1 Introduction

The split feasibility problem (SFP) is to find a point such that itself and its image under a linear transformation fall within two closed convex sets in the space and the image space, respectively. The SFP was introduced by Censor and Elfving in [1] to resolve the phase retrieval problem and provided a unified framework for vast application problems in various areas such as signal processing and image reconstruction [2, 3], intensity modulated radiation therapy [4] and systems biology [5, 6]. Various numerical algorithms have been developed to solve the SFP; see [2, 5, 7,8,9] and references therein. One of the most popular and practical algorithms is the CQ algorithm, which was proposed by Byrne [2, 7]. One can further refer to [10, 11] and references therein for the study of nonconvex feasibility problems.

The multiple-sets split feasibility problem (MSSFP) is a generalization of the SFP with the involved sets being the intersection of a series of closed convex sets. The MSSFP was introduced by Censor et al. [12] to resolve the intensity modulated radiation therapy treatment planning [4]. Many numerical algorithms have been developed to solve the MSSFP; see [12,13,14,15,16,17] and references therein. More precisely, Censor et al. in [12] proposed a projection gradient method, where the step size depends on the Lipschitz constant of the gradient operator. To avoid the inconvenience of calculating the Lipschitz constant, Zhao and Yang in [15] (also see [14]) introduced a self-adaptive projection method by adopting the Armijo-like search rule; Zhao and Yang in [16] further suggested a new self-adaptive method with an implementable dynamic step-size rule. Moreover, considering the MSSFP in Hilbert spaces, Xu in [13] proposed a fixed-point method with the step size relying on the Lipschitz constant of the gradient operator. Adopting the self-adaptive step-size rule, Wen et al. in [17] introduced the cyclic/simultaneous iteration methods for the MSSFP and proposed the relaxed cyclic/simultaneous iteration methods for the general MSSFP where the involved sets are given by the level sets of a series of convex functions. Global weak convergence results of these theorems were established therein. However, there are some gaps in the proofs of [17, Theorems 3.1 and 4.1] (as explained in Remarks 3.2 and 4.2), and some imperfections in [17, Theorems 3.2 and 4.2] (as explained in Remarks 3.3, 4.3 and Examples 3.1, 3.2).

In this paper, we consider a family of projection gradient methods for solving the MSSFP in Hilbert spaces, which include the cyclic/simultaneous iteration methods introduced in [17] as special cases. For the general case where the involved sets are given by level sets of a series of convex functions, the projections onto the level sets are not easily implemented in general. To avoid this difficulty, we introduce a family of relaxed projection gradient methods, in which the projections onto the approximated halfspaces are adopted in place of the ones onto the level sets; they cover the relaxed cyclic/simultaneous iteration methods introduced in [17]. Global weak convergence theorems of these methods will be established in the present paper. In particular, when applied to the cyclic/simultaneous iteration methods, our results fill the gaps and correct the imperfections that appeared in [17] mentioned above and hence extend and improve the corresponding results in [17]. This is also a motivation of this paper.

The rest of this paper is organized as follows. In Sect. 2, we introduce some basic notations and preliminary results that will be used in the remaining sections. In Sect. 3, we propose a family of projection gradient methods for solving the MSSFP and establish their global weak convergence. In Sect. 4, we investigate a family of relaxed projection gradient methods and explore their global weak convergence. Conclusions will be drawn in the last section.

2 Notations and Preliminary Results

The notations used in this paper are standard (cf. [18]). We consider a real Hilbert space \(\mathbb {H}\), associated with the inner product \(\langle \cdot ,\cdot \rangle \), and the induced norm \(\Vert \cdot \Vert \). As usual, \(\mathbb {N}^*\) denotes the set of all positive integers, and \(\mathbb {N}:=\mathbb {N}^*\cup \{0\}\). For \(\{x_n\}\subseteq \mathbb {H}\), \(x_n\rightharpoonup x\) denotes that \(\{x_n\}\) converges weakly to x. For \(C\subseteq \mathbb {H}\), the distance function of C and the projection operator onto C are denoted by \(\mathrm{d}_C:\mathbb {H}\rightarrow \mathbb {R}\) and \(\mathbb {P}_C:\mathbb {H}\rightarrow \mathbb {H}\), respectively, namely

$$\begin{aligned} \mathrm{d}_C(x) := \inf \{\Vert x - c\Vert : c \in C\}\quad \text{ for } \text{ each } x \in \mathbb {H}, \end{aligned}$$

and

$$\begin{aligned} \mathbb {P}_C(x) := \{ {\bar{x}}\in C : \Vert {\bar{x}} - x\Vert = \mathrm{d}_C(x)\} \quad \text{ for } \text{ each } x \in \mathbb {H}. \end{aligned}$$

Some useful properties about \(\mathbb {P}_C(\cdot )\) are presented in the following lemma; see [18].

Lemma 2.1

Let \(C \subseteq \mathbb {H}\) be nonempty, closed and convex. Then, the following assertions hold for any \(x,y \in \mathbb {H}\) and \(z \in C\):

  1. (i)

    \(\left\langle x-\mathbb {P}_C(x),z-\mathbb {P}_C(x) \right\rangle \le 0\);

  2. (ii)

    \(\Vert \mathbb {P}_C(x)-\mathbb {P}_C(y) \Vert ^2 \le \left\langle \mathbb {P}_C(x)-\mathbb {P}_C(y),x-y \right\rangle \);

  3. (iii)

    \(\Vert \mathbb {P}_C(x)-z \Vert ^2 \le \Vert x-z \Vert ^2-\Vert \mathbb {P}_C(x)-x \Vert ^2\).

