Characterization of H-monotone operators with applications to variational inclusions

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Computers and Mathematms with Apphcatlons 50 (2005) 329-337 www elsevmr com/locate/camwa

Characterization of H - M o n o t o n e Operators with Applications to Variational Inclusions LU-CHUAN Department

of Mathematms,

ZENG* Shanghai

Normal

Umvermty

Shanghai 200234, P R China S Y - M I N G GUU t D e p a r t m e n t of Business AdmmlstraUon, College of M a n a g e m e n t Yuan-Ze Umvermty, Chung-L1 City, Taoyuan Hsmn, Taiwan 330, R 0 C. iesmguu©s aturn, yzu. edu. tw

JEN-CHIH YAO $ D e p a r t m e n t of A p p h e d Mathematics, National Sun Yat-sen University Kaohsmng, Taiwan 804, R O C yao 3 c~math, nsysu, edu. tw

(Recewed and accepted June 2005) Abstract--This paper estabhshes necessary and sufficmnt conditions for operators to be Hmonotone Based on these conditions, we introduce a new iteratwe algorithm for solving a class of variational mclumons Strong convergence of this algorithm is estabhshed under appropriate assumptions on the parameters Estimate of its convergence rate is also prowded @ 2005 Elsevier Ltd All rights reserved

Keywords--H-monotone algorithm

operator, Resolvent operator techmque, Variatlonal inclusion, Iteratlve

1. I N T R O D U C T I O N Variational inequality was imtlally studied by Stampacchia [1] m 1964. Since then, it has been extensively studied because it plays a crucial role in the study of mechanics, physics, optimization and control, eeonomms and transportation equiIibrmm and engmeering sciences, etc. Thanks to its wide applications, the classical variational inequality has been well studied and generalized in various directions. The reader is referred to [2-8] and the references therein Among these generalizations, variational inclusion is of interest and importance. Recent development of the variational inequality is to design effMent iterative algorithms to compute approximate solutions for variational inequalities and their generalizations. Up to now, many authors have presented *This research was partlally supported by the Teaching and Research Award Fund for Outstanding Young Teachers in Higher Educatlon Instltutlons of MOE, China and the Dawn Program Foundatlon m Shanghal fAuthor to whom all correspondence should be addressed. This research was partially supported by a grant NSC 93-2213-E155-029 from the Natmnal Scmnce Council of Talwan :~Thls research was partlally supported by a grant from the National Scmnce Counell of Talwan 0898-1221/05/$ - see front matter @ 2005 Elsevmr Ltd All rights reserved d o r l 0 1016/j camwa 2005.06 001

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L.-C ZENGet al

330

implementable and significant numerical methods such as projection method and its variant forms, hnear approximation, descent method, Newton's method and the method based on auxiliary principle technique. In particular, the method based on the resolvent operator techmque is a generalization of projection method and has been widely used to solve variational inclusions; see, e.g., [9-18]. In 2003, Fang and Huang [19] introduced a new class of monotone operators which were called H-monotone operators. For an H-monotone operator, they established the definition and Lipschitz continuity for its resolvent operator. Furthermore, based on the resolvent operator technique, they constructed an iterative algorithm for approximating the solution of a class of varlationat inclusions involving H-monotone operators. Their results improved and extended many known results in the hterature. In this paper, under the assumption that H is strongly monotone, continuous and singlevalued, we first prove that a multivalued monotone operator is H-monotone if and only if it is maximal monotone. Subsequently, we define the resolvent operator associated with a strongly Hmonotone operator, prove its Lipschltz continuity, and estimate its Lipschitz constant. Further, we study the variational inclusion introduced in [19] with strongly H-monotone operators. We construct a new algorithm for solving this class of variational inclusions by using the resolvent operator technique. Thanks to our estimate of Lipschitz constant of the resolvent operator, our convergence criteria for the algorithm are very different from corresponding ones in [19]. Throughout this paper, we suppose that 7-/is a real Hilbert space endowed with a norm II U and an inner product (., .), respectively. Let 2~ denote the family of all the nonempty subsets of ~ . In what follows, we recall some concepts which will be used in the sequel. DEFINITION 1.1. Let T, H : 7~ -~ ~ be two single-valued operators. T is said to be (i) monotone if (Tx-Ty,

x - y > >O,

Vx, yCT-l;

