Hilbert space H with an inner product h., .i and the induced norm ||.||. Let T : C → C be a nonexpansive mapping with the set of fixed points F (T ). We begin with the following properties of nonexpansive mappings. Lemma 1 [9] Assume that T : C → C is a nonexpansive mapping. If T has a fixed point , then (i) F (T ) is a closed convex subset of H. (ii) I − T is demiclosed, i.e., whenever {xn } is a sequence in C weakly converging to some x ∈ C and the sequence {(I − T )xn } strongly converges to some y , it follows that (I − T )x = y.

4


Since C is a nonempty closed and convex subset of H, for every x ∈ H, there exists a unique element PC x, defined by PC x = arg min {ky − xk : y ∈ C} . The mapping PC : H → C is called the metric (orthogonal) projection of H onto C. It is also known that PC is firmly nonexpansive, or 1-inverse strongly monotone (1-ism), i.e., 2

hPC x − PC y, x − yi ≥ kPC x − PC yk . Besides, we have 2

2

2

kx − PC yk + kPC y − yk ≤ kx − yk .

(4)

Moreover, z = PC x if only if hx − z, z − yi ≥ 0,

∀y ∈ C.

(5)

A function f : C × C → < ∪ {+∞}, where C ⊂ H is a closed convex subset, such that f (x, x) = 0 for all x ∈ C is called a bifunction. Throughout this paper we consider bifunctions with the following properties: A1. f is pseudomonotone, i.e., for all x, y ∈ C, f (x, y) ≥ 0 ⇒ f (y, x) ≤ 0; A2. f is Lipschitz-type continuous, i.e., there exist two positive constants c1 , c2 such that f (x, y) + f (y, z) ≥ f (x, z) − c1 ||x − y||2 − c2 ||y − z||2 ,

∀x, y, z ∈ C;

A3. f is weakly continuous on C × C; A4. f (x, .) is convex and subdifferentiable on C for every fixed x ∈ C. A bifunction f is called monotone on C if for all x, y ∈ C, f (x, y) + f (y, x) ≤ 0. It is obvious that any monotone bifunction is a pseudomonotone one, but not vice versa. Recall that a mapping A : C → H is pseudomonotone if and only if the bifunction f (x, y) = hA(x), y − xi is pseudomonotone on C. The following statements will be needed in the next section. Lemma 2 [4] If the bifunction f satisfies Assumptions A1 − A4, then the solution set EP (f ) is weakly closed and convex. Lemma 3 [7] Let C be a convex subset of a real Hilbert space H and g : C → < be a convex and subdifferentiable function on C. Then, x∗ is a solution to the following convex problem min {g(x) : x ∈ C} if only if 0 ∈ ∂g(x∗ ) + NC (x∗ ), where ∂g(.) denotes the subdifferential of g and NC (x∗ ) is the normal cone of C at x∗ .


5

Lemma 4 [16] Let X be a uniformly convex Banach space, r be a positive number and Br (0) ⊂ X be a closed ball with center at origin and the radius r. Then, for any given subset {x1 , x2 , . . . , xN } ⊂ Br (0) and for any positive PN numbers λ1 , λ2 , . . . , λN with i=1 λi = 1, there exists a continuous, strictly increasing, and convex function g : [0, 2r) → [0, ∞) with g(0) = 0 such that, for any i, j ∈ {1, 2, . . . , N } with i < j,

N

2 N

X

X

2 ≤ λ x λk kxk k − λi λj g(||xi − xj ||).

