Open Access

Strong convergence theorems for the general split variational inclusion problem in Hilbert spaces

Fixed Point Theory and Applications20142014:171

https://doi.org/10.1186/1687-1812-2014-171

Received: 1 May 2014

Accepted: 19 July 2014

Published: 18 August 2014

Abstract

The purpose of this paper is to introduce and study a general split variational inclusion problem in the setting of infinite-dimensional Hilbert spaces. Under suitable conditions, we prove that the sequence generated by the proposed new algorithm converges strongly to a solution of the general split variational inclusion problem. As a particular case, we consider the algorithms for a split feasibility problem and a split optimization problem and give some strong convergence theorems for these problems in Hilbert spaces.

Keywords

general split variational inclusion problemsplit feasibility problemsplit optimization problemquasi-nonexpansive mappingzero pointresolvent mapping

1 Introduction

Let C and Q be nonempty closed convex subsets of real Hilbert spaces H 1 and H 2 , respectively. The split feasibility problem ( S F P ) is formulated as
to find  x C  and  A x Q ,
(1.1)

where A : H 1 H 2 is a bounded linear operator. In 1994, Censor and Elfving [1] first introduced the S F P in finite-dimensional Hilbert spaces for modeling inverse problems which arise from phase retrievals and in medical image reconstruction [2]. It has been found that the S F P can also be used in various disciplines such as image restoration, computer tomography and radiation therapy treatment planning [35]. The S F P in an infinite-dimensional real Hilbert space can be found in [2, 4, 610]. For comprehensive literature, bibliography and a survey on SFP, we refer to [11].

Assuming that the S F P is consistent, it is not hard to see that x C solves S F P if and only if it solves the fixed point equation
x = P C ( I γ A ( I P Q ) A ) x ,

where P C and P Q are the metric projection from H 1 onto C and from H 2 onto Q, respectively, γ > 0 is a positive constant, and A is the adjoint of A.

A popular algorithm to be used to solves the S F P (1.1) is due to Byrne’s CQ-algorithm [2]:
x k + 1 = P C ( I γ k A ( I P Q ) A ) x k , k 1 ,

where γ k ( 0 , 2 / λ ) with λ being the spectral radius of the operator A A .

On the other hand, let H be a real Hilbert space, and B be a set-valued mapping with domain D ( B ) : = { x H : B ( x ) } . Recall that B is called monotone, if u v , x , x y 0 for any u B x and v B y ; B is maximal monotone, if its graph { ( x , y ) : x D ( B ) , y B x } is not properly contained in the graph of any other monotone mapping. An important problem for set-valued monotone mappings is to find x H such that 0 B ( x ) . Here, x is called a zero point of B. A well-known method for approximating a zero point of a maximal monotone mapping defined in a real Hilbert space H is the proximal point algorithm first introduced by Martinet [12] and generated by Rockafellar [13]. This is an iterative procedure, which generates { x n } by x 1 = x H and
x n + 1 = J β n B x n , n 1 ,
(1.2)
where { β n } ( 0 , ) , B is a maximal monotone mapping in a real Hilbert space, and J r B is the resolvent mapping of B defined by J r B = ( I + r B ) 1 for each r > 0 . Rockafellar [13] proved that if the solution set B 1 ( 0 ) is nonempty and lim inf n β n > 0 , then the sequence { x n } in (1.2) converges weakly to an element of B 1 ( 0 ) . In particular, if B is the sub-differential ∂f of a proper convex and lower semicontinuous function f : H R , then (1.2) is reduced to
x n + 1 = argmin y H { f ( y ) + 1 2 β n y x n 2 } , n 1 .
(1.3)

In this case, { x n } converges weakly to a minimizer of f. Later, many researchers have studied the convergence problems of the proximal point algorithm in Hilbert spaces (see [1421] and the references therein).

Motivated by the works in [1417] and related literature, the purpose of this paper is to introduce and consider the following general split variational inclusion problem.

Let H 1 and H 2 be two real Hilbert spaces, B i : H 1 H 1 and K i : H 2 H 2 , i = 1 , 2 , be two families of set-valued maximal monotone mappings, A : H 1 H 2 be a linear and bounded operator, and A be the adjoint of A. The so-called general split variational inclusion problem is
to find  x H 1  such that  0 i = 1 B i ( x )  and  0 i = 1 K i ( A x ) .
(1.4)

The following examples are special cases of (GSVIP) (1.4).