(il) strictly monotone if T is monotone and (Tx - Ty, x - y> = 0 ~ x = y; (iii) strongly monotone Jf there exists some constant r > O, such that

_> rllx- yl[ 2,

Vx, y

(iv) strongly monotone with respect to H if there exists some constant ~/> 0, such that (Tx-Zy,

Hx-Hy)

>_71lx-yll 2,

Vx, y c H ;

(v) Llpschitz continuous if there exists some constant s > O, such that IITx - Tyll <_ llx - YlI.

Vx, y c

REMARK 1.1 (See [19].) If T and H are Lipschitz continuous with constants ~- and s, respectively, and T is strongly monotone with respect to H with constant 7, then ~/_< TS. DEFINITION 1.2. A multJvalued operator M ' ~ -~ 2~ is said to be (i) monotone if (x-y,u-v)>O,

Vu, vET-l,

xEMu,

ycMv;

(11) strongly monotone if there exists some constant ~ > O, such that

>>_vllu-vll 2,

Vu, v ~ , xCMu, ycMv, and (S+aM)(~) = ~ for aii A > 0,

(iil) maximal monotone i f M is monotone where I denotes the identity mapping on 7-/; (iv) maximal strongly monotone 1t: M is strongly monotone and (I + AM)(~) = H for ali A>0.

Characterization of H-Monotone Operators

331

REMARK 1.2. A multivalued operator M is maximal monotone if and only if M is monotone and there is no other monotone operator whose graph properly contains the graph G r ( M ) of M where G r ( M ) = {(u, x) 6 7{ x 7-/:x E M u } .

2. S T R O N G L Y

H-MONOTONE

OPERATORS

Recently, Fang and Huang [19] have mtroduced a new class of monotone operators, 1.e., Hmonotone operators and discussed some p r o p e m e s of this class of operators. DEFINITION 2.1. (See [19].) Let H : 7-{ ~ ~ be a single-valued operator and M : ~ -~ 2 ~ be a multivalued operator. M is said to be

(i) H-monotone ifM is monotone and (H + AM)(T/) = 7~ holds for every A > O; (i) strongly H-monotone if M is strongly monotone and (H + AM)(~) = • holds for every A>O. REMARK 2.1. If H = I, then the definition o f / - m o n o t o n e operators reduces to that of maximal monotone operators. As a matter of fact, the class of H - m o n o t o n e operators has close relation with that of maximal monotone operators In order to give a characterization of H-monotone operators, we need the following propositions and lemmas. PROPOSITION 2.1. (See [19] ) L e t H : TL ~ ~ be a single-valued strictly m o n o t o n e operator and M : 7-t ~ 2 n be an H - m o n o t o n e operator. Then M is m a x i m a l m o n o t o n e Now recall the notion of m-accretive operators. Let X be a real Banaeh space with a norm l[ " I], X * denote the dual space of X and let (x, f> denote the value of f E X * at x E X. For k c ( - o o , +co), a multivalued operator A : D ( A ) C X ~ 2 x is said to be k-accretive if for each x , y E D ( A ) there exists g(u - v) E Y(u - v), such that k[lu-vH

2,

VxEAu,

yEAv.

(21)

Here J : X ~ 2 x* is the normalized duality mapping defined by J ( x ) = { f E X * : {x,f> = Ilxll2 = IIftl2},