k k

k=1

k=1

3 Main results In this section, we propose three novel parallel hybrid extragradient algorithms for finding a common element of the set of solutions of equilibrium problems N for pseudomonotone bifunctions {fi }i=1 and the set of fixed points of nonexM pansive mappings {Sj }j=1 in a real Hilbert space H. In what follows, we assume that the solution set \ M F = ∩N ∩j=1 F (Sj ) i=1 EP (fi ) is nonempty and each bifunction fi (i = 1, . . . , N ) satisfies all the conditions A1 − A4. Observe that we can choose the same Lipschitz coefficients {c1 , c2 } for all bifunctions fi , i = 1, . . . , N. Indeed, condition A2 implies that fi (x, z)−fi (x, y)− fi (y, z) ≤ c1,i ||x − y||2 + c2,i ||y − z||2 ≤ c1 ||x − y||2 + c2 ||y − z||2 , where c1 = max {c1,i : i = 1, . . . , N } and c2 = max {c2,i : i = 1, . . . , N } . Hence, fi (x, y) + fi (y, z) ≥ fi (x, z) − c1 ||x − y||2 − c2 ||y − z||2 . Further, since F 6= ∅, by Lemmas 2.1, 2.2, the sets F (Sj ) j = 1, . . . , M and EP (fi ) i = 1, . . . , N are nonempty, closed and convex, hence the solution set F is a nonempty closed and convex subset of C. Thus, given any fixed element x0 ∈ C there exists a unique element x† := PF (x0 ). Algorithm 1 (Parallel Hybrid Mann-extragradient method) 1 1 0 Initialize x ∈ C, 0 < ρ < min 2c1 , 2c2 , n := 0 and the sequence {αk } ⊂ (0, 1) satisfying the condition lim supk→∞ αk < 1. Step 1. Solve N strong convex programs in parallel 1 yni = argmin{ρfi (xn , y) + ||xn − y||2 : y ∈ C} 2

i = 1, . . . , N.

Step 2. Solve N strong convex programs in parallel 1 zni = argmin{ρfi (yni , y) + ||xn − y||2 : y ∈ C} 2 Step 3. Find among zni ,

i = 1, . . . , N.

i = 1, . . . , N, the farthest element from xn , i.e.,

in = argmax{||zni − xn || : i = 1, . . . , N }, z¯n := znin .

6


Step 4. Find intermediate approximations ujn in parallel ujn = αn xn + (1 − αn )Sj z¯n , j = 1, . . . , M. Step 5. Find among ujn ,

j = 1, . . . , M, the farthest element from xn , i.e.,

jn = argmax{||ujn − xn || : j = 1, . . . , M }, u ¯n := ujnn . Step 6. Construct two closed convex subsets of C Cn = {v ∈ C : ||¯ un − v|| ≤ ||xn − v||}, Qn = {v ∈ C : hx0 − xn , v − xn i ≤ 0}. Step 7. The next approximation xn+1 is defined as the projection of x0 onto Cn ∩ Qn , i.e., xn+1 = PCn ∩Qn (x0 ). Step 8. If xn+1 = xn then stop. Otherwise, n := n + 1 and go to Step 1. For establishing the strong convergence of Algorithm 1, we need the following results. Lemma 5 [1, 17] Suppose that x∗ ∈ EP (fi ), and xn , yni , zni , i = 1, . . . , N, are defined as in Step 1 and Step 2 of Algorithm 1. Then ||zni − x∗ ||2 ≤ ||xn − x∗ ||2 − (1 − 2ρc1 )||yni − xn ||2 − (1 − 2ρc2 )||yni − zni ||2 . (6) Lemma 6 If Algorithm 1 reaches a step n ≥ 0, then F ⊂ Cn ∩ Qn and xn+1 is well-defined. Proof As mentioned above, the solution set F is closed and convex. Further, by definition, Cn and Qn are the intersections of halfspaces with the closed convex subset C, hence they are T closed and convex. Next, we verify that F ⊂ Cn Qn for all n ≥ 0. For every x∗ ∈ F , by the convexity of ||.||2 , the nonexpansiveness of Sj , and Lemma 5, we have ||¯ un − x∗ ||2 = ||αn xn + (1 − αn )Sjn z¯n − x∗ ||2 ≤ αn ||xn − x∗ ||2 + (1 − αn )||Sjn z¯n − x∗ ||2 ≤ αn ||xn − x∗ ||2 + (1 − αn )||¯ zn − x∗ ||2 ≤ αn ||xn − x∗ ||2 + (1 − αn )||xn − x∗ ||2 ≤ ||xn − x∗ ||2 .

