The regularized CQ algorithm without a priori knowledge of operator norm for solving the split feasibility problem.

The split feasibility problem (SFP) is finding a point [Formula: see text] such that [Formula: see text], where C and Q are nonempty closed convex subsets of Hilbert spaces [Formula: see text] and [Formula: see text], and [Formula: see text] is a bounded linear operator. Byrne's CQ algorithm is an effective algorithm to solve the SFP, but it needs to compute [Formula: see text], and sometimes [Formula: see text] is difficult to work out. López introduced a choice of stepsize [Formula: see text], [Formula: see text], [Formula: see text]. However, he only obtained weak convergence theorems. In order to overcome the drawbacks, in this paper, we first provide a regularized CQ algorithm without computing [Formula: see text] to find the minimum-norm solution of the SFP and then obtain a strong convergence theorem.

Introduction

Let and be real Hilbert spaces and let C and Q be nonempty closed convex subsets of and , and let be a bounded linear operator. Let and denote the sets of positive integers and real numbers.

In 1994, Censor and Elfving [1] came up with the split feasibility problem (SFP) in finite-dimensional Hilbert spaces. In infinite-dimensional Hilbert spaces, it can be formulated as where C and Q are nonempty closed convex subsets of and , and is a bounded linear operator. Suppose that SFP (1.1) is solvable, and let S denote its solution set. The SFP is widely applied to signal processing, image reconstruction and biomedical engineering [2-4].

So far, some authors have studied SFP (1.1) [5-17]. Others have also found a lot of algorithms to study the split equality fixed point problem and the minimization problem [18-20]. Byrne’s CQ algorithm is an effective method to solve SFP (1.1). A sequence , generated by the formula where the parameters , , and , is a set of orthogonal projections.

As is well-known, Cencor and Elfving’s algorithm needs to compute , and Byrne’s CQ algorithm needs to compute . However, they are difficult to calculate.

Consider the following convex minimization problem: where is differentiable and the gradient ∇f is L-Lipschitz with .

The gradient-projection algorithm [21] is the most effective method to solve (1.3). A sequence is generated by the recursive formula where the parameter . Then we know that Byrne’s CQ algorithm is a special case of the gradient-projection algorithm.

In Byrne’s CQ algorithm, depends on the operator norm . However, it is difficult to compute. In 2005, Yang [22] considered as follows: where and satisfies In 2012, López [23] introduced as follows: where . However, López’s algorithm only has weak convergence.

In 2013, Yao [24] introduced a self-adaptive method for the SFP and obtained a strong convergence theorem. However, the algorithm is difficult to work out.

In general, there are two types of algorithms to solve SFPs. One is the algorithm which depends on the norm of the operator. The other is the algorithm without a priori knowledge of the operator norm. The first type of algorithm needs to calculate , but is not easy to work out. The second type of algorithm also has a drawback. It always has weak convergence. If we want to obtain strong convergence, we have to use the composited iterative method, but then the algorithm is difficult to calculate. In order to overcome the drawbacks, we propose a new regularized CQ algorithm without a priori knowledge of the operator norm to solve the SFP and we obtain a strong convergence theorem.

Consider the following regularized minimization problem: where the regularization parameter . A sequence is generated by the formula where , , and , . Then, under suitable conditions, the sequence generated by (1.8) converges strongly to a point , where is the minimum-norm solution of SFP (1.1).

Preliminaries

In this part, we introduce some lemmas and some properties that are used in the rest of the paper. Throughout this paper, let and be real Hilbert spaces, be a bounded linear operator and I be the identity operator on or . If is a differentiable functional, then the gradient of f is denoted by ∇f. We use the sign ‘→’ to denote strong convergence and use the sign ‘⇀’ to denote weak convergence.

Definition 2.1

See [25]

Let D be a nonempty subset of H, and let . Then T is firmly nonexpansive if

Lemma 2.2

See [26]

Let be an operator. Then the following are equivalent:

T is firmly nonexpansive,

is firmly nonexpansive,

is nonexpansive,

, ,

.

Recall is an orthogonal projection, where C is a nonempty closed convex subset of H. Then to each point , the unique point satisfies the following property: also has the following characteristics.

Lemma 2.3

See [27]

For a given ,

, ,

, ,

, .

Lemma 2.4

See [28]

Let f be given by (1.4). Then

f is convex and differential,

, ,

f is w-lsc on H,

∇f is -Lipschitz: , .

Lemma 2.5

See [29]

Let be a sequence of nonnegative real numbers such that where is a sequence in and is a sequence in such that Then .

,

or .

Lemma 2.6

See [30]

Let be a sequence of real numbers such that there exists a subsequence of such that for all . Then there exists a nondecreasing sequence of such that and the following properties are satisfied by all (sufficiently large) numbers : In fact, is the largest number n in the set such that the condition holds.

Main results

In this paper, we always assume that is a real-valued convex function, where , the gradient , C and Q are nonempty closed convex subsets of real Hilbert spaces and , and is a bounded linear operator.

Algorithm 3.1

Choose an initial guess arbitrarily. Assume that the nth iterate has been constructed and . Then we calculate the th iterate via the formula where is chosen as follows: with . If , then is a solution of SFP (1.1) and the iterative process stops. Otherwise, we set and go to (3.1) to evaluate the next iterate .

Theorem 3.1

Suppose that and the parameters and satisfy the following conditions: Then the sequence generated by Algorithm 3.1 converges strongly to , where .

, , ,

for some small enough.

Proof

Let . Since minimization is an exactly fixed point of its projection mapping, we have and .

By (3.1) and the nonexpansivity of , we derive

Since is firmly nonexpansive, from Lemma 2.2, we deduce that is also firmly nonexpansive. Hence, we have

Note that . From (3.3), we obtain

By condition (ii), without loss of generality, we assume that for all . Thus from (3.2) and (3.4), we obtain Hence, is bounded.

Let . From (3.5), we deduce We consider the following two cases.

Case 1. One has for every large enough.

In this case, exists as finite and hence This, together with (3.6), implies that Since (where is a constant), we get Noting that is bounded, we deduce immediately that

Next, we prove that Since is bounded, there exists a subsequence satisfying and By the lower semicontinuity of f, we get So That is, ẑ is a minimizer of f, and . Therefore

Then we have Note that is bounded, and that . Thus by (3.8). From Lemma 2.5, we deduce that

Case 2. There exists a subsequence of such that By Lemma 2.6, there exists a strictly nondecreasing sequence of positive integers such that and the following properties are satisfied by all numbers :

We have Consequently, Hence,

By a similar argument to that of Case 1, we prove that where In particular, from (3.13), we get Since , we deduce that Then, from (3.14), we have Then Then, from (3.12), we deduce that Thus, from (3.11) and (3.16), we conclude that Therefore, . This completes the proof. □

Conclusion

Recently, the SFP has been studied extensively by many authors. However, some algorithms need to compute , and this is not an easy thing to work out. Others do not need to compute , but the algorithms always have weak convergence. If we want to obtain strong convergence theorems, the algorithms are complex and difficult to calculate. We try to get over the drawbacks. In this article, we use the regularized CQ algorithm without computing to find the minimum-norm solution of the SFP, where , . Then, under suitable conditions, the explicit strong convergence theorem is obtained.