Classical split variational inclusion problem. Let B : H 1 H 1 and K : H 2 H 2 be set-valued maximal monotone mappings. The so-called classical split variational inclusion problem (CSVIP) is
to find  x H 1  such that  0 B ( x )  and  0 K ( A x ) ,
(1.5)
which was introduced by Moudafi [17]. It is obvious that problem (1.5) is a special case of (GSVIP) (1.4). In [17], Moudafi proved that the iteration process
x n + 1 = J λ B ( x n + γ A ( J λ K I ) A x n )

converges weakly to a solution of problem (1.5), where λ and γ are given positive numbers.

Split optimization problem. Let f : H 1 R , g : H 2 R be two proper convex and lower semicontinuous functions. The so-called split optimization problem (SOP) is
to find  x H 1  such that  f ( x ) = min y H 1 f ( y )  and  g ( A x ) = min z H 2 g ( z ) .
(1.6)
Denote by B = ( f ) and K = ( g ) , then B and K both are maximal monotone mappings, and problem (1.6) is equivalent to the following classical split variational inclusion problem, i.e.:
to find  x H 1  such that  0 ( f ( x ) )  and  0 ( g ( A x ) ) .
(1.7)
Split feasibility problem. As in (1.1), let C and Q be two nonempty closed convex subsets of real Hilbert spaces H 1 and H 2 , respectively and A be the same as above. The split feasibility problem is
to find  x C  such  A x Q .
(1.8)

It is well known that this kind of problems was first introduced by Censor and Elfving [1] for modeling inverse problems arising from phase retrievals and in medical image reconstruction [2]. Also it can be used in various disciplines such as image restoration, computer tomography and radiation therapy treatment planning.

Let i C ( i Q ) be the indicator function of C (Q), i.e.,
i C ( x ) = { 0 , if  x C , + , if  x C ; i Q ( x ) = { 0 , if  x Q , + , if  x Q .
(1.9)
Then i C and i Q both are proper convex and lower semicontinuous functions, and its subdifferentials i C and i Q are maximal monotone operators. Consequently problem (1.8) is equivalent to the following ‘split optimization problem’ and ‘Moudafi’s classical split variational inclusion problem’, i.e.,
to find  x H 1  such that  i C ( x ) = min y H 1 i C ( y )  and  i Q ( A x ) = min z H 2 i Q ( z ) to find  x H 1  such that  0 ( i C ( x ) )  and  0 ( i Q ( A x ) ) .
(1.10)
For solving (GSVIP) (1.4), in our paper we propose the following iterative algorithms:
x n + 1 = α n x n + ξ n f ( x n ) + i = 1 γ n , i J β i B i [ x n λ n , i A ( I J β i K i ) A x n ] , n 0 ,
(1.11)

where f : H 1 H 1 is a contraction mapping with a contractive constant k ( 0 , 1 ) , { α n } , { ξ n } and { γ n , i } are sequence in [ 0 , 1 ] satisfying some conditions. Under suitable conditions, some strong convergence theorems for the sequence proposed by (1.11) to a solution for (GSVIP) (1.4) in Hilbert spaces are proved. As a particular case, we consider the algorithms for a split feasibility problem and a split optimization problem and give some strong convergence theorems for these problems in Hilbert spaces. Our results extend and improve the related results of Censor and Elfving [1], Byrne [2], Censor et al. [35], Rockafellar [13], Moudafi [14, 17], Eslamian and Latif [15], Eslamian [21], and Chuang [22].

2 Preliminaries

Throughout the paper, we denote by H a real Hilbert space, C be a nonempty closed and convex subset of H. F ( T ) denote by the set of fixed points of a mapping T. Let { x n } be a sequence in H and x H . Strong convergence of { x n } to x is denoted by x n x , and weak convergence of { x n } to x is denoted by x n x . For every point x H , there exists a unique nearest point in C, denoted by P C x . This point satisfies.
x P C x x y , y C .
The operator P C is called the metric projection. The metric projection P C is characterized by the fact that P C x C and
x P C x , P C x y 0 , x H , y C .
Recall that a mapping T : C H is said to be nonexpansive, if T x T y x y for every x , y C . T is said to be quasi-nonexpansive, if F ( T ) and T x p x p for every x C and p F ( T ) . It is easy to see that F ( T ) is a closed convex subset of C if T is a quasi-nonexpansive mapping. Besides, T is said to be a firmly nonexpansive, if
T x T y 2 x y , T x T y x , y C ; T x T y 2 x y 2 ( I T ) x ( I T ) y 2 x , y C .

Lemma 2.1 (demi-closed principle)

Let C be a nonempty closed convex subset of a real Hilbert space H. Let T : C H be a nonexpansive mapping, and let { x n } be a sequence in C. If x n w and lim n x n T x n = 0 , then T w = w .