(7)

Therefore, ||¯ un − x∗ || ≤ ||xn −Tx∗ || or x∗ ∈ Cn . Hence F ⊂ Cn for all n ≥ 0. Now we show that F ⊂ Cn Qn by induction. Indeed, we have F ⊂ C0 T as above. Besides, F ⊂ C = Q , hence F ⊂ C Q . Assume that F ⊂ 0 0 0 T Cn−1 Qn−1 for some n ≥ 1. From xn = PCn−1 T Qn−1 x0 and (5), we get \ hxn − z, x0 − xn i ≥ 0, ∀z ∈ Cn−1 Qn−1 . T Since F ⊂ Cn−1 Qn−1 , hxn − z, x0 − xn i ≥ 0 for all z ∈ F . This T together with the definition of Qn implies that F ⊂ Qn . Hence F ⊂ Cn Qn for all n ≥ 1. Since F and Cn ∩ Qn are nonempty closed convex subsets, PF x0 and xn+1 := PCn ∩Qn (x0 ) are well-defined.


7

Lemma 7 If Algorithm 1 finishes at a finite iteration n < ∞, then xn is a M common element of two sets ∩N i=1 EP (fi ) and ∩j=1 F (Sj ), i.e., xn ∈ F . Proof If xn+1 = xn then xn = xn+1 = PCn ∩Qn (x0 ) ∈ Cn . By the definition of Cn , ||¯ un − xn || ≤ ||xn − xn || = 0, hence u ¯n = xn . From the definition of jn , we obtain ujn = xn , ∀j = 1, . . . , M. This together with the relations ujn = αn xn + (1 − αn )Sj z¯n and 0 < αn < 1 implies that xn = Sj z¯n . Let x∗ ∈ F. By Lemma 5 and the nonexpansiveness of Sj , we get ||xn − x∗ ||2 = ||Sj z¯n − x∗ ||2 ≤ ||¯ zn − x∗ ||2 ≤ ||xn − x∗ ||2 − (1 − 2ρc1 )||ynin − xn ||2 − (1 − 2ρc2 )||ynin − z¯n ||2 . Therefore (1 − 2ρc1 )||ynin − xn ||2 + (1 − 2ρc2 )||ynin − z¯n ||2 ≤ 0. o n Since 0 < ρ < min 2c11 , 2c12 , from the last inequality we obtain xn = ynin = z¯n . Therefore xn = Sj z¯n = Sj xn or xn ∈ F (Sj ) for all j = 1, . . . , M . Moreover, from the relation xn = z¯n and the definition of in , we also get xn = zni for all i = 1, . . . , N . This together with the inequality (6) implies that xn = yni for all i = 1, . . . , N . Thus, 1 xn = argmin{ρfi (xn , y) + ||xn − y||2 : y ∈ C}. 2 By [13, Proposition 2.1], from the last relation we conclude that xn ∈ EP (fi ) for all i = 1, . . . , N, hence xn ∈ F . Lemma 7 is proved. Lemma 8 Let {xn } , yni , zni , ujn be (infinite) sequences generated by Algorithm 1. Then, there hold the relations lim ||xn+1 − xn || = lim ||xn − ujn || = lim ||xn − zni || = lim ||xn − yni || = 0,

n→∞

n→∞

n→∞

n→∞

and limn→∞ ||xn − Sj xn || = 0. Proof From the definition of Qn and (5), we see that xn = PQn x0 . Therefore, for every u ∈ F ⊂ Qn , we get 2

2

2

2

kxn − x0 k ≤ ku − x0 k − ku − xn k ≤ ku − x0 k .

(8)

This implies that the sequence un }, {xn } is bounded. From (7), the sequence {¯ and hence, the sequence ujn are also bounded. Observing that xn+1 = PCn T Qn x0 ∈ Qn , xn = PQn x0 , from (4) we have 2

2

2

2

kxn − x0 k ≤ kxn+1 − x0 k − kxn+1 − xn k ≤ kxn+1 − x0 k .