Recall that a mapping \(T: \mathbb {H} \rightarrow \mathbb {H}\) is said to be nonexpansive, if

$$\begin{aligned} \Vert Tx-Ty \Vert \le \Vert x-y \Vert \quad \text{ for } \text{ each } x,y \in \mathbb {H}. \end{aligned}$$

Clearly, it follows from Lemma  2.1(ii) that \(\mathbb {P}_C\) is nonexpansive. The following demiclosedness principle of a nonexpansive mapping is well known in Hilbert spaces; see [18, Theorem 4.17]. Let \(\mathbb {I}\) be the identity operator in \(\mathbb {H}\), and let \(\mathrm{Fix}(T)\) denote the set of all fixed points of T, namely

$$\begin{aligned} \mathrm{Fix}(T):=\{x\in \mathbb {H}: T(x)=x\}. \end{aligned}$$

Lemma 2.2

Let \(C \subseteq \mathbb {H}\) be nonempty, closed and convex, and let \(T: C \rightarrow C\) be a nonexpansive mapping with \(\mathrm{Fix}(T)\ne \emptyset \). If \(\{x_n\}\subseteq C\) converges weakly to x and \(\{(\mathbb {I}-T)(x_n)\}\) converges strongly to y, then \((\mathbb {I}-T)(x)=y\). In particular, if \(y=0\), then \(x=T(x)\).

The following lemma (cf. [18, Theorem 5.5]) will be useful in convergence analysis of projection gradient methods.

Lemma 2.3

Let \(C \subseteq \mathbb {H}\) be nonempty, closed and convex. Let \(\left\{ x_n\right\} \subseteq \mathbb {H}\) satisfy that \(\lim _{n\rightarrow \infty } \Vert x_n-x\Vert \) exists for each \(x \in C\) and that each weak cluster point of \(\left\{ x_n\right\} \) belongs to C. Then, \(\left\{ x_n\right\} \) converges weakly to a point in C.

For a convex function \(f:\mathbb {H}\rightarrow \mathbb {R}\), the subdifferential of f at x is denoted by \(\partial f(x)\) and defined by

$$\begin{aligned} \partial f(x):=\{v\in \mathbb {H}:f(x)+\langle v,y-x\rangle \le f(y) \text{ for } \text{ each } y\in \mathbb {H}\}. \end{aligned}$$

f is said to be subdifferentiable at \(x\in \mathbb {H}\) if \(\partial f(x)\ne \emptyset \). It is shown in [18, Proposition 16.3] that the convex function is subdifferentiable everywhere.

Let \(\mathbb {H}_1\) and \(\mathbb {H}_2\) be two real Hilbert spaces, and let \(A: \mathbb {H}_1 \rightarrow \mathbb {H}_2\) be a bounded linear operator. Let \(t,r\in \mathbb {N}^*\), and consider two families of closed convex sets in \(\mathbb {H}_1\) and \(\mathbb {H}_2\): \(\{C_i: i=1,\dots ,t\}\) and \(\{Q_j: j=1,\dots ,r\}\), respectively. In the present paper, we consider the following multiple-sets split feasibility problem (MSSFP): Find a point \(x^*\in \mathbb {H}_1\) such that

$$\begin{aligned} x^* \in \bigcap _{i=1}^t C_i \quad \text{ and } \quad Ax^* \in \bigcap _{j=1}^r Q_j. \end{aligned}$$
(1)

In particular, if \(t=r=1\), then the MSSFP is reduced to the classical split feasibility problem (SFP). Let S denote the solution set of problem (1). Throughout the paper, we always assume that the MSSFP is consistent, that is, \(S\ne \emptyset \).

Let \(\beta _j > 0\) for \(j=1,\dots , r\). We recall that the proximity function associated with (1), which was introduced in Xu [13]:

$$\begin{aligned} p(x) :=\frac{1}{2} \sum _{j=1}^r \beta _j \Vert Ax-\mathbb {P}_{Q_j}(Ax) \Vert ^2 \quad \text{ for } \text{ each } x\in \mathbb {H}_1. \end{aligned}$$
(2)

The following lemma describes the property of \(p(\cdot )\); see [13, p. 2028].

Lemma 2.4

The function \(p(\cdot )\) is differentiable with its gradient given by

$$\begin{aligned} \nabla p(\cdot )=\sum _{j=1}^r \beta _j A^*(\mathbb {I}-\mathbb {P}_{Q_j})(A\cdot ), \end{aligned}$$
(3)

and the Lipschitz constant of \(\nabla p(\cdot )\) is \(L:=\Vert A\Vert ^2\sum _{j=1}^r\beta _j\).

We end this section by the following lemma, which will be useful in the sequel.

Lemma 2.5

Let \(x\in \mathbb {H}_1\), \(z\in S\), and \(\lambda >0\). Then,

$$\begin{aligned} \Vert x-\lambda \nabla p(x)-z\Vert ^2\le \Vert x-z\Vert ^2+\lambda ^2\Vert \nabla p(x)\Vert ^2-4\lambda p(x). \end{aligned}$$
(4)

Proof

Note that

$$\begin{aligned} \Vert x-\lambda \nabla p(x)-z\Vert ^2 =\Vert x-z\Vert ^2+\lambda ^2\Vert \nabla p(x)\Vert ^2-2\lambda \left\langle \nabla p(x),x-z\right\rangle . \end{aligned}$$
(5)

One has from (3) that

$$\begin{aligned} \langle \nabla p(x), x-z \rangle = \sum _{j=1}^r\beta _j \left\langle (\mathbb {I}-\mathbb {P}_{Q_j})(Ax),Ax-Az \right\rangle . \end{aligned}$$
(6)