Lemma 2.2 [23]

Let H be a (real) Hilbert space. Then for all x , y H ,
x + y 2 x 2 + 2 y , x + y .
(2.1)

Lemma 2.3 [24]

Let H be a Hilbert space and let { x n } be a sequence in H. Then, for any given sequence { λ n } ( 0 , 1 ) with n = 1 λ n = 1 and for any positive integers i, j with i < j ,
n = 1 λ n x n 2 n = 1 λ n x n 2 λ i λ j x i x j 2 .
(2.2)
Lemma 2.4 Let { a n } be a sequence of nonnegative real numbers, { b n } be a sequence of real numbers in ( 0 , 1 ) with n = 1 b n = , { u n } be a sequence of nonnegative real numbers with n = 1 u n < , { t n } be a real numbers with lim sup n t n 0 . If
a n + 1 ( 1 b n ) a n + b n t n + u n , for each  n 1 ,

then lim n a n = 0 .

Lemma 2.5 [25]

Let { a n } be a sequence of real numbers such that there exists a subsequence { n i } of { n } such that a n i < a n i + 1 for all i N . Then there exists a nondecreasing sequence { m k } N such that m k , a m k a m k + 1 and a k a m k + 1 are satisfied by all (sufficiently large) numbers k N . In fact, m k = max { j k : a j < a j + 1 } .

Lemma 2.6 [22]

Let H be a real Hilbert space, B : H 2 H be a set-valued maximal monotone mapping, β > 0 , and let J β B be the resolvent mapping of B.
  1. (i)

    For each β > 0 , J β B is a single-valued and firmly nonexpansive mapping;

     
  2. (ii)

    D ( J β B ) = H and F ( J β B ) = B 1 ( 0 ) : = { x D ( B ) : 0 B x } ;

     
  3. (iii)

    ( I J β B ) is a firmly nonexpansive mapping for each β > 0 ;

     
  4. (iv)
    suppose that B 1 ( 0 ) , then for each x H , each x B 1 ( 0 ) and each β > 0
    x J β B x 2 + J β B x x x x 2 ;
     
  5. (v)

    suppose that B 1 ( 0 ) . Then x J β B x , J β B x w 0 for each x H and each w B 1 ( 0 ) , and each β > 0 .

     
Lemma 2.7 Let H 1 , H 2 be two real Hilbert spaces, A : H 1 H 2 be a linear bounded operator and A be the adjoint of A. Let B : H 2 2 2 H be a set-valued maximal monotone mapping, β > 0 , and let J β B be the resolvent mapping of B, then
  1. (i)

    ( I J β B ) A x ( I J β B ) A y 2 ( I J β B ) A x ( I J β B ) A y , A x A y ;

     
  2. (ii)

    A ( I J β B ) A x A ( I J β B ) A y 2 A 2 ( I J β B ) A x ( I J β B ) A y , A x A y ;

     
  3. (iii)

    if ρ ( 0 , 2 A 2 ) , then ( I ρ A ( I J β B ) A ) is a nonexpansive mapping.

     

Proof By Lemma 2.6(iii), the mapping ( I J β B ) is firmly nonexpansive, hence the conclusions (i) and (ii) are obvious.

Now we prove the conclusion (iii).

In fact, for any x , y H 1 , it follows from the conclusions (i) and (ii) that
( I ρ A ( I J β B ) A ) x ( I ρ A ( I J β B ) A ) y 2 = x y 2 2 ρ x y , A ( I J β B ) A x A ( I J β B ) A y + ρ 2 A ( I J β B ) A x A ( I J β B ) A y 2 x y 2 2 ρ A x A y , ( I J β B ) A x ( I J β B ) A y + ρ 2 A 2 ( I J β B ) A x ( I J β B ) A y 2 x y 2 ρ ( 2 ρ A 2 ) ( I J β B ) A x ( I J β B ) A y 2 x y 2 ( since  ρ ( 2 ρ A 2 ) 0 ) .

This completes the proof of Lemma 2.7. □

3 Main results

The following lemma will be used in proving our main results.

Lemma 3.1 Let H 1 and H 2 be two real Hilbert spaces, A : H 1 H 2 be a linear and bounded operator, and A be the adjoint of A. Let B i : H 1 2 H 1 , and K i : H 2 2 H 2 , i = 1 , 2 ,  , be two families of set-valued maximal monotone mappings, and let β > 0 and γ > 0 . If Ω (the solution set of (GSVIP) (1.4)), then x H 1 is a solution of (GSVIP) (1.4) if and only if for each i 1 , for each γ > 0 and for each β > 0
x = J β B i ( x γ A ( I J β K i ) A x ) .
(3.1)
Proof Indeed, if x is a solution of (GSVIP) (1.4), then for each i 1 , γ > 0 and β > 0 ,
x B i 1 ( 0 ) and A x K i 1 ( 0 ) , i.e. , x = J β B i x  and  A x = J β K i A x .