(9)

8


Thus, the sequence {kxn − x0 k} is nondecreasing, hence there exists the limit of the sequence {kxn − x0 k}. From (9) we obtain 2

2

2

kxn+1 − xn k ≤ kxn+1 − x0 k − kxn − x0 k . Letting n → ∞, we find lim kxn+1 − xn k = 0.

n→∞

(10)

Since xn+1 ∈ Cn , ||¯ un − xn+1 || ≤ kxn+1 − xn k. Thus ||¯ un − xn || ≤ ||¯ un − xn+1 || + ||xn+1 − xn || ≤ 2||xn+1 − xn ||. The last inequality together with (10) implies that ||¯ un − xn || → 0 as n → ∞. From the definition of jn , we conclude that

(11) lim ujn − xn = 0 n→∞

for all j = 1, . . . , M . Moreover, Lemma 5 shows that for any fixed x∗ ∈ F, we have ||ujn − x∗ ||2 = ||αn xn + (1 − αn )Sj z¯n − x∗ ||2 ≤ αn ||xn − x∗ ||2 + (1 − αn )||Sj z¯n − x∗ ||2 ≤ αn ||xn − x∗ ||2 + (1 − αn )||¯ zn − x∗ ||2 ≤ ||xn − x∗ ||2 − (1 − αn )|| (1 − 2ρc1 )||ynin − xn ||2 + (1 − 2ρc2 )||ynin − z¯n ||2 . Therefore (1 − αn )(1 − 2ρc1 )||ynin − xn ||2 +(1 − 2ρc2 )||ynin − z¯n ||2 ≤ ||xn − x∗ ||2 − ||ujn − x∗ ||2 = ||xn − x∗ || − ||ujn − x∗ || ||xn − x∗ || + ||ujn − x∗ || ≤ ||xn − ujn || ||xn − x∗ || + ||ujn − x∗ || . (12) Using the last inequality together with (11) and taking into account the bound edness of two sequences ujn , {xn } as well as the condition lim supn→∞ αn < 1, we come to the relations

lim ynin − xn = lim ynin − z¯n = 0

n→∞

n→∞

(13)

for all i = 1, . . . , N . From ||¯ zn − xn || ≤ ||¯ zn − ynin || + ||ynin − xn || and (13), we obtain limn→∞ k¯ zn − xn k = 0. By the definition of in , we get

lim zni − xn = 0

n→∞

(14)

for all i = 1, . . . , N . From Lemma 5 and (14), arguing similarly to (12) we obtain

lim yni − xn = 0 (15) n→∞


9

for all i = 1, . . . , N . On the other hand, since ujn = αn xn + (1 − αn )Sj z¯n , we have ||ujn − xn || = (1 − αn )||Sj z¯n − xn || = (1 − αn )||(Sj xn − xn ) + (Sj z¯n − Sj xn )|| ≥ (1 − αn ) (||Sj xn − xn || − ||Sj z¯n − Sj xn ||) ≥ (1 − αn ) (||Sj xn − xn || − ||¯ zn − xn ||) . Therefore ||Sj xn − xn || ≤ ||¯ zn − xn || +

1 ||uj − xn ||. 1 − αn n

The last inequality together with (11), (14) and the condition lim supn→∞ αn < 1 implies that lim kSj xn − xn k = 0,

n→∞

(16)

for all j = 1, . . . , M . The proof of Lemma 8 is complete. Lemma 9 Let {xn } be sequence generated by Suppose ¯ is TAlgorithm 1. T T that x N M a weak limit point of {xn }. Then x ¯∈F = i=1 EP (fi ) j=1 F (Sj ) , i.e., x ¯ is a common element of the set of solutions of equilibrium problems N for bifunctions {fi }i=1 and the set of fixed points of nonexpansive mappings M {Sj }j=1 . Proof From Lemma 8 we see that {xn } is bounded. Then there exists a subsequence of {xn } converging weakly to x ¯. For the sake of simplicity, we denote the weakly convergent subsequence again by {xn } , i.e., xn * x ¯. From (16) TM and the demiclosedness of I − Sj , we have x ¯ ∈ F (Sj ). Hence, x ¯ ∈ j=1 F (Sj ). Noting that 1 yni = argmin{ρfi (xn , y) + ||xn − y||2 : y ∈ C}, 2 by Lemma 3, we obtain 1 2 0 ∈ ∂2 ρfi (xn , y) + ||xn − y|| (yni ) + NC (yni ). 2 ¯ ∈ NC (yni ) such that Therefore, there exists w ∈ ∂2 fi (xn , yni ) and w ρw + xn − yni + w ¯ = 0. (17)

