Convergence Theorems of Approximate Proximal Point Algorithm for Zeroes of Maximal Monotone Operators in Hilbert Spaces 1

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1 Int. Journal of Math. Analysis, Vol. 1, 2007, no. 4, Convergence Theorems of Approximate Proximal Point Algorithm for Zeroes of Maximal Monotone Operators in Hilbert Spaces 1 Haiyun Zhou Institute of Nonlinear Anal. North China Electric Power University Baoding , P.R. China Li Wei School of Math. and Stat. Hebei University of Economics and Business Shijiazhuang , P.R. China Bin Tan Institute of Appl. Math. Ordnance Engineering College Shijiazhuang , P.R. China Abstract In this paper, we introduce two kinds of iterative algorithms for finding zeroes of maximal monotone operators, and establish strong and weak convergence theorems of the modified proximal point algorithms. By virtue of the established theorems, we consider the problem of finding a minimizer of a convex function. Mathematics Subject Classification: Primary 47H17; Secondary 47H05, 47H10 Keywords: Maximal monotone operator; modified approximate proximal point algorithm; strong convergence; weak convergence 1 Supported by the National Science Foundation of China(Grant No ).

2 176 Haiyun Zhou, Li Wei and Bin Tan 1 Introduction Let H be a real Hilbert space with the inner product, and norm.a set T H H is called a monotone operator on H if T has the following property: (x, y), (x,y ) T x x,y y 0. (1) A monotone operator T on H is said to be maximal monotone if it is not properly contained in any other monotone operator on H. Equivalently, a monotone operator T is maximal monotone if R(I +tt ) = H for all t > 0. For a maximal monotone operator T, we can defined the resolvent of T by J t =(I + tt ) 1, t > 0. (2) It is well known that J t : H D(T ) is nonexpansive. Also we can define the Yosida approximation T t by T t = 1 t (I J t), t > 0. (3) We know that T t x TJ t x for all x H and T t x Tx for all x D(T ), where Tx = inf{ y : y Tx}, and T 1 0=F (J t ) for all t>0. Throughout this paper, we assume that Ω is a nonempty closed convex subset of a real Hilbert space H and T :Ω 2 H be a maximal monotone operator with T 1 0. It is well known that, for any u H, there exists uniquely y 0 Ω such that u y 0 = inf{ u y : y Ω}. (4) When a mapping P Ω : H Ω is defined by P Ω u = y 0, we call P Ω the metric projection of H onto Ω. The metric projection P Ω of H onto Ω has the following basic properties: (i) P Ω x x, x P Ω x 0 for all x Ω and x H; (ii) P Ω x P Ω y 2 x y, P Ω x P Ω y for all x, y H; (iii) P Ω x P Ω y x y for all x, y H; and (iv)x n x 0 weakly and Px n y 0 strongly imply that Px 0 = y 0. To find zeroes of maximal monotone operators is the central and important topics in nonlinear functional analysis. A classical method to solve the following set-valued equation 0 Tz, (5) where T :Ω 2 H is a maximal monotone operator, is the proximal point algorithm, which, starting with any point x 0 H, updates x n+1 iteratively conforming to the following recursion x n x n+1 + β n Tx n+1, n 0, (6)