where (., .) denotes the generalized duality paring. It is an immediate consequence of the HahnBanach theorem t h a t Y(x) is nonempty for each x E X. Moreover, it is known t h a t Y is singlevalued if and only if X is smooth. For k > 0 in inequality (2.1), we say that A is strongly accretive while for k = 0, A is simply called accretive. In addition, if the range of I + AA is precisely X for all A > 0, where I is the identity mapping on X, then A is said to be m-accretive. In particular, if X = 7-/a real Hilbert space, then the definitions of strong accretiveness, accretiveness and maccretiveness reduce to the ones of strong monotonieity, monotonieity and maximal monotonieity, respectively. Recently, Jung and Morales [20] proved the following deep and important result. PROPOSITION 2.2. (See [20].) L e t X be a s m o o t h Banach space, A : D ( A ) C X -~ 2 x be m-accretive, and S : D ( S ) c X --* X be continuous and strongly accretive with D ( A ) C D ( S ) . T h e n for each z E X , the equation z 6 S x + A A x has a unique solution x~ for A > 0 COROLLARY 2.1. L e t 7-{ be a real Hilbert space. L e t M : 7-{ --~ 2 7~ be a m a x i m a l monotone multlvalued operator and H : 7-{ --~ 7~ be a strongly monotone, continuous and single-valued operator T h e n for each z 6 7-{ the equation z E H x + A M x has a unique solution x~ for A > 0

REMARK 2.2. If H : 7/ --* 7-{ is a strongly monotone, continuous, single-valued operator and M : 7-/ --* 2 4 is a maximal monotone multivalued operator, then from Corollary 2.1 we know that the operator (H + AM) -1 is single-valued. Hence, we can define the resolvent operator RH M,X : 7-/--* 7-{ as follows: RH,~(u) = ( H + A M ) - I ( u ) ,

Vu E 7-/.

We are ready to give a characterization for the class of H - m o n o t o n e operators.

(2.2)

332

L -C ZENG st al

THEOREM 2.1 Let H ~f --* 1-t be a strongly monotone, continuous and single-valued operator. Then a multivalued operator M : ~ --* 2 ~ is H - m o n o t o n e if and only if M is m a x i m a / m o n o t o n e . PROOF. At first, let M 7-I -~ 2 ~ be H-monotone. Since H : 7{ -~ 7-/is strongly monotone, H is strictly monotone. Thus, it follows from Proposition 2.1 that M is maximal monotone Conversely, suppose that M is maximal monotone. Then M is monotone. Note that H is a strongly monotone, eontmuous and single-valued operator. Hence, it follows from Corollary 2.1 that for each z E H the equation z E H x + A M x has a unique solution x~ for A > 0. This impiies that ( H + AM)(7-/) = 7-{ holds for every A > 0. Therefore, M is H-monotone. I COROLLARY 2.2. Let H : ~ ~ ~ be a strongly monotone, continuous and single-valued operator. Then a multivalued operator M : 7-( --* 2 7t is strongly H - m o n o t o n e if and only ff M is maxima/ strongly monotone Let H be continuous and strongly monotone and M be maximal strongly monotone. Now we prove the Lipschitz continuity of the resolvent operator / { H a defined by (2.2) and estimate its Lipschitz constant. THEOREM 2.2. Let H : ~ ~ 7{ be continuous and strongly monotone with constant 7 Let M ' ~ ~ 2 ~ be maximal strongly monotone with constant r]. Then the resolvent operator R H x : 7-I ~ 7-{ is Lipschitz continuous ruth constant 1/(3' + Ar~), i.e., Vu, v E "H.

PROOF

Let u, v be any given points in 7-(. It follows from (2.2) that RH,),(u) = ( H + A M ) - I ( u )

and

RH¢,(v) = ( H + A M ) - I ( v ) .

and

1 (v - H (RH,),(v))) E M (RH,x(v))

This implies that 1 (u - H ( R H ) , ( u ) ) ) E M (RH,;~(u))

Since M is strongly monotone, we have ,

R

,

(v)ll

-<

1 (u - g ( R H x(u)) -- (v -- H ( R H x(v))),RHM,),(U) -- RH;~(v))

= 1 (u- vl

( H (RH,),Cu)) -- H ( R H , x ( v ) ) ) , R H x ( u ) - RH,),(v))

It follows that

+ a,

:

-

+

-

,

and hence, 1

This completes the proof

Vu, v C'H

I

C h a r a c t e r i z a t i o n of H - M o n o t o n e O p e r a t o r s

3. V A R I A T I O N A L

333

INCLUSIONS

In this section, we consider a class of variatmnal inclusions involving strong H - m o n o t o n e operators in Hilbert spaces. We construct a new iterative algorithm for approximating solutions of this class of variational inclusions by using the resolvent operator technique Let A, H : ~ -+ 7-/be two single-valued operators and M : %/-+ 2 4 be a multivalued operator. Consider the following variational inclusion' find u C ~ , such t h a t