For \(j=1,\dots ,r\), it follows from Lemma 2.1(i) that

$$\begin{aligned} \begin{aligned}&\left\langle (\mathbb {I}-\mathbb {P}_{Q_j}) Ax,Ax-Az \right\rangle \\&\quad =\Vert Ax-\mathbb {P}_{Q_j}(Ax)\Vert ^2+\left\langle Ax-\mathbb {P}_{Q_j}(Ax),\mathbb {P}_{Q_j}(Ax)-Az \right\rangle \\&\quad \ge \Vert Ax-\mathbb {P}_{Q_j}(Ax)\Vert ^2. \end{aligned} \end{aligned}$$

Combining this with (6) yields that

$$\begin{aligned} \left\langle \nabla p(x), x-z\right\rangle \ge \sum _{j=1}^r\beta _j\Vert Ax-\mathbb {P}_{Q_j}(Ax)\Vert ^2=2p(x). \end{aligned}$$

This, together with (5), implies (4). The proof is complete. \(\square \)

3 Projection Gradient Methods for MSSFP

In this section, for the case where the projections onto \(C_i\) and \(Q_j\) are easily implemented (e.g., they are halfspaces or box constraints), we propose a family of projection gradient methods for solving the MSSFP (1) and investigate its global weak convergence. Write \(I:=\left\{ 1,\dots ,t\right\} \) and recall \(p(\cdot )\) defined by (2). Throughout the paper, we adopt the convention that \(\frac{0}{0}:=0\). The projection gradient method for solving the MSSFP is formally stated as follows.

Algorithm 3.1

  • Step 0 Choose \(x_0\in \mathbb {H}_1\), \(\delta >0\), \(0< {\underline{\rho }}< {\overline{\rho }} <4\), and set \(n:=0\).

  • Step 1 Determine the step size \(\lambda _n\) by

    $$\begin{aligned} \lambda _n := \frac{\rho _n p(x_n)}{\Vert \nabla p(x_n)\Vert ^2}, \text{ where } {\underline{\rho }}\le \rho _n \le {\overline{\rho }}. \end{aligned}$$
    (7)

  • Step 2 Take the weights \(\{\omega _i^n\in {\mathbb {R}}_+: i\in I \}\) such that

    $$\begin{aligned} \sum _{i=1}^t\omega _i^n=1 \quad \text{ and }\quad \inf _{i\in I_n}\omega _i^n>\delta , \text{ where } I_n:=\left\{ i\in I : \omega _i^n>0\right\} . \end{aligned}$$
    (8)

  • Step 3 Set

    $$\begin{aligned} x_{n+1} := \sum _{i=1}^t\omega _i^n\mathbb {P}_{C_i}(x_n-\lambda _n \nabla p(x_n)). \end{aligned}$$

  • Step 4 Set \(n := n+1\) and go to Step 1.

Remark 3.1

Algorithm 3.1 includes the cyclic/simultaneous iteration methods introduced in [17] as special cases. In particular, we choose the weights as

$$\begin{aligned} \omega _i^n: = {\left\{ \begin{array}{ll} 1,&{}\text{ if } i=(n \text{ mod } t)+1,\\ 0,&{}\text {otherwise}. \end{array}\right. } \end{aligned}$$
(9)

Then, \(\{x_n\}\) can be regarded as a sequence generated by the cyclic iteration method. Let \(\{w_i\}\) be defined in [17, Theorem 3.2], and set

$$\begin{aligned} \omega _i^n: = w_i\quad \text{ for } \text{ each } i=1,\dots ,t. \end{aligned}$$
(10)

Then, \(\{x_n\}\) is identical to a sequence generated by the simultaneous iteration method.

The notion of set control was introduced in [19, 20] for feasibility problems, which is useful in convergence analysis of projection algorithms; see [8, 21, 22]. Here, we introduce a practical set control scheme for Algorithm 3.1, which will be useful in its convergence analysis.

Definition 3.1

Let \(\{x_n\}\) be a sequence generated by Algorithm 3.1, and let \(q\in \mathbb {N}^*\). We say that \(\{x_n\}\) satisfies the q-intermittent set control, if

$$\begin{aligned} I_n\cup \dots \cup I_{n+q-1}=I\quad \text{ for } \text{ each } n\in \mathbb {N}. \end{aligned}$$

The main result of this section is presented in the following theorem, which establishes the global weak convergence property of the sequence generated by Algorithm 3.1. Recall that S is the solution set of (1).

Theorem 3.1

Let \(\{x_n\}\) be a sequence generated by Algorithm 3.1. Suppose that there exists \(q\in \mathbb {N}^*\) such that \(\{x_n\}\) satisfies the q-intermittent set control. Then, \(\{x_n\}\) converges weakly to a solution of (1).

Proof

Let \(z\in S\) and \(u_n:=x_n-\lambda _n \nabla p(x_n)\). In view of Algorithm 3.1, we obtain by the convexity of \(\Vert \cdot \Vert ^2\) and Lemma 2.1(iii) that

$$\begin{aligned} \begin{aligned} \Vert x_{n+1}-z\Vert ^2&= \left\| \sum _{i=1}^t\omega _i^n(\mathbb {P}_{C_i}(u_n)-z)\right\| ^2\\&\le \sum _{i=1}^t\omega _i^n\Vert \mathbb {P}_{C_i}(u_n)-z\Vert ^2\\&\le \sum _{i=1}^t\omega _i^n(\Vert u_n-z\Vert ^2-\Vert \mathbb {P}_{C_i}(u_n)-u_n\Vert ^2)\\&\le \Vert u_n-z\Vert ^2- \Vert x_{n+1}-u_n\Vert ^2. \end{aligned} \end{aligned}$$
(11)