3 Convergence theorems 177 where {β n } [β, ), β>0, is a sequence of real numbers. However, as pointed out in [3], the ideal form of the method is often impractical since, in many cases, to solve the problem (6) exactly is either impossible or the same difficult as the original problem (5). Therefore, one of the most interesting and important problems in the theory of maximal monotone operators is to find an efficient iterative algorithm to compute approximately zeroes of T. In Rockafellar [8] gave an inexact variant of the method x n + e n+1 x n+1 + β n Tx n+1, n 0, (7) where {e n } is regarded as an error sequence. This method is called an inexact proximal point algorithm. It was shown that, if n=0 e n < +, then {x n } defined by (7) converges weakly to a zero of T provided that T 1 0. Furthermore, Rockafellar[8] posed an open question of whether the sequence generated by (7)converges strongly or not. This question was solved by Güller [4], who introduced an example for which the sequence generated by (7) converges weakly but not strongly. Question Is it possible to construct an iterative scheme based on approximating proximal point algorithm which converges strongly to a fixed point of a maximal monotone operator T in Hilbert spaces? It is our purpose in this paper to give an affirmative answer to the above question by introducing a new iterative scheme in Hilbert spaces. For this purpose, we need the following lemmas. 2 Preliminaries The first lemma is well known which can be found in some standard functional analysis context books. Lemma 2.1 For all x, y H and λ [0, 1], λx +(1 λ)y 2 = λ x 2 +(1 λ) y 2 λ(1 λ) x y 2. Lemma 2.2 (see Bruck [1, Lemma 1]) For all u H, lim t J t u exists and is the point of T 1 0 nearest to u. Lemma 2.3 (see Eckstein [3, Lemma 2]) For any given x n H and β n > 0, there exists x n Ω and e n H conforming to the following set-valued mapping equation (SVME): x n + e n x n + β n T x n. (8)

4 178 Haiyun Zhou, Li Wei and Bin Tan Furthermore, for any p T 1 0, we have x n p, x n x n + e n x n x n,x n x n + e n (9) and x n e n p 2 x n p 2 x n x n 2 + e n 2. (10) Lemma 2.4 (see Liu [7, Lemma 1.1] Let {a n }, {b n } and {c n } be three real sequences satisfying a n+1 (1 t n )a n + b n + c n, n 0, (11) where {t n } [0, 1], n=0 t n =, b n = (t n ) and n=0 c n <. Then a n 0 as n. 3 Main Results Now we give our main results in this paper. Theorem 3.1 Let H be a real Hilbert space, Ω be a nonempty closed convex subset of H and T :Ω 2 H be a maximal monotone operator with T 1 0. Let P Ω be the metric projection of H onto Ω. For any given x n H and β n > 0, find x n Ω and e n H conforming to the (SVME)(8), where {β n } (0, + ) with β n + as n, and e n η n x n x n with sup n 0 η n = η<1. Let {α n } be a real sequence in [0, 1] satisfying the following control conditions: (i) α n 0(n ); and (ii) n=0 α n =. For any fixed anchor u Ω, for arbitrary initial value x 0 H, let the sequence {x n } be generated by the following manner: x n+1 = α n u +(1 α n )P Ω ( x n e n ), n 0, (12) then {x n } converges strongly to a fixed point z of T, where z = lim t J t u,if and only if e n 0asn. Proof. ( Necessity )Assume that x n z as n, where z T 1 0; then it follows from (8) that x n z x n z + e n x n z + η n x n x n (1 + η n ) x n z + η n x n z, (13)

5 Convergence theorems 179 which implies that x n z 1+η n 1 η n x n z, and hence x n z as n, consequently, e n 0asn. ( Sufficiency )The proof is finished by three steps. Claim 1. {x n } is bounded. Fix p T 1 0 and set M = max{ u p 2, x 0 p 2 }. First, we prove that, for all n 0, x n p 2 M. (14) When n =0, (14) is true. Now, assume that (14) holds for some n 0. We shall prove that (14) holds for n + 1. By using (12), Lemmas 2.1 and 2.3, we have x n+1 p 2 = α n u p 2 +(1 α n ) P Ω ( x n e n ) p 2 By induction, we assert that α n (1 α n ) u P Ω ( x n e n ) 2 α n M +(1 α n ) x n e n p 2 α n M +(1 α n ) x n p 2 α n M +(1 α n )M = M. (15) x n p 2 M for all n 0. This implies that {x n } is bounded and so is {J βn x n }. Claim 2. lim sup n u z, x n+1 z 0, where z = lim t J t u, which is guaranteed by Lemma 2.2. Since T is monotone and β n + (n ), we have, for any t>0, u J t u, J βn x n J t u = t TJ t u, J t u J βn x n = t TJ t u TJ βn x n,j t u J βn x n t TJ βn x n,j t u J βn x n t x n J βn x n,j t u J βn x n β n 0 (n ) (16) and hence lim sup u J t u, J βn x n J t u 0. (17) n