Since w ¯ ∈ NC (yni ), w, ¯ y − yni ≤ 0 for all y ∈ C. This together with (17) implies that

ρ w, y − yni ≥ yni − xn , y − yni (18) for all y ∈ C. Since w ∈ ∂2 fi (xn , yni ),

fi (xn , y) − fi (xn , yni ) ≥ w, y − yni , ∀y ∈ C.

(19)

10


From (18) and (19), we get

ρ fi (xn , y) − fi (xn , yni ) ≥ yni − xn , y − yni , ∀y ∈ C.

(20)

Since xn * x ¯ and ||xn − yni || → 0 as n → ∞, we find yni * x ¯. Letting n → ∞ in (20) and using assumption A3, we conclude that fi (¯ x, y) ≥ 0 for all TN y ∈ C (i=1,. . . ,N). Thus, x ¯ ∈ i=1 EP (fi ), hence x ¯ ∈ F . The proof of Lemma 9 is complete. Theorem 1 Let C be a nonempty closed convex subset of a real Hilbert space N H. Suppose that {fi }i=1 is a finite family of bifunctions satisfying conditions M A1 − A4 and {Sj }j=1 is a finite family of nonexpansive mappings on C. Moreover, suppose that the solution set F is nonempty. Then, the (infinite) sequence {xn } generated by Algorithm 1 converges strongly to x† = PF x0 . Proof It is followed directly from T Lemma 6 that the sets F, Cn , Qn are closed convex subsets of C and F ⊂ Cn Qn for all n ≥ 0. Moreover, from Lemma 8 we see that the sequence {xn } is bounded. Suppose that x ¯ is any weak limit point of {xn } and xnj * x ¯. By Lemma 9, x ¯ ∈ F . We now show that the sequence {xn } converges strongly to x† := PF x0 . Indeed, from x† ∈ F and (8), we obtain ||xnj − x0 || ≤ ||x† − x0 ||. The last inequality together with xnj * x ¯ and the weak lower semicontinuity of the norm ||.|| implies that ||¯ x − x0 || ≤ lim inf ||xnj − x0 || ≤ lim sup ||xnj − x0 || ≤ ||x† − x0 ||. j→∞

j→∞

By the definition of x† , x ¯ = x† and limj→∞ ||xnj − x0 || = ||x† − x0 ||. Therefore † limj→∞ ||xnj || = ||x ||. By the Kadec-Klee property of the Hilbert space H, we have xnj → x† as j → ∞. Since x ¯ = x† is any weak limit point of {xn }, the sequence {xn } converges strongly to x† := PF x0 . The proof of Theorem 1 is complete. Corollary 1 Let C be a nonempty closed convex subset of a real Hilbert space N H. Suppose that {fi }i=1 is a finite family of bifunctions satisfying conditions TN A1 − A4, and the set F = i=1 EP (fi ) is nonempty. Let {xn } be the sequence generated in the following manner:  x0 ∈ C0 := C, Q0 := C,   1  i 2  y  n = argmin{ρfi (xn , y) + 2 ||xn − y|| : y ∈ C} i = 1, . . . , N,   1 i i 2   zn = argmin{ρfi (yn , y) + 2 ||xn − y|| : y ∈ C} i = 1, . . . , N, in = argmax{||zni − xn || : i = 1, . . . , N }, z¯n := znin ,   zn − v|| ≤ ||xn − v||},  Cn = {v ∈ C : ||¯    Q = {v ∈ C : hx n 0 − xn , v − xn i ≤ 0},    x T x0 , n ≥ 0, n+1 = PCn Qn where 0 < ρ < min 2c11 , 2c12 . Then the sequence {xn } converges strongly to x† = P F x0 .