This, together with (7) and (4), implies that

$$\begin{aligned} \begin{aligned} \Vert x_{n+1}-z\Vert ^2&\le \Vert x_n-z\Vert ^2+ \lambda _n^2\Vert \nabla p(x_n)\Vert ^2-4\lambda _n p(x_n)\\&= \Vert x_n-z\Vert ^2-\rho _n(4-\rho _n)\frac{p^2(x_n)}{\Vert \nabla p(x_n)\Vert ^2}. \end{aligned} \end{aligned}$$
(12)

Since \(0< \rho _n <4\), it follows that \(\Vert x_{n+1}-z\Vert \le \Vert x_{n}-z\Vert \), which shows that \(\left\{ x_n\right\} \) is a bounded sequence and \(\lim _{n\rightarrow \infty }\Vert x_{n}-z\Vert \) exists. Let \(x^*\) be a weak cluster point of \(\left\{ x_n\right\} \). That is, there exists a subsequence \(\left\{ x_{n_l}\right\} \) such that \(x_{n_l}\rightharpoonup {x^*}\). Below, we will show that \({x^*}\in S\). Granting this, one has from Lemma 2.3 that \(\left\{ x_n\right\} \) converges weakly to a point in S; hence the conclusion follows.

To complete the proof, it suffices to show that \(x^*\in S\). By (12) and the fact that \(0<{\underline{\rho }}\le \rho _n \le {\overline{\rho }} <4\), we obtain

$$\begin{aligned} {\underline{\rho }}(4- {\overline{\rho }})\frac{p^2(x_n)}{\Vert \nabla p(x_n)\Vert ^2}\le \Vert x_{n}-z\Vert ^2-\Vert x_{n+1}-z\Vert ^2. \end{aligned}$$

Letting \(n\rightarrow \infty \), one has

$$\begin{aligned} \lim _{n\rightarrow \infty }\frac{p^2(x_n)}{\Vert \nabla p(x_n)\Vert ^2}=0. \end{aligned}$$
(13)

Since \(\nabla p\) is Lipschitz continuous (cf. Lemma 2.4) and \(\left\{ x_n\right\} \) is bounded, \(\left\{ \Vert \nabla p(x_n)\Vert \right\} \) is bounded. Hence, we conclude from (13) that \(\lim _{n\rightarrow \infty }p(x_n)=0\) and so by (2) that

$$\begin{aligned} \underset{n\rightarrow \infty }{\lim }\Vert Ax_n-\mathbb {P}_{Q_j}(Ax_n)\Vert = 0 \quad \text{ for } j=1,\dots ,r. \end{aligned}$$

Since \(x_{n_l}\rightharpoonup {x^*}\) and \(\mathrm{d}_{Q_j}(A\cdot )\) is lower semicontinuous (thanks to its convexity), it follows that

$$\begin{aligned} \mathrm{d}_{Q_j}(A{x^*})\le \underset{l\rightarrow \infty }{\liminf }\;\mathrm{d}_{Q_j}(Ax_{n_l})= 0, \end{aligned}$$

which implies \(A{x^*}\in Q_j\) for \(j=1,\dots ,r\); consequently, \(A{x^*}\in \bigcap _{j=1}^rQ_j\).

Finally, it remains to show that \({x^*}\in \bigcap _{i=1}^tC_i\). Note that \(x_{n_l}\rightharpoonup {x^*}\) and it follows from (7) and (13) that

$$\begin{aligned} \lim _{n\rightarrow \infty } \lambda _n\Vert \nabla p(x_n)\Vert =0. \end{aligned}$$
(14)

This, together with the definition of \(u_n\), implies that \(u_{n_l}\rightharpoonup {x^*}\) and

$$\begin{aligned} \lim _{n\rightarrow \infty }\Vert u_n-z\Vert ^2= \lim _{n\rightarrow \infty }\Vert x_n-z\Vert ^2. \end{aligned}$$
(15)

Observing from Lemma 2.1(iii) and the convexity of \(\Vert \cdot \Vert ^2\), we obtain

$$\begin{aligned} \begin{aligned} \sum _{i=1}^t\omega _i^n \Vert \mathbb {P}_{C_i}(u_n)-u_n \Vert ^2&\le \Vert u_n-z \Vert ^2-\sum _{i=1}^t\omega _i^n \Vert \mathbb {P}_{C_i}(u_n)-z \Vert ^2\\&\le \Vert u_n-z \Vert ^2- \Vert x_{n+1}-z \Vert ^2. \end{aligned} \end{aligned}$$

Combining this with (15) yields

$$\begin{aligned} \lim _{n\rightarrow \infty }\sum _{i=1}^t\omega _i^n\Vert \mathbb {P}_{C_i}(u_n)-u_n\Vert ^2=0. \end{aligned}$$
(16)

Fix \(i\in I\) and \(n_l\in {\mathbb {N}}\). Since \(\{x_n\}\) satisfies the q-intermittent set control, one has

$$\begin{aligned} I_{n_l}\cup \dots \cup I_{{n_l}+q-1}=I. \end{aligned}$$

Hence, there exists \(q_i^l \in \left\{ 0,1,\dots ,q-1 \right\} \) such that \(i\in I_{{n_l}+q_i^l}\) and so

$$\begin{aligned} \omega _{i}^{n_l+q_i^l}>\delta >0. \end{aligned}$$

Let \({{ {\widetilde{n}}_l}}:=n_l+q_i^l\). Note that, for each \({ {\widetilde{n}}_l}\),

$$\begin{aligned} \begin{aligned} \delta \Vert \mathbb {P}_{C_{i}}(u_{{{{\widetilde{n}}_l}}})-u_{{{ {\widetilde{n}}_l}}}\Vert&\le \sum _{i=1}^{t}\omega _i^{{{ {\widetilde{n}}_l}}}\Vert \mathbb {P}_{C_{i}}(u_{{{ {\widetilde{n}}_l}}})-u_{{{ {\widetilde{n}}_l}}}\Vert , \\ \end{aligned} \end{aligned}$$

which, together with (16), implies

$$\begin{aligned} \lim _{l\rightarrow \infty }\Vert \mathbb {P}_{C_{i}}(u_{{{ {\widetilde{n}}_l}}})-u_{{{ {\widetilde{n}}_l}}}\Vert = 0. \end{aligned}$$
(17)