6 180 Haiyun Zhou, Li Wei and Bin Tan Note that J βn (x n + e n ) J βn x n e n 0asn and so it follows from (17) that lim sup u J t u, J βn (x n + e n ) J t u 0. (18) n Note that P Ω ( x n e n ) J βn (x n +e n ) e n 0asn and so it follows from (18) that lim sup u J t u, P Ω ( x n e n ) J t u 0. (19) n Since α n 0asn, from (12), we have It follows from (19) and (20) that x n+1 P Ω ( x n e n ) 0 (n ). (20) lim sup u J t u, x n+1 J t u 0 t>0, (21) n and so, from z = lim t J t u and (21), lim sup u z, x n+1 z 0. (22) n Claim 3. x n z as n. Observe that and so which implies that (1 α n )(P Ω ( x n e n ) z) =(x n+1 z) α n (u z) (1 α n ) 2 P Ω ( x n e n ) P Ω z 2 x n+1 z 2 2α n u z, x n+1 z, (23) x n+1 z 2 (1 α n ) x n e n z 2 +2α n u z, x n+1 z. It follows from Lemma 2.3 and (22) that x n+1 z 2 (1 α n ) x n z 2 (1 α n ) x n x n 2 +(1 α n ) e n 2 +2α n u z, x n+1 z (1 α n ) x n z 2 +2α n u z, x n+1 z. (24) Set σ n = max{ u z, x n+1 z, 0}. Then σ n 0asn. Set a n = x n z 2 and b n =2α n σ n. Then (24) reduces to a n+1 (1 α n )a n + b n, n 0,

7 Convergence theorems 181 where n=0 α n =, and b n = (α n ). Thus it follows from Lemma 2.4 that a n 0asn 0, i.e., x n z T 1 0asn. This completes the proof. Remark 1. Iterative algorithm (10) is new, which produces a strongly convergent sequence to a zero of maximal monotone operator T ; while Rockafellar s algorithm (6) only yields a weakly convergent sequence, and requires that the error sequence {e n } satisfy n=0 e n < (see, [8]). Remark 2. The maximal monotonicity of T is only used to guarantee the existence of solutions of (SVME)(8) for any given x n H and β n > 0. If we assume that T :Ω 2 H is monotone (need not be maximal) and T satisfies the range condition: D(T )=Ω r>0 R(I + rt), then, for any given x n Ω and β n > 0, we may find x n Ω and e n H satisfying (SVME)(8). Furthermore, Lemma 2.2 also holds for u Ω and hence Theorem 3.1 still holds for monotone operators which satisfy the range condition. Following the proof lines of Theorem 3.1, we can prove the following: Corollary 3.1 Let H be a real Hilbert space, Ω be a nonempty closed convex subset of H and S :Ω Ω be a continuous and pseudo-contractive mapping with a fixed point in Ω. For any given x n Ω and β n > 0, there exist x n Ω and e n H such that x n + e n =(1+β n ) x n β n S x n, (25) where β n (n ) and {e n } satisfies the conditions that e n 0 as n, and e n η n x n x n with sup n 0 η n = η<1. Let{α n } (0, 1] be a real sequence such that α n 0(n ) and n=0 α n =. For any fixed u Ω, define the sequence {x n } iteratively as follows: x n+1 = α n u +(1 α n )P Ω ( x n e n ), n 0, (26) then {x n } converges strongly to a fixed point z of S, where z = lim t J t u, and J t =(I + t(i S)) 1 for all t>0. Proof. Let T = I S. Then T :Ω H is continuous, monotone and satisfies the range condition: D(T )=Ω r>0 R(I + rt). (27) Now, we only need to verify the last assertion. operator G :Ω Ωby Gx = For any y Ω, define an t 1+t Sx + 1 y. (28) 1+t