This implies that x = J β B i ( x γ A x ( I J β K i ) A x ) .

Conversely, if x solves (3.1), by Lemma 2.6(v), we have
x ( x γ A ( I J β K i ) A x ) , y x 0 , y B i 1 ( 0 ) .
Hence we have
( I J β K i ) A x , A y A x 0 , y B i 1 ( 0 ) .
(3.2)
On the other hand, by Lemma 2.6(v) again
( A x J β K i A x , J β K i A x v 0 , v K i 1 ( 0 ) .
(3.3)
Adding up (3.2) and (3.3), we have
A x J β K i A x , J β K i A x + A y A x v 0 , y B i 1 ( 0 ) ,  and  v K i 1 ( 0 ) .
Simplifying it, we have
A x J β K i A x 2 A x J β K i A x , A y v 0 , y B i 1 ( 0 ) ,  and  v K i 1 ( 0 ) .
(3.4)
By the assumption that Ω . Taking w Ω , hence for each i 1 w B i 1 ( 0 ) and A w K i 1 ( 0 ) . In (3.4), taking y = w and v = A w , then we have
A x J β K i A x 2 = 0 .

This implies that A x = J β K i A x , and so A x K i 1 ( 0 ) for each i 1 . Hence from (3.1), x = J β B i x , i.e., x B i 1 ( 0 ) . Hence x is a solution of (GSVIP)(1.4).

This completes the proof of Lemma 3.1. □

We are now in a position to prove the following main result.

Theorem 3.2 Let H 1 , H 2 , A, A , { B i } , { K i } , Ω be the same as in Lemma  3.1. Let f : H 1 H 1 be a contractive mapping with contractive constant k ( 0 , 1 ) . Let { α n } , { ξ n } , { γ n , i } be the sequences in ( 0 , 1 ) with α n + ξ n + i = 1 γ n , i = 1 , for each n 0 . Let { β i } be a sequence in ( 0 , ) , and { λ n , i } be a sequence in ( 0 , 2 A 2 ) . Let { x n } be the sequence defined by (1.11). If Ω and the following conditions are satisfied:
  1. (i)

    lim n ξ n = 0 , and n = 0 ξ n = ;

     
  2. (ii)

    lim inf n α n γ n , i > 0 for each i 1 ;

     
  3. (iii)

    0 < lim inf n λ n , i lim sup n λ n , i < 2 A 2 ,

     

then x n x Ω where x = P Ω f ( x ) , where P Ω is the metric projection from H 1 onto Ω.

Proof (I) First we prove that { x n } is bounded.

In fact, letting z Ω , by Lemma 3.1, for each i 1 ,
z = J β i B i [ z λ n , i A ( I J β i K i ) A z ] .
Hence it follows from Lemma 2.7(iii) that for each i 1 and each n 1 we have
x n + 1 z = α n x n + ξ n f ( x n ) + i = 1 γ n , i J β i B i [ x n λ n , i A ( I J β i K i ) A x n ] z α n x n z + ξ n f ( x n ) z + i = 1 γ n , i J β i B i [ x n λ n , i A ( I J β i K i ) A x n ] z α n x n z + ξ n f ( x n ) z + i = 1 γ n , i J β i B i [ x n λ n , i A ( I J β i K i ) A x n ] z α n x n z + ξ n f ( x n ) z + i = 1 γ n , i x n z = ( 1 ξ n ) x n z + ξ n f ( x n ) z ( 1 ξ n ) x n z + ξ n f ( x n ) f ( z ) + ξ n f ( z ) z ( 1 ξ n ( 1 k ) ) x n z + ξ n ( 1 k ) 1 k f ( z ) z max { x n z , 1 1 k f ( z ) z } .
By induction, we can prove that
x n z max { x 0 z , 1 1 k f ( z ) z } , n 0 .
(3.5)

This implies that { x n } is bounded, so is { f ( x n ) } .

(II) Now we prove that for each j 1
α n γ n , j x n J β i B i [ x n λ n , i A ( I J β i K i ) A x n ] 2 x n z 2 x n + 1 z 2 + ξ n f ( x n ) z 2 , for each  i 1 .
(3.6)
Indeed, it follows from Lemma 2.3 that for any positive j 1
x n + 1 z 2 = α n x n + ξ n f ( x n ) + i = 1 γ n , i J β i B i [ x n λ n , i A ( I J β i K i ) A x n ] z 2 α n x n z 2 + ξ n f ( x n ) z 2 + i = 1 γ n , i J β i B i [ x n λ n , i A ( I J β i K i ) A x n ] z 2 α n γ n , j x n J β i B i [ x n λ n , i A ( I J β i K i ) A x n ] 2 ( 1 ξ n ) x n z 2 + ξ n f ( x n ) z 2 α n γ n , j x n J β i B i [ x n λ n , i A ( I J β i K i ) A x n ] 2 .

