Strong convergence of a relaxed CQ algorithm for the split feasibility problem

The split feasibility problem (SFP) is finding a point in a given closed convex subset of a Hilbert space such that its image under a bounded linear operator belongs to a given closed convex subset of another Hilbert space. The most popular iterative method is Byrne’s CQ algorithm. López et al. proposed a relaxed CQ algorithm for solving SFP where the two closed convex sets are both level sets of convex functions. This algorithm can be implemented easily since it computes projections onto half-spaces and has no need to know a priori the norm of the bounded linear operator. However, their algorithm has only weak convergence in the setting of infinite-dimensional Hilbert spaces. In this paper, we introduce a new relaxed CQ algorithm such that the strong convergence is guaranteed. Our result extends and improves the corresponding results of López et al. and some others.

MSC:90C25, 90C30, 47J25.

The split feasibility problem (SFP) was proposed by Censer and Elfving in [1]. It can mathematically be formulated as the problem of finding a point x satisfying the property:

where A is a given M\times N real matrix (where A^{\ast } is the transpose of A), C and Q are nonempty, closed and convex subsets in {\mathbb{R}}^N and {\mathbb{R}}^M, respectively. This problem has received much attention [2] due to its applications in signal processing and image reconstruction, with particular progress in intensity-modulated radiation therapy [3–5], and many other applied fields.

We assume the SFP (1.1) is consistent, and let S be the solution set, i.e.,

It is easily seen that S is closed convex. Many iterative methods can be used to solve the SFP (1.1); see [6–16]. Byrne [6, 17] was among others the first to propose the so-called CQ algorithm, which is defined as follows:

where {\tau }_n\in (0,\frac{2}{{\parallel A\parallel }^2}), and P_C and P_Q are the orthogonal projections onto the sets C and Q, respectively. Compared with Censer and Elfving’ algorithm [1], the Byrne’ algorithm is easily executed since it only deal with orthogonal projections with no need to compute matrix inverses.

The CQ algorithm (1.2) can be obtained from optimization. In fact, if we introduce the (convex) objective function

and analyze the minimization problem

then the CQ algorithm (1.2) comes immediately as a special case of the gradient projection algorithm (GPA)(For more details about the GPA, the reader is referred to [18]). Since the convex objective f(x) is differentiable, and has a Lipschitz gradient, which is given by

the GPA for solving the minimization problem (1.4) generates a sequence (x_n) recursively as

where the stepsize {\tau }_n is chosen in the interval (0,\frac{2}{L}), where L is the Lipschitz constant of ∇f.

Observe that in algorithms (1.2) and (1.6) mentioned above, in order to implement the CQ algorithm, one has to compute the operator (matrix) norm \parallel A\parallel, which is in general not an easy work in practice. To overcome this difficulty, some authors proposed different adaptive choices of selecting the stepsize {\tau }_n (see [6, 14, 19]). For instance, very recently López et al. introduced a new way of selecting the stepsize [19] as follows:

The computation of a projection onto a general closed convex subset is generally difficult. To overcome this difficulty, Fukushima [20] suggested a so-called relaxed projection method to calculate the projection onto a level set of a convex function by computing a sequence of projections onto half-spaces containing the original level set. In the setting of finite-dimensional Hilbert spaces, this idea was followed by Yang [13], who introduced the relaxed CQ algorithms for solving SFP (1.1) where the closed convex subsets C and Q are level sets of convex functions.

Recently, for the purpose of generality, the SFP (1.1) is studied in a more general setting. For instance, Xu [12] and López et al. [19] considered the SFP (1.1) in infinite-dimensional Hilbert spaces (i.e., the finite-dimensional Euclidean spaces {\mathbb{R}}^N and {\mathbb{R}}^M are replaced with general Hilbert spaces). Very recently, López et al. proposed a relaxed CQ algorithm with a new adaptive way of determining the stepsize sequence ({\tau }_n) for solving the SFP (1.1) where the closed convex subsets C and Q are level sets of convex functions. This algorithm can be implemented easily since it computes projections onto half-spaces and has no need to know a priori the norm of the bounded linear operator. However, their algorithm has only weak convergence in the setting of infinite-dimensional Hilbert spaces. In this paper, we introduce a new relaxed CQ algorithm such that the strong convergence is guaranteed in infinite-dimensional Hilbert spaces. Our result extends and improves the corresponding results of López et al. and some others.