8 182 Haiyun Zhou, Li Wei and Bin Tan Then G : Ω Ω is continuous and strongly pseudo-contractive. By Corollary 1 of Deimling[2], G has a unique fixed point x in Ω, i.e., x = t Sx + 1 y, 1+t 1+t which implies that y R(I + tt ) for all t>0. In particular, for any given x n Ω and β n > 0, there exist x n Ω and e n H such that which means that x n + e n = x n + β n T x n, n 0, (29) x n + e n =(1+β n ) x n β n S x n, n 0, (30) and the relation (25) follows. The reminder of proof is same as in the corresponding part of Theorem 3.1. This completes the proof. Remark 3. In Corollary 1.1, we do not know whether the continuity assumption on S can be dropped or not. Remark 4. In Theorem 3.1, if the operator T is defined on the whole space H, then the metric projection mapping P Ω is not need. Remark 5. Our convergence results are different from those obtained by Kamimura et al. [6]. Theorem 3.2 Let H be a real Hilbert space, Ω be a nonempty closed convex subset of H and T :Ω 2 H be a maximal monotone operator with T 1 0. For any given x n H and β n > 0, let x n Ω be the solution of the (SVME)(8), that is, x n + e n x n + β n T x n, n 0, where lim inf n β n > 0 and e n η n x n x n with sup n 0 η n = η<1. Let {α n } be a sequence in [0, 1] with lim sup n α n < 1 and define the sequence {x n } iteratively as follows: { x0 Ω, (31) x n+1 = α n x n +(1 α n )P Ω ( x n e n ), n 0, then {x n } converges weakly to a point p T 1 0. Proof. Claim 1. {x n } is bounded. Since T 1 0, we can take some w T 1 0. By using (31), Lemmas 2.1 and 2.3, we obtain x n+1 w 2 = α n x n w 2 +(1 α n ) P Ω ( x n e n ) w 2 α n (1 α n ) x n P Ω ( x n e n ) 2 α n x n w 2 +(1 α n ) x n e n w 2 α n x n w 2 +(1 α n ) x n w 2 (1 α n )(1 η 2 ) x n x n 2 = x n w 2 (1 α n )(1 η 2 ) x n x n 2 x n w 2 (32)

9 Convergence theorems 183 and so (32) implies that lim n x n w 2 exists. Therefore, {x n } is bounded. Claim 2. x n J βn x n 0asn. It follows from (32) that (1 α n )(1 η 2 ) x n x n 2 x n w x n+1 w 2 and so the last inequality together with lim sup n α n < 1 implies that x n x n 0 (n ). Hence x n J βn x n 0asn. Claim 3. {x n } converges weakly to a point p T 1 0asn. Set y n = J βn x n and let p H be a weakly subsequential limit of {x n } such that {x nj } converges weakly to a point p as j. Thus it follows that {y nj } converges weakly to p as j. Observe that y n J 1 y n = (I J 1 )y n = T 1 y n inf{ z : z Ty n } T βn x n = n y n. (33) β n By assumption lim inf n β n > 0, we have y n J 1 y n 0 (n ). Since J 1 is nonexpansive, by Browder s demi-closedness principle, we assert that p F (J 1 )=T 1 0. Now, Opial s condition of H guarantees that {x n } converges weakly to p T 1 (0) as n. This completes the proof. Following the proof lines of Theorem 3.2 and Corollary 3.1, we have the following: Corollary 3.2 Let H be a real Hilbert space, Ω be a nonempty closed convex subset of H and U :Ω Ω be a continuous and pseudo-contractive mapping with a fixed point. Set T = I U. For any given x n Ω and β n > 0, find x n Ω and e n H such that x n + e n =(1+β n ) x n β n U x n, n 0. Define the sequence {x n } iteratively as follows: { x0 Ω, x n+1 = α n x n +(1 α n )P Ω ( x n e n ), n 0, where {α n } [0, 1] with lim sup n α n < 1, {β n } (0, + ) with lim inf n β n > 0 and {e n } H with e n η n x n x n and sup n 0 η n = η<1. Then {x n } converges weakly to a fixed point p of U.