11

Corollary 2 Let C be a nonempty closed convex subset of a real Hilbert space N H. Suppose that {Ai }i=1 is a finite family of pseudomonotone and L-Lipschitz TN continuous mappings from C to H such that F = i=1 V IP (Ai , C) is nonempty. Let {xn } be the sequence generated in the following manner:  x0 ∈ C0 := C, Q0 := C,     yni = PC (xn − ρAi (xn )) i = 1, . . . , N,    i i  i = 1, . . . , N,  zn = PC xn − ρAi (yn ) i in = argmax{||zn − xn || : i = 1, . . . , N }, z¯n := znin ,   Cn = {v ∈ C : ||¯ zn − v|| ≤ ||xn − v||},     Qn = {v ∈ C : hx0 − xn , v − xn i ≤ 0},    x T x0 , n ≥ 0, n+1 = PCn Qn where 0 < ρ
nmax for some chosen sufficiently large number nmax , then stop. Theorem 2 Let C be a nonempty closed convex subset of a real Hilbert space N H. Suppose that {fi }i=1 is a finite family of bifunctions satisfying conditions M A1−A4, and {Sj }j=1 is a finite family of nonexpansive mappings on C. Moreover, suppose that the solution set F is nonempty. Then, the sequence {xn } generated by the Algorithm 2 converges strongly to x† = PF x0 .


13

Proof Arguing similarly as in the proof of Lemma 6 and Theorem 1, we conclude that F, Cn , Qn are closed and convex. Besides, F ⊂ Cn ∩Qn for all n ≥ 0. Moreover, the sequence {xn } is bounded and lim ||xn+1 − xn || = 0.

(21)

n→∞

Since xn+1 ∈ Cn+1 , ||¯ un − xn+1 ||2 ≤ αn ||x0 − xn+1 ||2 + (1 − αn )||xn − xn+1 ||2 . Letting n → ∞, from (21), limn→∞ αn = 0 and the boundedness of {xn }, we obtain lim ||¯ un − xn+1 || = 0.

n→∞

Proving similarly to (11) and (12), we get lim ||ujn − xn || = 0,

n→∞

j = 1, . . . , M,

and (1 − αn )(1 − 2ρc1 )||ynin − xn ||2 + (1 − 2ρc2 )||ynin − z¯n ||2 ≤ αn (||x0 − x∗ ||2 − ||xn − x∗ ||2 ) +||xn − ujn || ||xn − x∗ || + ||ujn − x∗ ||

(22)

for each x∗ ∈ F . Letting n → ∞ in (22), one has zn − xn || = 0, lim ||ynin − xn || = lim ||¯

n→∞

n→∞

j = 1, . . . , N,

Repeating the proof of (14) and (15), we get lim ||yni − xn || = lim ||zni − xn || = 0,

n→∞

n→∞

i = 1, . . . , N.

Using ujn = αn x0 + (1 − αn )Sj z¯n , by a straightforward computation, we obtain ||Sj xn − xn || ≤ ||¯ zn − xn || +

αn 1 ||uj − xn || + ||x0 − xn ||, 1 − αn n 1 − αn

which implies that limn→∞ ||Sj xn −xn || = 0. The rest of the proof of Theorem 2 is similar to the arguments in the proofs of Lemma 9 and Theorem 1. Next replacing Steps 4 and 5 in Algorithm 1, consisting of a Mann’s iteration and a parallel splitting-up step, by an iteration step involving a convex combination of the identity mapping I and the mappings Sj , j = 1, . . . , N , we come to the following algorithm.

14


Algorithm 3 (Parallel hybrid iteration-extragradient method) 1 1 0 Initialize: x ∈ C, 0 < ρ < min 2c1 , 2c2 , n := 0 and the positive sequences PM ∞ {αk,l }k=1 (l = 0, . . . , M ) satisfying the conditions: 0 ≤ αk,j ≤ 1, j=0 αk,j = 1, lim inf k→∞ αk,0 αk,l > 0 for all l = 1, . . . , M . Step 1. Solve N strong convex programs in parallel 1 yni = argmin{ρfi (xn , y) + ||xn − y||2 : y ∈ C} 2

i = 1, . . . , N.