Simplifying it, (3.6) is proved.

By the assumption that Ω , and it is easy to prove that Ω is closed and convex. This implies that P Ω is well defined. Again since P Ω f : H 1 Ω is a contraction mapping with contractive constant k ( 0 , 1 ) , there exists a unique x Ω such that x = P Ω f x . Since x Ω , it solves (GSVIP) (1.4). By Lemma 3.1,
x = J β j B j ( x λ n , j A ( I J β j K j ) A x ) , j 1 , n 0 .
(3.7)
  1. (III)

    Now we prove that x n x .

     

In order to prove that x n x (as n ), we consider two cases.

Case 1. Assume that { x n x } is a monotone sequence. In other words, for n 0 large enough, { x n x } n n 0 is either nondecreasing or non-increasing. Since { x n x } is bounded, { x n x } is convergence. Again since lim n ξ n = 0 , and { f ( x n ) } is bounded, from (3.6) we get
lim n α n γ n , j x n J β i B i [ x n λ n , i A ( I J β i K i ) A x n ] 2 = 0 .
By condition (ii), we obtain
lim n x n J β i B i [ x n λ n , i A ( I J β i K i ) A x n ] = 0 .
(3.8)
Now we prove that
lim sup n f ( x ) x , x n x 0 .
(3.9)
To show this inequality, we choose a subsequence { x n k } of { x n } such that x n k w , λ n k , i λ i ( 0 , 2 A 2 ) for each i 1 , and
lim sup n f ( x ) x , x n x = lim n k f ( x ) x , x n k x .
(3.10)
It follows from (3.8) that
J β i B i [ x n λ i A ( I J β i K i ) A x n ] x n J β i B i [ x n λ i A ( I J β i K i ) A x n ] J β i B i [ x n λ n , i A ( I J β i K i ) A x n ] + J β i B i [ x n λ n , i A ( I J β i K i ) A x n ] x n [ x n λ i A ( I J β i K i ) A x n ] [ x n λ n , i A ( I J β i K i ) A x n ] + J β i B i [ x n λ n , i A ( I J β i K i ) A x n ] x n | λ i λ n , i | A ( I J β i K i ) A x n + J β i B i [ x n λ n , i A ( I J β i K i ) A x n ] x n 0 ( as  n ) .
For each i 1 , J β i B i [ I λ i A ( I J β i K i ) A ] is a nonexpansive mapping. Thus from Lemma 2.1, w = J β i B i [ I λ i A ( I J β i K i ) A ] w . By Lemma 3.1 w Ω , i.e., w is a solution of (GSVIP) (1.4). Consequently we have
lim sup n f ( x ) x , x n x = lim n k f ( x ) x , x n k x = f ( x ) x , w x 0 .
  1. (IV)

    Finally, we prove that x n P Ω f ( x ) .

     
In fact, from Lemma 2.2 we have
x n + 1 x 2 α n ( x n x ) + i = 1 γ n , i J β i B i [ x n λ n , i A ( I J β i K i ) A x n ] x 2 + 2 ξ n f ( x n ) x , x n + 1 x ( 1 ξ n ) 2 x n x 2 + 2 ξ n f ( x n ) f ( x ) , x n + 1 x + 2 ξ n f ( x ) x , x n + 1 x ( 1 ξ n ) 2 x n x 2 + 2 ξ n k x n x x n + 1 x + 2 ξ n f ( x ) x , x n + 1 x ( 1 ξ n ) 2 x n x 2 + ξ n k { x n + 1 x 2 + x n x 2 } + 2 ξ n f ( x ) x , x n + 1 x .
Simplifying it, we have
x n + 1 x 2 ( 1 ξ n ) 2 + ξ n k 1 ξ n k x n x 2 + 2 ξ n 1 ξ n k f ( x ) x , x n + 1 x 1 2 ξ n + ξ n k 1 ξ n k x n x 2 + ξ n 2 1 ξ n k x n x 2 + 2 ξ n 1 ξ n k f ( x ) x , x n + 1 x ( 1 η n ) x n x 2 + η n δ n , n 0 ,

where δ n = ξ n M 2 ( 1 k ) + 1 1 k f ( x ) x , x n + 1 x , M = sup n 0 x n x 2 , and η n = 2 ( 1 k ) ξ n 1 ξ n k . It is easy to see that η n 0 , n = 1 η n = , and lim sup n δ n 0 . Hence by Lemma 2.4, the sequence { x n } converges strongly to x = P Ω f ( x ) .