The rest of this paper is organized as follows. Some useful lemmas are listed in Section 2. In Section 3, the strong convergence of the new relaxed CQ algorithm of this paper is proved.

Throughout the rest of this paper, we denote by H or K a Hilbert space, A is a bounded linear operator from H to K, and by I the identity operator on H or K. If f:H\to \mathbb{R} is a differentiable function, then we denote by ∇f the gradient of the function f. We will also use the notations:

• → denotes strong convergence.

• ⇀ denotes weak convergence.

• {\omega }_w(x_n)=\{x\mid \mathrm{\exists }\{x_{n_k}\}\subset \{x_n\}\text{ such that }{x}_{n_k}\rightharpoonup x\} denotes the weak ω-limit set of \{x_n\}.

Recall that a mapping T:H\to H is said to be nonexpansive if

T:H\to H is said to be firmly nonexpansive if, for x,y\in H,

Recall that a function f:H\to \mathbb{R} is called convex if

A differentiable function f is convex if and only if there holds the inequality:

Recall that an element g\in H is said to be a subgradient of f:H\to \mathbb{R} at x if

This relation is called the subdifferentiable inequality.

A function f:H\to \mathbb{R} is said to be subdifferentiable at x, if it has at least one subgradient at x. The set of subgradients of f at the point x is called the subdifferentiable of f at x, and it is denoted by \partial f(x). A function f is called subdifferentiable, if it is subdifferentiable at all x\in H. If a function f is differentiable and convex, then its gradient and subgradient coincide.

A function f:H\to \mathbb{R} is said to be weakly lower semi-continuous (w-lsc) at x if x_n\rightharpoonup x implies

We know that the orthogonal projection P_C from H onto a nonempty closed convex subset C\subset H is a typical example of a firmly nonexpansive mapping, which is defined by

It is well known that P_C is characterized by the inequality (for x\in H)

The following lemma is not hard to prove (see [17, 21]).

Lemma 2.1 Let f be given as in (1.3). Then

1. (i)

   f is convex and differential.

2. (ii)

   \mathrm{\nabla }f(x)=A^{\ast }(I-P_Q)Ax, x\in H.

3. (iii)

   f is w-lsc on H.

4. (iv)

   ∇f is {\parallel A\parallel }^2-Lipschitz: \parallel \mathrm{\nabla }f(x)-\mathrm{\nabla }f(y)\parallel \le {\parallel A\parallel }^2\parallel x-y\parallel, x,y\in H.

The following are characterizations of firmly nanexpansive mappings (see [22]).

Lemma 2.2 Let T:H\to H be an operator. The following statements are equivalent.

1. (i)

   T is firmly nonexpansive.

2. (ii)

   {\parallel Tx-Ty\parallel }^2\le 〈x-y,Tx-Ty〉, x,y\in H.

3. (iii)

   I-T is firmly nonexpansive.

Lemma 2.3 [23]

Assume ({\alpha }_n) is a sequence of nonnegative real numbers such that

where ({\gamma }_n) is a sequence in (0,1), and ({\sigma }_n) is a sequence in ℝ such that

1. (i)

   {\sum }_{n=0}^{\mathrm{\infty }}{\gamma }_n=\mathrm{\infty }.

2. (ii)

   {lim\hspace{0.17em}sup}_{n\to \mathrm{\infty }}{\sigma }_n\le 0, or .

Then {lim}_{n\to \mathrm{\infty }}{\alpha }_n=0.

In this section, we turn to consider a new relaxed CQ algorithm in the setting of infinite-dimensional Hilbert spaces for solving SFP (1.1) where the closed convex subsets C and Q are of the particular structure, i.e. level sets of convex functions given as follows:

where c:H\to \mathbb{R} and q:K\to \mathbb{R} are convex functions. We assume that both c and q are subdifferentiable on H and K, respectively, and that ∂c and ∂q are bounded operators (i.e., bounded on bounded sets). By the way, we mention that every convex function defined on a finite-dimensional Hilbert space is subdifferentiable and its subdifferential operator is a bounded operator (see [24]).

Set

where {\xi }_n\in \partial c(x_n), and

where {\zeta }_n\in \partial q(A{x}_n).

Obviously, C_n and Q_n are half-spaces and it is easy to verify that C_n\supset C and Q_n\supset Q hold for every n\ge 0 from the subdifferentiable inequality. In what follows, we define

where Q_n is given as in (3.3). We then have

Firstly, we recall the relaxed CQ algorithm of López et al. [19] for solving the SFP (1.1) where C and Q are given in (3.1) as follows.