0 E A(u) + M(u).

(3.1)

SPECIAL CASES. (I) When M is maximal monotone problem (3.1) has been studied (2) If M = a~ where O~ denotes continuous functional ~ : 7-( -~ problem: find u E ~, such that

and A is strongly monotone and Lipschitz continuous, by Huang [15]. the subdifferential of a proper, convex and lower semiR U {+oo}, then problem (3.1) reduces to the following

( A ( u ) , ~ - ~) + v(~) - ~(~) > O,

(3 2)

vve~,

which is called a nonlinear variational inequality and has been studied by many authors; see, for example, [2-5,18,21]. (3) If M = 06K where 6K is the indicator functlon of a nonempty, closed and convex subset/( of ~, then problem (3.1) reduces to the following problem: find u E/(, such that { A ( u ) , v - u } >_0,

VvEK,

(3.3)

which is the classical variational inequality; see, e.g., [1,7,22]. H From the definition of RM,,X , we have the following result

LEMMA 3.1. Let H : ~ --+ ~ be a strongly monotone and continuous operator and M . ~ -~ 24 be maximal monotone. Then u E 7-I is a solution of problem (3.1) ff and only if u

:

RH,~

[H(u)

-

AA(u)],

for some I > 0. Based on L e m m a 3.1, we construct the following iterative algorithm for solving problem (3.1). ALGORITHM 3.1. Let {c~} and {/3~} be two sequences in [0, 1] For any u0 E 7-/, the iterative sequence {u~} C 7-/is defined by tLn+ 1 = (1 -- Oln)lt n -~ OLnt~,A[H(~/n) vn = (1 - / 5 n ) u n

+/3~RH,a[H(un)

-- A A ( v ~ ) ] , - AA(un)],

(3.4) n = 0,1,

..

W h e n oLn : 1, /~n = 0, Vn >_ 0, Algorithm 3 1 reduces imme&ately to Algorithm 3 1 in [19]. For easy reference, we present it here as follows

ALGORITHM 3.2.

(See [19].) Un+l

For any uo E ~ , the iteratlve sequence fun} c ~ is defined by = ~I:~H A [ H ( u n )

-

AA(u,0],

n = 0, 1, ..

(3.5)

L -C ZENG et al.

334

THEOREM 3.1

L e t {c~} and {fl~} be employed by Algorithm 3.1 w~th }-~n~=o~ = oo Let H : T{ --. ~ be a strongly monotone and Lipschitz continuous operator with constants 7 and -c, respectively L e t A . T{ --~ 7-I be Lipschitz continuous and strongIy m o n o t o n e with respect to H with constants s and r~ respectively. A s s u m e that M : 7-~ --+ 2 ~ N a strongly H - m o n o t o n e operator with constant 77 and that there exists some constant A > O, such t h a t

22--~+ ~ ,I < ~/(~ + ~ ) 2 _~ _ _ ~2)(~ _ ~ ) ,

(,) Then the iterative sequence {un} generated by Algorithm 3 1 converges strongly to the unique solution u* E ~ o f problem (3.1). Moreover, we have for all n > 0

9-~0

where k = ( 1 / ( " / ÷ MI))V7 -2 - 2Ar + A2s 2. In partJcular, i f we take (~n : [ ( [ + n ) / ( [ + n + 1) 2, Vn > 0, where [ = 2/(1 - k), then we have the foIlowing convergence rate estimate:

PROOF. First note that H is strongly monotone and Lipschitz continuous Since M is strongly H-monotone, it follows from Corollary 2 2 that M is maximal strongly monotone. Thus, from Remark 2.2 we know that the resolvent operator R H M,.~ = ( H + h M ) -1 is well defined. Next we divide the proof into three steps. STEP 1. We claim that problem (3.1) has a unique solution u* ~ 7-{. Indeed, we define the mapping F : 7{ --~ 7{ as follows' F ( u ) = R ~ , ~ [ H ( u ) - hA(u)],

Vu E 7-t.