Step 2. Solve N strong convex programs in parallel 1 zni = argmin{ρfi (yni , y) + ||xn − y||2 : y ∈ C} 2 Step 3. Find among zni ,

i = 1, . . . , N.

i = 1, . . . , N, the farthest element from xn , i.e.,

in = argmax{||zni − xn || : i = 1, . . . , N }, z¯n := znin . Step 4. Compute in parallel ujn := Sj z¯n ; j = 1, . . . , M, and put un = αn,0 xn +

M X

αn,j ujn .

j=1

Step 5. Construct two closed convex subsets of C Cn = {v ∈ C : ||un − v|| ≤ ||xn − v||}, Qn = {v ∈ C : hx0 − xn , v − xn i ≤ 0}. Step 6. The next approximation xn+1 is determined as the projection of x0 onto Cn ∩ Qn , i.e., xn+1 = PCn ∩Qn (x0 ). Step 7. If xn+1 = xn then stop. Otherwise, n := n + 1 and go to Step 1. Remark 3 Arguing similarly as in the proof of Lemma 7, we can prove that if Algorithm 3 finishes at a finite iteration n < ∞, then xn ∈ F , i.e., xn is a common element of the set of solutions of equilibrium problems and the set of fixed points of nonexpansive mappings. Theorem 3 Let C be a nonempty closed convex subset of a real Hilbert space N H. Suppose that {fi }i=1 is a finite family of bifunctions satisfying conditions M A1−A4, and {Sj }j=1 is a finite family of nonexpansive mappings on C. Moreover, suppose that the solution set F is nonempty. Then, the (infinite) sequence {xn } generated by the Algorithm 3 converges strongly to x† = PF x0 .


15

Proof Arguing similarly as in the proof of Theorem 1, we T can conclude that F, Cn , Qn are closed convex subsets of C. Besides, F ⊂ Cn Qn and lim ||xn+1 − xn || = lim ||yni − xn || = lim ||zni − xn || = lim ||un − xn || = 0

n→∞

n→∞

n→∞

n→∞

(23) for all i = 1, . . . , N . For every x∗ ∈ F , by Lemmas 4 and 5, we have ∗ 2

||un − x ||

= ||αn,0 xn +

M X

αn,j Sj z¯n − x∗ ||2

j=1

= ||αn,0 (xn − x∗ ) +

M X

αn,j (Sj z¯n − x∗ )||2

j=1

≤ αn,0 ||xn − x∗ ||2 +

M X

αn,j ||Sj z¯n − x∗ ||2 − αn,0 αn,l g(||Sl z¯n − xn ||)

j=1

≤ αn,0 ||xn − x∗ ||2 +

M X

αn,j ||¯ zn − x∗ ||2 − αn,0 αn,l g(||Sl z¯n − xn ||)

j=1

≤ αn,0 ||xn − x∗ ||2 +

M X

αn,j ||xn − x∗ ||2 − αn,0 αn,l g(||Sl z¯n − xn ||)

j=1 ∗ 2

≤ ||xn − x || − αn,0 αn,l g(||Sl z¯n − xn ||). Therefore αn,0 αn,l g(||Sl z¯n − xn ||) ≤ ||xn − x∗ ||2 − ||un − x∗ ||2 ≤ (||xn − x∗ || − ||un − x∗ ||) (||xn − x∗ || + ||un − x∗ ||) ≤ ||xn − un || (||xn − x∗ || + ||un − x∗ ||) . The last inequality together with (23), lim inf n→∞ αn,0 αn,l > 0 and the boundedness of {xn } , {un } imply that limn→∞ g(||Sl z¯n − xn ||) = 0. Hence lim ||Sl z¯n − xn || = 0.

n→∞

(24)

Moreover, from (23), (24) and ||Sl xn − xn || ≤ ||Sl xn − Sl z¯n || + ||Sl z¯n − xn || ≤ ||xn − z¯n || + ||Sl z¯n − xn || we obtain lim ||Sl xn − xn || = 0

n→∞

for all l = 1, . . . , M . The same argument as in the proofs of Lemma 9 and Theorem 1 shows that the sequence {xn } converges strongly to x† := PF x0 . The proof of Theorem 3 is complete. Remark 4 Putting M = N = 1 in Theorems 1 and 3, we obtain the corresponding result announced in [1, Theorem 3.3].