Algorithm 3.1 Choose an arbitrary initial guess x_0. Assume x_n has been constructed. If \mathrm{\nabla }{f}_n(x_n)=0, then stop; otherwise, continue and construct x_{n+1} via the formula

where C_n is given as (3.2), and

López et al. proved that under some certain conditions the sequence (x_n) generated by Algorithm 3.1 converges weakly to a solution of the SFP (1.1). Since the projections onto half-spaces C_n and Q_n have closed forms and {\tau }_n is obtained adaptively via the formula (3.4) (no need to know a priori the norm of operator A), the above relaxed CQ algorithm 3.1 is implementable. But the weak convergence is its a weakness. To overcome this weakness, inspired by Algorithm 3.1, we will introduce a new relaxed CQ algorithm for solving the SFP (1.1) where C and Q are given in (3.1) so that the strong convergence is guaranteed.

It is well known that Halpern’s algorithm has a strong convergence for finding a fixed point of a nonexpansive mapping [25, 26]. Then we are in a position to give our algorithm. The algorithm given below is referred to as a Halpern-type algorithm [27].

Algorithm 3.2 Let u\in H, and start an initial guess x_0\in H arbitrarily. Assume that the n th iterate x_n has been constructed. If \mathrm{\nabla }{f}_n(x_n)=0, then stop (x_n is a approximate solution of SFP (1.1)). Otherwise, continue and calculate the (n+1)th iterate x_{n+1} via the formula:

where the sequence ({\alpha }_n)\subset (0,1) and ({\tau }_n) and ({\rho }_n) are given as in (3.4).

The convergence result of Algorithm 3.2 is stated in the next theorem.

Theorem 3.3 Assume that ({\alpha }_n) and ({\rho }_n) satisfy the assumptions:

(a1) {lim}_{n\to \mathrm{\infty }}{\alpha }_n=0 and {\sum }_{n=0}^{\mathrm{\infty }}{\alpha }_n=\mathrm{\infty }.

(a2) {inf}_n{\rho }_n(4-{\rho }_n)>0.

Then the sequence (x_n) generated by Algorithm  3.2 converges in norm to P_{S}u.

Proof We may assume that the sequence (x_n) is infinite, that is, Algorithm 3.2 does not terminate in a finite number of iterations. Thus \mathrm{\nabla }{f}_n(x_n)\ne 0 for all n\ge 0. Recall that S is the solution set of the SFP (1.1),

In the consistent case of the SFP (1.1), S is nonempty, closed and convex. Thus, the metric projection P_S is well-defined. We set z=P_{S}u. Since z\in S\subset C_n and the projection operator P_{C_n} is nonexpansive, we obtain

Note that I-P_{Q_n} is firmly nonexpansive and \mathrm{\nabla }{f}_n(z)=0, it is deduced from Lemma 2.2 that

which implies that

Thus, we have

Now we prove (x_n) is bounded. Indeed, we have from (3.6) that

and consequently

It turns out that

and inductively

and this means that (x_n) is bounded. Since {\alpha }_n\to 0, with no loss of generality, we may assume that there is \sigma >0 so that {\rho }_n(4-{\rho }_n)(1-{\alpha }_n)\ge \sigma for all n. Setting s_n={\parallel x_n-z\parallel }^2, from the last inequality of (3.6), we get the following inequality:

Now, following an idea in [28], we prove s_n\to 0 by distinguishing two cases.

Case 1: (s_n) is eventually decreasing (i.e. there exists k\ge 0 such that s_n>s_{n+1} holds for all n\ge k). In this case, (s_n) must be convergent, and from (3.7) it follows that

where M>0 is a constant such that 2\parallel x_{n+1}-z\parallel \parallel u-z\parallel \le M for all n\in \mathbb{N}. Using the condition (a1), we have from (3.8) that \frac{f_n^2(x_n)}{{\parallel \mathrm{\nabla }{f}_n(x_n)\parallel }^2}\to 0. Thus, to verify that f_n(x_n)\to 0, it suffices to show that (\mathrm{\nabla }f(x_n)) is bounded. In fact, it follows from Lemma 2.1 that (noting that \mathrm{\nabla }{f}_n(z)=0 due to z\in S)

This implies that \parallel \mathrm{\nabla }{f}_n(x_n)\parallel is bounded and it yields f_n(x_n)\to 0, namely \parallel (I-P_{Q_n})A{x}_n\parallel \to 0.