Below, we will show that \( u_{{{ {\widetilde{n}}_l}}} \rightharpoonup {x^*}\). Granting this, it follows from (17) and Lemma 2.2 (with \(\mathbb {P}_{C_i}\) in place of T) that \({x^*}\in C_{i}\). As \(i\in I\) is arbitrary, one has \({x^*}\in \bigcap _{i=1}^t C_{i}\), as desired.

To proceed, we obtain by (11) and (15) that

$$\begin{aligned} \lim _{n\rightarrow \infty }\Vert x_{n+1}-u_n\Vert ^2=0. \end{aligned}$$
(18)

Note by the definition of \(u_n\) that

$$\begin{aligned} \Vert x_{n+1}-u_n\Vert ^2=\Vert x_{n+1}-x_n\Vert ^2+\lambda _n^2\Vert \nabla p(x_n)\Vert ^2+2\left\langle x_{n+1}-x_n,\lambda _n\nabla p(x_n)\right\rangle . \end{aligned}$$

Letting \(n\rightarrow \infty \), we get from (18) and (14) that \(\lim _{n\rightarrow \infty }\Vert x_{n+1}-x_n\Vert =0\). Then, it follows from the definition of \(u_n\) and (14) that

$$\begin{aligned} \lim _{n\rightarrow \infty }\Vert u_{n+1}-u_n\Vert =0. \end{aligned}$$
(19)

Let \(v\in \mathbb {H}_1\). For each \(n_l\in {\mathbb {N}}\), one has

$$\begin{aligned} \left\langle u_{{{ {\widetilde{n}}_l}}}-{x^*},v \right\rangle =\left\langle u_{{{ {\widetilde{n}}_l}}}-u_{{{ {\widetilde{n}}_l}}-1},v \right\rangle +\dots +\left\langle u_{n_l}-{x^*},v \right\rangle . \end{aligned}$$

Since \( u_{n_l}\rightharpoonup {x^*}\) and \({\widetilde{n}}_l-n_l\le q\), it follows from (19) that

$$\begin{aligned} \lim _{l\rightarrow \infty }\left\langle u_{{{ {\widetilde{n}}_l}}}-{x^*},v \right\rangle =0, \end{aligned}$$

and thus, \( u_{{{ {\widetilde{n}}_l}}} \rightharpoonup {x^*}\) because \(v\in \mathbb {H}_1\) is arbitrary. The proof is complete. \(\square \)

By Remark 3.1 and Theorem 3.1, the convergence results of the iteration projection methods proposed in [17] can be directly obtained as follows.

Corollary 3.1

Let \(\{x_n\}\) be a sequence generated by Algorithm 3.1 with \(\{\omega _i^n\}\) given by (9). Then, \(\{x_n\}\) converges weakly to a solution of (1).

Proof

One can verify by (9) that \(I_n\cup \dots \cup I_{n+t-1}=I\) for each \(n\in \mathbb {N}\); consequently, \(\{x_n\}\) satisfies the t-intermittent set control. Hence, Theorem 3.1 is applicable and the conclusion follows. \(\square \)

Corollary 3.2

Let \(\{w_i\}_{i=1}^t\subset ]0,1[\) be such that \(\sum _{i=1}^tw_i=1\). Let \(\{x_n\}\) be a sequence generated by Algorithm 3.1 with \(\{\omega _i^n\}\) given by (10). Then, \(\{x_n\}\) converges weakly to a solution of (1).

Proof

By the assumption, it is easy to see by (10) that \(I_n=I\) for each \(n\in \mathbb {N}\), namely, \(\{x_n\}\) satisfies the 1-intermittent set control. Thus, Theorem 3.1 is applicable and the conclusion follows. \(\square \)

Remark 3.2

As mentioned in Remark 3.1, a sequence generated by Algorithm 3.1 with \(\{\omega _i^n\}\) given by (9) is identical to the one by the cyclic iteration method, whose weak convergence result was shown in [17, Theorem 3.1]. However, there is some gap in the proof of [17, Theorem 3.1]. More precisely, the authors claimed that “Notice that the pool\(\{1,\dots ,t\}\)is finite, then for any\(i\in \{1,\dots ,t\}\), we can choose a subsequence\(\{n_{k_l}\}\subset \{n_k\}\)such that\((n_{k_l}~\mathrm{mod}~t)=i\),” whereas this is not true if the subsequence is \(\{n_{k}:n_{k}:=tk+i_0\}\) for some fixed \(i_0\) or some other relevant cases. Corollary 3.1 fills the gap, and Theorem 3.1 extends the result of [17, Theorem 3.1].

Remark 3.3

As mentioned in Remark 3.1, a sequence generated by Algorithm 3.1 with \(\{\omega _i^n\}\) given by (10) is identical to the one by the simultaneous iteration method, which was discussed in [17, Theorem 3.2]. In particular, the condition that \(\{w_i\}_{i=1}^t\subset [0,1]\) is assumed in [17, Theorem 3.2]. As shown in Examples 3.1 and 3.2 below, [17, Theorem 3.2] may fail to (weakly) converge to a solution of (1) if \(\{w_i\}_{i=1}^t\subset [0,1]\) is assumed. Our Corollary 3.2 and Theorem 3.1 correct and extend [17, Theorem 3.2], respectively.