Case 2. Assume that { x n x } is not a monotone sequence. Then, by Lemma 2.3, we can define a sequence of positive integers: { τ ( n ) } , n n 0 (where n 0 large enough) by
τ ( n ) = max { k n : x k x x k + 1 x } .
(3.11)
Clearly { τ ( n ) } is a nondecreasing sequence such that τ ( n ) as n , and for all n n 0
x τ ( n ) x x τ ( n ) + 1 x .
(3.12)
Therefore { x τ ( n ) x } is a nondecreasing sequence. According to Case (1), lim n x τ ( n ) x = 0 and lim n x τ ( n ) + 1 x = 0 . Hence we have
0 x n x max { x n x , x τ ( n ) x } x τ ( n ) + 1 x 0 , as  n .

This implies that x n x and x = P Ω f ( x ) is a solution of (GSVIP) (1.4).

This completes the proof of Theorem 3.2. □

In Theorem 3.2, if B i = B and K i = K , for each i 1 , where B : H 1 2 H 1 and K : H 2 2 H 2 are two set-valued maximal monotone mappings, then from Theorem 3.2 we have the following.

Theorem 3.3 Let H 1 , H 2 , A, A , B, K, Ω, f be the same as in Theorem  3.2. Let { α n } , { ξ n } , { γ n } be the sequence in ( 0 , 1 ) with α n + ξ n + γ n = 1 for each n 0 . Let β > 0 be any given positive number, and { λ n } be a sequence in ( 0 , 2 A 2 ) . Let { x n } be the sequence defined by
x n + 1 = α n x n + ξ n f ( x n ) + γ n J β B [ x n λ n A ( I J β K ) A x n ] , n 0 .
(3.13)
If Ω and the following conditions are satisfied:
  1. (i)

    lim n ξ n = 0 , and n = 0 ξ n = ;

     
  2. (ii)

    lim inf n α n γ n > 0 ;

     
  3. (iii)

    0 < lim inf n λ n lim sup n λ n < 2 A 2 ,

     

then x n x Ω where x = P Ω f ( x ) .

4 Applications

In this section we shall utilize the results presented in Theorem 3.2 and Theorem 3.3 to study some problems.

4.1 Application to split optimization problem

Let H 1 and H 2 be two real Hilbert spaces. Let h : H 1 R and g : H 2 R be two proper, convex and lower semicontinuous functions, and A : H 1 H 2 be a linear and bounded operators. The so-called split optimization problem (SOP) is
to find  x H 1  such that  h ( x ) = min y H 1 h ( y )  and  g ( A x ) = min z H 2 g ( z ) .
(4.1)
Denote by h = B and g = K . It is know that B : H 1 2 H 1 (resp. K : H 2 2 H 2 ) is a maximal monotone mapping, so we can define the resolvent J β B = ( I + β B ) 1 and J β K = ( I + β K ) 1 , where β > 0 . Since x and A x is a minimum of h on H 1 and g on H 2 , respectively, for any given β > 0 , we have
x B 1 ( 0 ) = F ( J β B ) , and A x K 1 ( 0 ) = F ( J β K ) .
(4.2)

This implies that the (SOP) (4.1) is equivalent to the split variational inclusion problem (SVIP) (4.2). From Theorem 3.3 we have the following.

Theorem 4.1 Let H 1 , H 2 , A, B, K, h, g be the same as above. Let f, { α n } , { ξ n } , { γ n } be the same as in Theorem  3.3. Let β > 0 be any given positive number, and { λ n } be a sequence in ( 0 , 2 A 2 ) . Let { x n } be a sequence generated by x 0 H 1
{ y n = argmin z H 2 { g ( z ) + 1 2 β z A x n 2 } , z n = x n λ n A ( A x n y n ) , w n = argmin y H 1 { h ( y ) + 1 2 β y z n 2 } , x n + 1 = α n x n + ξ n f ( x n ) + γ n w n , n 0 .
(4.3)
If Ω 1 , the solution set of the split optimization problem (4.1), and the following conditions are satisfied:
  1. (i)

    lim n ξ n = 0 , and n = 0 ξ n = ;

     
  2. (ii)

    lim inf n α n γ n > 0 ;

     
  3. (iii)

    0 < lim inf n λ n lim sup n λ n < 2 A 2 ,

     

then x n x Ω 1 where x = P Ω 1 f ( x ) .