16


4 Numerical experiment Let H = u ¯n ≥ 0, by the proof of Theorem 1, 0 ∈ Cn , i.e., |¯ un | ≤ |xn |, hence 0 ≤ u ¯n < xn . This together with the definitions of Cn and Qn lead us to the following formulas: xn + u ¯n ; Cn = 0, 2 Qn = [0, xn ]. Therefore

xn + u ¯n . Cn ∩ Qn = 0, min xn , 2

Since u ¯n ≤ xn , we find

xn +¯ un 2

≤ xn . So xn + u ¯n . Cn ∩ Qn = 0, 2

From the definition of xn+1 we obtain xn+1 =

xn + u ¯n . 2

Thus we come to the following algorithm: Initialize x0 := 1; n := 1; ρ := 1/5; αn := 1/n; := 10−5 ; ξi := i/(N + 1), i = 1, . . . , N ; N := 2 × 106 ; M := 3 × 106 . Step 1. Find the intermediate approximations yni in parallel (i = 1, . . . , N ). xn if 0 ≤ xn ≤ ξi , i yn = xn − ρ[exp(xn − ξi ) + sin(xn − ξi ) − 1] if ξi < xn ≤ 1. Step 2. Find the intermediate approximations zni in parallel (i = 1, . . . , N ). xn if 0 ≤ yni ≤ ξi , zni = xn − ρ[exp(yni − ξi ) + sin(yni − ξi ) − 1] if ξi < yni ≤ 1. Step 3. Find the element z¯n which is farthest from xn among zni , i = 1, . . . , N . in = arg max |zni − xn | : i = 1, . . . , N , z¯n = znin . Step 4. Find the intermediate approximations ujn in parallel ujn = αn xn + (1 − αn )

z¯nj sinj−1 (¯ zn ) , j = 1, . . . , M. 2j − 1

18


Step 5. Find the element u ¯n which is farthest from xn among ujn , j = 1, . . . , M . jn = arg max |ujn − xn | : j = 1, . . . , M , u ¯n = znjn . Step 6. If |¯ un − xn | ≤ then stop. Otherwise go to Step 7. un . Step 7. xn+1 = xn +¯ 2 Step 8. If |xn+1 − xn | ≤ then stop. Otherwise n := n + 1 and go to Step 1.

The numerical experiment is performed on a LINUX cluster 1350 with 8 computing nodes. Each node contains two Intel Xeon dual core 3.2 GHz, 2GBRam. All the programs are written in C. For given tolerances we compare execution time of the parallel hybrid Mannextragradient method (PHMEM) in parallel and sequential modes. We use the following notations: PHMEM T OL Tp Ts

The parallel hybrid Mann-extragradient method Tolerance kxk − x∗ k Time for PHMEM’s execution in parallel mode (2CPUs - in seconds) Time for PHMEM’s execution in sequential mode (in seconds)

Table 1 Experiment with αn =

1 . n

T OL 10−5 10−6 10−8

PHMEM Tp 5.23 5.86 7.57

Ts 9.98 11.25 14.33

According to the above experiment, in the most favourable cases the speed up and the efficiency of the parallel hybrid Mann-extragradient method are Sp = Ts /Tp ≈ 2; Ep = Sp /2 ≈ 1, respectively. Concluding remarks In this paper we proposed three parallel hybrid extragradients methods for finding a common element of the set of solutions of equilibrium problems for N pseudomonotone bifunctions {fi }i=1 and the set of fixed points of nonexpanM sive mappings {Sj }j=1 in Hilbert spaces, namely: – a parallel hybrid Mann-extragradient method; – a parallel hybrid Halpern-extragradient method, and – a parallel hybrid iteration-extragradient method. The efficiency of the proposed parallel algorithms is verified by a simple numerical experiment on computing clusters.


19

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