Then it follows from Theorem 2.2 that for each u, v E H, IIF(~) - F(~)II = I [ R ~ , ~ ( H ( ~ ) - h A ( u ) ) - RH,:~(H(v) - hA(v))][ / \ 1 <

(3 6)

By assumptions,

IhH(u)

-

H(v) - A(A(u)

-

A(v))l]

2 :

liH(u)

-

H(v)ll

+ h211A(u)

-

2 -

2A(A(u) - A(v), H(u) - H(v))

A(v)ll 2

(3.7)

<_ (¢2 _ 2h~ + h2s ~) I1~ - ~11:. Combining (3.6) with (3 7) yields ]IF(u) - F(v)ll ~ ~11~ - vii,

(3.8)

where k = (1/(7 + h~))~/~ -2 - 2Ar + A2s 2. From (,) and (3.8), we know that 0 _< k < 1, and thus, F is a contraction. By the Banach Contraction Principle, we conclude that F has a unique fixed point u* E ~ . This implies that u* = F(u*) = R ~ , x [ H ( u * ) - hA(u*)],

Characterization of H-Monotone Operators

335

and hence, u* is a unique solution of the equation u = R~4,x[H(u ) -hA(u)]. L e m m a 3.1 it follows that u* is the unique solution of problem (3.1).

Therefore, from

STEP 2. We claim t h a t the sequence {u=} converges strongly to the unique solution u* of problem (3.1). Indeed from (3.4), L e m m a 3.1 and Theorem 2.2, we obtain I1~+~

- ~*11 = I1(1 - ~n)(~

- ¢)

+ ~

(R~,~,[H(v~)

- hA(v,,)]

_< (1 - ~,~)ll~,, - ,,*11 + ~,, = (1-

- hA(

- u*)II

o)] -

'*11

o~n)]l~,,-~*11

II

[H(vn) -

-- p~H M,A[rH/l'Lt

II

(3 9)

_< ( 1 - ~,,)11,,,,-~*1]

+ ~n ((7 -:1 by))IlH(vn)- H(u*)_< (1 - ~ ) l l ~ n

h(A(vn)-

A(u*))ll

- .kA(u~)]

- u*)II

- - u * l l ÷ ~nkllv,~ -- ~*11,

where k = (1/(7 + hq))x/~ -2 - 2hr + h2s 2. Furthermore, observe t h a t

II(1 - 9,~)(~,~ - ~*) + zn (R~,x[H(un)

<_ (1 --

- ~,dll~n - u*ll + ~,~ I l R ~ , x [ H ( u n )

-

AA(u,~)] - u* II

M,),[H(u* )-hA(~*)]II

RH

(7-~h~) l]g(un)-g(u*)

_< ( 1 - ~ , , ) l l u n - u * [ l + ~ n

(3.1o)

-,~(A(un) - A(u*))[[ = (1 - (1 - k)~,,)ll~,~

-~*11

-< I1~,,- u*ll. Hence, it follows from (3.9) and (3.10) that

I1~,~+~ - ~ * 1 t ~ (1 - ~n)ll~n --~*11 ÷ ~ , ~ k l l ~ --~*11 = (1 - (1 - k)~,dll~n -~*11 _< (1 - (1 - k),%,)...

(1 - (1 - k)O~o)ll~o -~*11

(3.11)

n

= E(1

I1~o - ~*11,

- (1 - k ) o g )

3=0

SinceO_
and ~ n = o c~n = 0% we have n

E(1 n=0

- (1 - k)C~n) = limoo E ( 1 - (1 - k)cb) = 0, 3=0

which, hence, implies that {un} converges strongly to u*. STEP 3. We claim t h a t there holds the convergence rate estimate ]lum - u*lt = O(1/m) for o~n = [([+ n)/([+ n + 1) 2, Vn _> 0, where l = 2/(1 - k). Indeed, by making use of (3 11), we

336

L -C ZENG et al.

derive for all n > 0

- *11

(1

-

(1

-

k)

)llun

-

u*ll

([-~n71)2/ II~n-

~*li

and hence, [ + n + 1) 2 Summing this inequality from n = 0 to m - 1 (m _> 1), we obtain

Thus,

from which it follows that

This completes the proof.

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