Proof Since h = B , g : = K , and y n = argmin z H 2 { g ( z ) + 1 2 β z A x n 2 } , we have
0 [ K ( z ) + 1 β ( z A x n ) ] z = y n , i.e. , A x n ( β K + I ) ( y n ) .
This implies that
y n = J β K ( A x n ) .
(4.4)
Similarly, from (4.3), we have
w n = J β B ( z n ) .
(4.5)
From (4.3)-(4.5), we have
w n = J β B ( x n λ n A ( I J β K ) A x n ) .
(4.6)
Therefore (4.3) can be rewritten as
x n + 1 = α n x n + ξ n f ( x n ) + γ n J β B ( x n λ n A ( I J β K ) A x n ) , n 0 .
(4.7)

The conclusion of Theorem 4.1 can be obtained from Theorem 3.3 immediately. □

4.2 Application to split feasibility problem

Let C H 1 and Q H 2 be two nonempty closed convex subsets and A : H 1 H 2 be a bounded linear operator. Now we consider the following split feasibility problem, i.e.: to find
x C  such that  A x Q .
(4.8)
Let i C and i Q be the indicator functions of C and Q defined by (1.9). Let N C ( u ) be the normal cone at u H 1 defined by
N C ( u ) = { z H 1 : z , v u 0 , v C } .
Since i C and i Q both are proper convex and lower semicontinuous functions on H 1 and H 2 , respectively, and the subdifferential i C of i C (resp. i Q of i Q ) is a maximal monotone operator, we can define the resolvents J β i C of i C and J β i Q of i Q by
J β i C ( x ) = ( I + β i C ) 1 ( x ) , x H 1 , J β i Q ( x ) = ( I + β i Q ) 1 ( x ) , x H 2 ,
where β > 0 . By definition, we know that
i C ( x ) = { z H 1 : i C ( x ) + z , y x i C ( y ) , y H 1 } = { z H 1 : z , y x 0 , y C } = N C ( x ) , x C .
Hence, for each β > 0 , we have
u = J β i C ( x ) x u β N C ( u ) x u , y u 0 , y C u = P C ( x ) .
This implies that J β i C = P C . Similarly J β i Q = P Q . Taking h ( x ) = i C ( x ) and g ( x ) = i Q ( x ) in (4.1), then the (SFP) (4.8) is equivalent to the following split optimization problem:
to find  x H 1  such that  i C ( x ) = min y H 1 i C ( y )  and  i Q ( A x ) = min z H 2 i Q ( z ) .
(4.9)

Hence, the following result can be obtained from Theorem 4.1 immediately.

Theorem 4.2 Let H 1 , H 2 , A, A , i C , i Q be the same as above. Let f, { α n } , { ξ n } , { γ n } be the same as in Theorem  4.1. Let { λ n } be a sequence in ( 0 , 2 A 2 ) . Let { x n } be the sequence defined by
x n + 1 = α n x n + ξ n f ( x n ) + γ n P C [ x n λ n A ( I P Q ) A x n ] , n 0 .
(4.10)
If the solution set of the split optimization problem (4.4) Ω 2 , and the following conditions are satisfied:
  1. (i)

    lim n ξ n = 0 , and n = 0 ξ n = ;

     
  2. (ii)

    lim inf n α n γ n > 0 ;

     
  3. (iii)

    0 < lim inf n λ n lim sup n λ n < 2 A 2 ,

     

then x n x Ω 2 where x = P Ω 2 f ( x ) .

Remark 4.3 Theorem 4.2 extends and improves the main results in Censor and Elfving [1] and Byrne [2].

Declarations

Acknowledgements

The authors would like to express their thanks to the referees and the editors for their kind and helpful comments and advice. This work was supported by the National Natural Science Foundation of China (Grant No. 11361070).

Authors’ Affiliations

(1)
College of Statistics and Mathematics, Yunnan University of Finance and Economics