10 184 Haiyun Zhou, Li Wei and Bin Tan 3. Applications We can apply Theorems 3.1 and 3.2 to find a minimizer of a convex function f. Let H be a real Hilbert space and f : H (, ] be a proper convex lower semi-continuous function. Then the subdifferential f of f is defined as follows: for all z H. f(z) ={v H : f(y) f(z)+ y z, v, y H} Theorem 3.3 Let H be a real Hilbert space and f : H (, ] aproper convex lower semi-continuous function. Let {β n } be a sequence in (0, ) with β n (n ) and {e n } a sequence in H with e n 0 as n, and e n η n x n x n with sup n 0 η n = η<1. Let x n be the solution of the (SVME)(8) with T replacing by f, that is, for any given x n H, x n + e n x n + β n f( x n ), n 0. Let {α n } [0, 1] such that α n 0(n ) and n=0 α n =. For any fixed u H, let {x n } be the sequence generated by u, x 0 H, { x n = arg min z H f(z)+ 1 2β n z x n e n 2}, x n+1 = α n u +(1 α n )( x n e n ), n 0. If f 1 0, then {x n } converges strongly to the minimizer of f nearest to u. Proof. Sice f : H (, ) is a proper convex lower semi-continuous function, by Rockafellar [8,Theorem 1], the subdifferential f of f is a maximal monotone operator. Noting that { 1 x n = arg min f(z)+ z x n e n 2} z H 2β n is equivalent to the following: we have 0 f( x n )+ 1 β n ( x n x n e n ), x n + e n x n + β n f( x n ), n 0. Therefore, using Theorem 3.1, we have the desired conclusion. This completes the proof.

11 Convergence theorems 185 Theorem 3.4 Let H be a real Hilbert space and f : H (, ] be a proper convex lower semi-continuous function. Let {α n } be a sequence in [0, 1] with lim sup n α n < 1, {β n } be a sequence in (0, ) with liminf n β n > 0 and {e n } be a sequence in H with e n η n x n x n and sup n 0 η n = η<1. Let x n be the solution of the (SVME)(8) with T replacing by f, that is, for any given x n H and β n > 0, there exist x n H and e n H such that x n + e n x n + β n f( x n ), n 0. Let {x n } be the sequence generated by u, x 0 H, { x n = arg min z H f(z)+ 1 2β n z x n e n 2}, x n=1 = α n x n +(1 α n )( x n e n ), n 0. If f 1 0, then {x n } converges weakly to the minimizer of f nearest to u. Proof. As shown in the proof lines of Theorem 3.3, f : H H is a maximal monotone operator and so the conclusion of Theorem 3.4 follows from Theorem 3.2. References [1] R. E. Bruck, A strongly convergent iterative method for the solution of 0 Ux for a maximal monotone operator U in Hilbert space, J. Math. Anal. Appl. 48 (1974), [2] K. Deimling, Zeros of accretive operators, Manuscripta Math. 13(1974), [3] J. Eckstein, Approximate iterations in Bregman-function-based proximal point algorithms, Math. Programming, 83 (1998), [4] O. Güller, On the convergence of the proximal point algorithm for convex minimization, SIAM,J.Control and Optim. 29 (1991), [5] B. S. He, L. Z. Liao and Z. H. Yang, A new approximate proximal point algorithm for maximal monotone operators, Since in China (Series A), 32 (2002), [6] S. Kamimura, S. H. Khan and W. Takahashi, Iterative schemes for approximating solutions of relations involving accretive operators in Banach spaces, Fixed Point Theory and Applications, Editors: Y. J. Cho, J. K. Kim and S. M. Kang, Nova Science Publishers, New York, 2003.

12 186 Haiyun Zhou, Li Wei and Bin Tan [7] L. S. Liu, Ishikawa and Mann iterative process with errors for nonlinear strongly accretive mappings in Banach spaces, J. Math. Anal. Appl. 194(1995), [8] R. T. Rockafellar, Monotone operators and the proximal point algorithm, SIAM J. Control Optim.14 (1976), [9] R. M. Wittmann, Approximation of fixed points of nonexpansine mappings, Arch. Math. 58 (1992), Received: September 2, 2006