We end this section by providing two counter examples of [17, Theorem 3.2], in which a sequence generated by the simultaneous iteration method with \(\{w_i\}_{i=1}^t\subset [0,1]\) fails to (weakly) converge to a solution of (1). It should be remarked that \(\nabla p(x_n)=0\) and \(\nabla p(x_n)\ne 0\) for each \(n\in {\mathbb {N}}\) in Examples 3.1 and 3.2, respectively.

Example 3.1

Consider problem (1) with \(t:=3\), \(r:=1\), \(A:=\mathbb {I}\) (the identity matrix in \({\mathbb {R}}^2\)) and

$$\begin{aligned}&C_1:=[0,4] \times [0,4],\; C_2:=[2,6] \times [2,6],\; C_3:=[1,3] \times [2,6],\; \\&Q:=[0,4] \times [2,6]. \end{aligned}$$

Clearly, the solution set of this problem is \(S=[2,3] \times [2,4]\). Recall in [17, Theorem 3.2] that the simultaneous iteration method is presented as

$$\begin{aligned} x_{n+1}:= \sum _{i=1}^3\omega _i\mathbb {P}_{C_i}(x_n-\lambda _n \nabla p(x_n))\quad \text{ for } \text{ each } n\ge 0, \end{aligned}$$
(20)

where \(\omega _i \in [0,1]\) such that \( \omega _1+ \omega _2+\omega _3= 1\), and \(\lambda _n\) is given by (7). Select parameters \(\beta =1\), \(\omega _1=\omega _2=\frac{1}{2}\), \(\omega _3=0\) and \(\rho _n\equiv 2\), and take an initial point \(x_0=(4,6)^\top \). Then, by (2) and (7), one has \(\nabla p(x_n)=0\) and \(\lambda _n=0\) for each \(n\in {\mathbb {N}}\), and thus, \(\{x_n\}\) is well generated and

$$\begin{aligned} x_n=\left( 4,4+\frac{1}{2^{n-1}}\right) ^\top \quad \text{ for } \text{ each } n\in {\mathbb {N}}. \end{aligned}$$

Consequently, \(\{x_n\}\) converges to \((4,4)^\top \notin S\). Hence, [17, Theorem 3.2] fails in this example.

Example 3.2

Consider problem (1) with \(t=r=2\), \(A:=\mathbb {I}\) and

$$\begin{aligned} C_1&:=\{ (x_1,x_2) \in {\mathbb {R}}^2 : -1\le x_1 \le 1, x_1=x_2\},\\ C_2&:=\{ (x_1,x_2) \in {\mathbb {R}}^2 : -2\le x_1 \le 0, -2\le x_2 \le 0, -2\le x_1+x_2\le -1\},\\ Q_1&:=[-1,0] \times [-1,1],\quad Q_2:=[-1,1] \times [-1,0]. \end{aligned}$$

Then, the solution set of (1) is \(S=\{(x_1,x_2) \in {\mathbb {R}}^2 : -1\le x_1 \le -\frac{1}{2}, x_1=x_2\}\). Select parameters \(\beta _1=\beta _2=\frac{1}{2}\), \(\omega _1=1\), \(\omega _2=0\) and \(\rho _n\equiv 1\), and take an initial point \(x_0=(1,1)^\top \). Then, by (2) and (20), one has \(\nabla p(x_n)\ne 0\) and \(\lambda _n\) is well defined for each \(n\in {\mathbb {N}}\), and thus, \(\{x_n\}\) is well generated and

$$\begin{aligned} x_n=(2^{-n},2^{-n})^\top \quad \text{ for } \text{ each } n\in {\mathbb {N}}. \end{aligned}$$

Consequently, \(\{x_n\}\) converges to the origin, which is not a solution of (1). Hence, [17, Theorem 3.2] fails in this example.

4 Relaxed Projection Gradient Methods for MSSFP

In this section, we consider a general case of the MSSFP, where \(\left\{ C_i\right\} _{i=1}^t\) and \(\left\{ Q_j\right\} _{j=1}^r\) in (1) are given by level sets of convex functions. Throughout this section, we assume that each \(c_i: \mathbb {H}_1 \rightarrow {\mathbb {R}}\) and \(q_j: \mathbb {H}_2 \rightarrow {\mathbb {R}}\) are convex functions and the sets \(C_i\) and \(Q_j\) are given, respectively, by

$$\begin{aligned} C_i:=\left\{ x \in \mathbb {H}_1 :c_i(x)\le 0 \right\} \quad \text{ and } \quad Q_j:=\left\{ y \in \mathbb {H}_2 :q_j(y)\le 0 \right\} . \end{aligned}$$
(21)

We assume that \(\partial c_i\) and \(\partial q_j\) are bounded operators (i.e., bounded on any bounded set) for each \(i=1,\dots ,t\) and \(j=1,\dots ,r\).