References

  1. Censor Y, Elfving T: A multiprojection algorithm using Bregman projections in a product space. Numer. Algorithms 1994, 8: 221–239. 10.1007/BF02142692View ArticleMathSciNetGoogle Scholar
  2. Byrne C: Iterative oblique projection onto convex subsets and the split feasibility problem. Inverse Probl. 2002, 18: 441–453. 10.1088/0266-5611/18/2/310View ArticleMathSciNetGoogle Scholar
  3. Censor Y, Bortfeld T, Martin N, Trofimov A: A unified approach for inversion problem in intensity-modulated radiation therapy. Phys. Med. Biol. 2006, 51: 2353–2365. 10.1088/0031-9155/51/10/001View ArticleGoogle Scholar
  4. Censor Y, Elfving T, Kopf N, Bortfeld T: The multiple-sets split feasibility problem and its applications. Inverse Probl. 2005, 21: 2071–2084. 10.1088/0266-5611/21/6/017View ArticleMathSciNetGoogle Scholar
  5. Censor Y, Motova A, Segal A: Perturbed projections and subgradient projections for the multiple-sets split feasibility problem. J. Math. Anal. Appl. 2007, 327: 1244–1256. 10.1016/j.jmaa.2006.05.010View ArticleMathSciNetGoogle Scholar
  6. Xu HK: A variable Krasnosel’skii-Mann algorithm and the multiple-sets split feasibility problem. Inverse Probl. 2006, 22: 2021–2034. 10.1088/0266-5611/22/6/007View ArticleGoogle Scholar
  7. Yang Q: The relaxed CQ algorithm for solving the split feasibility problem. Inverse Probl. 2004, 20: 1261–1266. 10.1088/0266-5611/20/4/014View ArticleGoogle Scholar
  8. Zhao J, Yang Q: Several solution methods for the split feasibility problem. Inverse Probl. 2005, 21: 1791–1799. 10.1088/0266-5611/21/5/017View ArticleGoogle Scholar
  9. Chang SS, Cho YJ, Kim JK, Zhang WB, Yang L: Multiple-set split feasibility problems for asymptotically strict pseudocontractions. Abstr. Appl. Anal. 2012., 2012: Article ID 491760 10:1155/2012/491760Google Scholar
  10. Chang SS, Wang L, Tang YK, Yang L: The split common fixed point problem for total asymptotically strictly pseudocontractive mappings. J. Appl. Math. 2012., 2012: Article ID 385638 10.1155/2012.385638Google Scholar
  11. Ansari QH, Rehan A: Split feasibility and fixed point problems. In Nonlinear Analysis: Approximation Theory, Optimization and Applications. Birkhäuser, New Delhi; 2014:281–322.Google Scholar
  12. Martinet B: Régularisation d’inéquations variationnelles par approximations successives. Rev. Fr. Autom. Inform. Rech. Opér. 1970, 4: 154–158.MathSciNetGoogle Scholar
  13. Rockafellar RT: Monotone operators and the proximal point algorithm. SIAM J. Control Optim. 1976, 14: 877–898. 10.1137/0314056View ArticleMathSciNetGoogle Scholar
  14. Moudafi A: A relaxed alternating CQ algorithm for convex feasibility problems. Nonlinear Anal. 2013, 79: 117–121.View ArticleMathSciNetGoogle Scholar
  15. Eslamian M, Latif A: General split feasibility problems in Hilbert spaces. Abstr. Appl. Anal. 2013., 2013: Article ID 805104Google Scholar
  16. Chen RD, Wang J, Zhang HW: General split equality problems in Hilbert spaces. Fixed Point Theory Appl. 2014., 2014: Article ID 35Google Scholar
  17. Moudafi A: Split monotone variational inclusions. J. Optim. Theory Appl. 2011, 150: 275–283. 10.1007/s10957-011-9814-6View ArticleMathSciNetGoogle Scholar
  18. Güler O: On the convergence of the proximal point algorithm for convex minimization. SIAM J. Control Optim. 1991, 29: 403–419. 10.1137/0329022View ArticleMathSciNetGoogle Scholar
  19. Kamimura S, Takahashi W: Strong convergence of a proximal-type algorithm in a Banach space. SIAM J. Optim. 2002, 13: 938–945. 10.1137/S105262340139611XView ArticleMathSciNetGoogle Scholar
  20. Solodov MV, Svaiter BF: Forcing strong convergence of proximal point iterations in a Hilbert space. Math. Program. 2000, 87: 189–202.MathSciNetGoogle Scholar
  21. Eslamian M: Rockafellar’s proximal point algorithm for a finite family of monotone operators. Sci. Bull. ‘Politeh.’ Univ. Buchar., Ser. A, Appl. Math. Phys. 2014, 76(1):43–50.MathSciNetGoogle Scholar
  22. Chuang C-S: Strong convergence theorems for the split variational inclusion problem in Hilbert spaces. Fixed Point Theory Appl. 2013., 2013: Article ID 350Google Scholar
  23. Chang SS: On Chidume’s open questions and approximate solutions for multi-valued strongly accretive mapping equations in Banach spaces. J. Math. Anal. Appl. 1997, 216: 94–111. 10.1006/jmaa.1997.5661View ArticleMathSciNetGoogle Scholar
  24. Chang S-S, Kim JK, Wang XR: Modified block iterative algorithm for solving convex feasibility problems in Banach spaces. J. Inequal. Appl. 2010., 2010: Article ID 869684Google Scholar
  25. Maingé P-E: Strong convergence of projected subgradient methods for nonsmooth and nonstrictly convex minimization. Set-Valued Anal. 2008, 16(7–8):899–912. 10.1007/s11228-008-0102-zView ArticleMathSciNetGoogle Scholar

Copyright

© Chang and Wang; licensee Springer. 2014

This article is published under license to BioMed Central Ltd. This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly credited.