Since ∂q is bounded on bounded sets, there exists a constant \eta >0 such that \parallel {\zeta }_n\parallel \le \eta for all n\ge 0. From (3.3) and the trivial fact that P_{Q_n}(A{x}_n)\in Q_n, it follows that

If x^{\ast }\in {\omega }_w(x_n), and (x_{n_k}) is a subsequence of (x_n) such that x_{n_k}\rightharpoonup x^{\ast }, then the w-lsc of q and (3.9) imply that

It turns out that A{x}^{\ast }\in Q. Next, we turn to prove x^{\ast }\in C. For convenience, we set y_n:={\alpha }_{n}u+(1-{\alpha }_n)(x_n-{\tau }_n\mathrm{\nabla }{f}_n(x_n)). In fact, since the P_{C_n} is firmly nonexpansive, it concludes that

On the other hand, we have

and

Noting that (x_n) is bounded, we have from (3.10)-(3.12) that

where M is some positive constant. Clearly, from (3.13), it turns out that

Thus, we assert that \parallel (I-P_{C_n})x_n\parallel \to 0 due to the fact that s_n-s_{n+1}+({\alpha }_n+\frac{f_n(x_n)}{\parallel \mathrm{\nabla }{f}_n(x_n)\parallel })M\to 0. Moreover, by the definition of C_n, we obtain

where δ is a constant such that \parallel {\xi }_n\parallel \le \delta for all n\ge 0. The w-lsc of c then implies that

Consequently, x^{\ast }\in C, and hence {\omega }_w(x_n)\subset S. Furthermore, due to (2.2), we get

Taking into account of (3.7), we have

Applying Lemma 2.3 to (3.16), we obtain s_n\to 0.

Case 2: (s_n) is not eventually decreasing, that is, we can find an integer n_0 such that s_{n_0}\le s_{n_0+1}. Now we define

It is easy to see that V_n is nonempty and satisfies V_n\subseteq V_{n+1}. Let

It is clear that \psi (n)\to \mathrm{\infty } as n\to \mathrm{\infty } (otherwise, (s_n) is eventually decreasing). It is also clear that s_{\psi (n)}\le s_{\psi (n)+1} for all n>n_0. Moreover,

In fact, if \psi (n)=n, then the inequity (3.17) is trivial; if , from the definition of \psi (n), there exists some i\in \mathbb{N} such that \psi (n)+i=n, we deduce that

and the inequity (3.17) holds again. Since s_{\psi (n)}\le s_{\psi (n)+1} for all n>n_0, it follows from (3.8) that

so that f_{\psi (n)}(x_{\psi (n)})\to 0 as n\to \mathrm{\infty } (noting that \parallel \mathrm{\nabla }{f}_{\psi (n)}(x_{\psi (n)})\parallel is bounded). By the same argument to the proof in case 1, we have {\omega }_w(x_{\psi (n)})\subset S. On the other hand, noting s_{\psi (n)}\le s_{\psi (n)+1} again, we have from (3.12) and (3.14) that

where M is a positive constant. Letting n\to \mathrm{\infty } yields that

from which one can deduce that

Since s_{\psi (n)}\le s_{\psi (n)+1}, it follows from (3.7) that

Combining (3.19) and (3.20) yields

and hence s_{\psi (n)}\to 0, which together with (3.18) implies that

which, together with (3.17), in turn implies that s_n\to 0, that is, x_n\to z. □

Remark 3.4 Since u can be chosen in H arbitrarily, one can compute the minimum-norm solution of SFP (1.1) where C and Q are given in (3.1) by taking u=0 in Algorithm 3.2 whether 0\in C or 0\notin C.

The authors wish to thank the referees for their helpful comments, which notably improved the presentation of this manuscript. This work was supported by the Fundamental Research Funds for the Central Universities (ZXH2012K001) and in part by the Foundation of Tianjin Key Lab for Advanced Signal Processing.

Affiliations

1. College of Science, Civil Aviation University of China, Tianjin, 300300, China

   • Songnian He
   •  & Ziyi Zhao

2. Tianjin Key Laboratory for Advanced Signal Processing, Civil Aviation University of China, Tianjin, 300300, China

   • Songnian He

Corresponding author

Correspondence to Songnian He.

Competing interests

The authors declare that they have no competing interests.

Authors’ contributions

All the authors read and approved the final manuscript.

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