In this situation, the projections onto \(C_i\) and \(Q_j\) are not easily implemented in general. To avoid this difficulty, we introduce a family of relaxed projection gradient methods, in which the projections onto the approximated halfspaces are adopted in place of the projections onto \(C_i\) and \(Q_j\) (as in Algorithm 3.1). In particular, letting \(n\in {\mathbb {N}}\), \(i\in \{1,\dots ,t\}\) and \(j\in \{1,\dots ,r\}\), we use \(C_i^n\) and \(Q_j^n\) to denote the approximated halfspaces of \(C_i\) and \(Q_j\) at \(x_n\), i.e.,

$$\begin{aligned} C_i^n :=\left\{ x \in \mathbb {H}_1 :c_i(x_n)+\left\langle \xi _i^n,x-x_n \right\rangle \le 0 \right\} , \quad \text{ where } \xi _i^n \in \partial c_i(x_n), \end{aligned}$$

and

$$\begin{aligned} Q_j^n :=\left\{ y \in \mathbb {H}_2 :q_j(Ax_n)+\left\langle \eta _j^n,y-Ax_n \right\rangle \le 0 \right\} ,\quad \text{ where } \eta _j^n \in \partial q_j(Ax_n), \end{aligned}$$

respectively. By the definition of subgradient, it follows that \(C_i \subseteq C_i^n\) and \(Q_j \subseteq Q_j^n\). As in [17], letting \(\beta _j > 0\) for \(j=1,\dots , r\), we define the following (relaxed) proximity function

$$\begin{aligned} p_n(x) :=\frac{1}{2} \sum _{j=1}^r \beta _j \Vert Ax-\mathbb {P}_{Q_j^n}(Ax) \Vert ^2\quad \text{ for } \text{ each } x\in \mathbb {H}_1. \end{aligned}$$
(22)

It follows from Lemma 2.4 that \(p_n(\cdot )\) is differentiable with its gradient given by

$$\begin{aligned} \nabla p_n(x)=\sum _{j=1}^r \beta _j A^*(\mathbb {I}-\mathbb {P}_{Q_j^n})(Ax). \end{aligned}$$

Recall that \(I:=\left\{ 1,\dots ,t\right\} \). The relaxed projection gradient method for solving the MSSFP (1) with \(\{C_i\}\) and \(\{Q_j\}\) given by (21) is formulated as follows.

Algorithm 4.1

  • Step 0 Choose \(x_0\in \mathbb {H}_1\), \(\delta >0\), \(0< \underline{{\rho }}< \overline{{\rho }} <4\), and set \(n:=0\).

  • Step 1 Choose the step size \(\lambda _n\) by

    $$\begin{aligned} \lambda _n := \frac{\rho _n p_n(x_n)}{\Vert \nabla p_n(x_n)\Vert ^2}, \text{ where } {\underline{\rho }}\le \rho _n \le {\overline{\rho }}. \end{aligned}$$

  • Step 2 Take the weights \(\{\omega _i^n\in {\mathbb {R}}_+: i\in I \}\) satisfying (8).

  • Step 3 Set

    $$\begin{aligned} x_{n+1}: = \sum _{i=1}^t\omega _i^n\mathbb {P}_{C_i^n}(x_n-\lambda _n \nabla p_n(x_n)). \end{aligned}$$

  • Step 4 Set \(n := n+1\) and go to Step 1.

Remark 4.1

Algorithm 4.1 includes the relaxed cyclic/simultaneous iteration methods introduced in [17] as special cases where the weights \(\{\omega _i^n: i\in I \}\) are chosen by (9) and (10), respectively.

The main result of this section is stated in the following theorem, in which we prove the global weak convergence of the sequence generated by Algorithm 4.1. Recall that S is the solution of (1).

Theorem 4.1

Let \(\{x_n\}\) be a sequence generated by Algorithm 4.1. Suppose that there exists \(q\in \mathbb {N}^*\) such that \(\{x_n\}\) satisfies the q-intermittent set control. Then, \(\{x_n\}\) converges weakly to a solution of (1).

Proof

Let \(z\in S\) and \(u_n:=x_n-\lambda _n \nabla p_n(x_n)\). Following the same lines as in the beginning of the proof of Theorem  3.1, we can deduce that \(\lim \nolimits _{n\rightarrow \infty }\Vert x_{n}-z\Vert \) exists and

$$\begin{aligned} \lim _{n\rightarrow \infty }p_n(x_n)=0\quad \text{ and } \quad \lim _{n\rightarrow \infty }\Vert u_{n}-x_n\Vert =0. \end{aligned}$$
(23)

Let \(x^*\) be a weak cluster point of \(\left\{ x_n\right\} \), namely, there exists a subsequence \(\left\{ x_{n_k}\right\} \) such that \(x_{n_k}\rightharpoonup {x^*}\). Below, we will show that \({x^*}\in S\). Granting this, one has from Lemma 2.3 that \(\left\{ x_n\right\} \) converges weakly to a point in S; hence the conclusion follows.

To accomplish the proof, it suffices to show that \({x^*}\in S\). Note by (22) and the first equality of (23) that

$$\begin{aligned} \underset{n\rightarrow \infty }{\lim }\Vert Ax_n-\mathbb {P}_{Q_j^n}(Ax_n)\Vert = 0 \quad \text{ for } j=1,\dots ,r. \end{aligned}$$
(24)

By assumption that \(\partial q_j\) is bounded on bounded set and noting that \(\left\{ x_n\right\} \) is bounded, there exists \(\eta >0\) such that \(\Vert \eta _j^n \Vert \le \eta \) for each \(n\in {\mathbb {N}}\) and \(j=1,\dots ,r\). Hence, one has by the definition of \(Q_j^n\) that

$$\begin{aligned} q_j(Ax_n)\le \langle \eta _j^n,Ax_n-\mathbb {P}_{Q_j^n}(Ax_n) \rangle \le \eta \Vert Ax_n-\mathbb {P}_{Q_j^n}(Ax_n)\Vert . \end{aligned}$$
(25)

By the convexity of \(q_j\), it is weakly lower semicontinuous, and then, we obtain from (24) and (25) that \(q_j(A{x^*})\le {\liminf }_{k\rightarrow \infty } q_j(Ax_{n_k})\le 0\), which says \(A{x^*}\in Q_j\) for \(j=1,\dots ,r\); consequently, \(A{x^*}\in \bigcap _{j=1}^rQ_j.\)

Finally, it remains to show that \(x^*\in \bigcap _{i=1}^tC_i\). Fix \(i\in I\). Following the same arguments as in the proof of Theorem  3.1 [cf. (18)], we can show that there exists a subsequence \(\{{\widetilde{n}}_k\}\) such that

$$\begin{aligned} \lim _{k\rightarrow \infty }\Vert \mathbb {P}_{C_{i}^{{ {\widetilde{n}}_k}} }(u_{{ {\widetilde{n}}_k}} )-u_{{ {\widetilde{n}}_k}} \Vert = 0. \end{aligned}$$
(26)

and \( x_{{ {\widetilde{n}}_k}} \rightharpoonup {x^*}\). Note that there exists \(\zeta >0\) such that \(\Vert \xi _{i}^{{ {\widetilde{n}}_k}} \Vert \le \zeta \) for each \(k\in \mathbb {N}\) (by the assumption that \(\partial c_{i}\) is bounded on bounded set); consequently, one has by the definition of \(C_j^n\) that

$$\begin{aligned} \begin{aligned} c_{i}(x_{{ {\widetilde{n}}_k}} )&\le \langle \xi _{i}^{{ {\widetilde{n}}_k}} ,x_{{ {\widetilde{n}}_k}} -\mathbb {P}_{C_{i}^{{ {\widetilde{n}}_k}} }(u_{{ {\widetilde{n}}_k}} )\rangle \\&\le \zeta (\Vert x_{{ {\widetilde{n}}_k}} - u_{{{ {\widetilde{n}}_l}}} \Vert +\Vert u_{{ {\widetilde{n}}_k}} -\mathbb {P}_{C_{i}^{{ {\widetilde{n}}_k}} }(u_{{ {\widetilde{n}}_k}} ) \Vert ). \end{aligned} \end{aligned}$$

By the lower semicontinuity of \(c_i(\cdot )\) (thanks to its convexity) and \(x_{{\widetilde{n}}_k} \rightharpoonup {x^*}\), this, together with (26) and the second equality of (23), yields that

$$\begin{aligned} c_{i}({x^*})\le \underset{k\rightarrow \infty }{\liminf }\ c_{i}(x_{{ {\widetilde{n}}_k}} )\le 0; \end{aligned}$$

consequently, \({x^*}\in C_{i}\). As \(i\in I\) is arbitrary, it follows that \({x^*}\in \bigcap _{i=1}^t C_{i}\), as desired. The proof is complete. \(\square \)

By applying Remark 4.1 and Theorem 4.1, the convergence results of the relaxed cyclic/simultaneous iteration methods proposed in [17] are directly obtained as follows.

Corollary 4.1

Let \(\{x_n\}\) be a sequence generated by Algorithm 4.1 with \(\{\omega _i^n\}\) given by (9). Then, \(\{x_n\}\) converges weakly to a solution of (1).

Corollary 4.2

Let \(\{w_i\}_{i=1}^t\subset ]0,1[\) such that \(\sum _{i=1}^tw_i=1\). Let \(\{x_n\}\) be a sequence generated by Algorithm 4.1 with \(\{\omega _i^n\}\) given by (10). Then, \(\{x_n\}\) converges weakly to a solution of (1).

Remark 4.2

As mentioned in Remark 4.1, a sequence generated by Algorithm 4.1 with \(\{\omega _i^n\}\) given by (9) is identical to the one by the relaxed cyclic iteration method, whose weak convergence result was shown in [17, Theorem 4.1]. However, there is some gap in the proof of [17, Theorem 4.1]. More precisely, the authors claimed that “Since the pool of convex sets\(\{C_i\}_{i=1}^t\)is finite. For any\(i\in \{1,\dots ,t\}\), we can choose a subsequence\(\{n_{k_l}\}\subset \{n_k\}\)such that\((n_{k_l}~\mathrm{mod}~t)=i\),” whereas this is not true if the subsequence is \(\{n_{k}:n_{k}:=tk+i_0\}\) for some fixed \(i_0\) or some other relevant cases. Our Corollary 4.1 fills the gap, and Theorem 4.1 extends the result in [17, Theorem 4.1].

Remark 4.3

As mentioned in Remark 4.1, a sequence generated by Algorithm 4.1 with \(\{\omega _i^n\}\) given by (10) is identical to the one by the relaxed simultaneous iteration method, which was discussed in [17, Theorem 4.2]. The condition \(\{w_i\}_{i=1}^t\subset [0,1]\) is assumed in [17, Theorem 4.2]; however, the convergence may fail under this assumption (see Example 3.1). Hence, our Corollary 4.2 and Theorem 4.1 correct and extend [17, Theorem 4.2], respectively.

5 Conclusions

The multiple-sets split feasibility problem (MSSFP) is a generalization of the split feasibility problem, which has attracted a great amount of attention in numerical algorithms and has been widely applied in various fields. In the present paper, a family of projection gradient methods for the MSSFP were proposed for the cases where the projections onto the involved sets are easily implemented, which include the cyclic/simultaneous iteration methods introduced in [17] as special cases. For the general case where the involved sets are given by level sets of convex functions, we proposed a family of relaxed projection gradient methods (with projections onto the approximated halfspaces in place of the ones onto the level sets), which cover the relaxed cyclic/simultaneous iteration method introduced in [17] as special cases. Global weak convergence theorems of these methods were established in the present paper. In particular, as their direct applications, our results fill some gaps and imperfections and improve/extend the corresponding results in [17].