An Extrapolated Iterative Algorithm for Multiple-Set Split Feasibility Problem

Abstract

The multiple-set split feasibility problem (MSSFP), as a generalization of the split feasibility problem, is to find a point in the intersection of a family of closed convex sets in one space such that its image under a linear transformation will be in the intersection of another family of closed convex sets in the image space. Censor et al. (2005) proposed a method for solving the multiple-set split feasibility problem (MSSFP), whose efficiency depends heavily on the step size, a fixed constant related to the Lipschitz constant of ∇𝑝(𝑥) which may be slow. In this paper, we present an accelerated algorithm by introducing an extrapolated factor to solve the multiple-set split feasibility problem. The framework encompasses the algorithm presented by Censor et al. (2005). The convergence of the method is investigated, and numerical experiments are provided to illustrate the benefits of the extrapolation.

1. Introduction

The multiple-set split feasibility problem (MSSFP) is to find a point𝑥∈𝐶=^𝑡_𝑖=1𝐶_𝑖suchthat𝐴𝑥∈𝑄=^𝑟_𝑗=1𝑄_𝑗, (1.1) where 𝑡 and 𝑟 are positive integers, 𝐶_𝑖⊂ℜ^𝑁,𝑖=1,…,𝑡 and 𝑄_𝑗⊂ℜ^𝑀,𝑗=1,…,𝑟 are closed convex, 𝐴 is an 𝑀×𝑁 real matrix. When 𝑡=𝑟=1, the problem becomes to find a point 𝑥∈𝐶 and 𝐴𝑥∈𝑄, which is just the two-set split feasibility problem (SFP, for short). SFP was originally introduced in [1] allowing for constraints both in the domain and range of a linear operator. Many methods have been developed for solving the SFP, for example, the basic CQ algorithm proposed by Byrne [2], the relaxed CQ algorithm presented by Yang [3] and the KM-CQ-like algorithm developed by Dang and Gao [4]. The MSSFP, formulated in [5], arises in the field of intensity-modulated radiation therapy when one attempts to describe physical does constraints and equivalent uniform does (EUD) constraints within a single model, see [6]. Censor et al. generalized the CQ algorithm [2] to solve the MSSFP [5] to get the following iterative process:𝑥^𝑘+1=𝑥^𝑘−𝑠𝐿⋅∑^𝑡_𝑖=1𝛼_𝑖𝑥^𝑘−𝑃_𝐶𝑖𝑥^𝑘+∑^𝑟_𝑗=1𝛽_𝑗𝐴^𝑇𝐴𝑥^𝑘−𝑃_𝑄𝑗𝐴𝑥^𝑘, (1.2) where 0<𝑠<2, 𝐿=∑^𝑡_𝑖=1𝛼_𝑖+𝜌(𝐴^𝑇𝐴)∑^𝑟_𝑗=1𝛽_𝑗 and 𝜌(𝐴^𝑇𝐴) is the spectral radius of 𝐴^𝑇𝐴, and 𝛼_𝑖>0,𝛽_𝑗>0, for all 𝑖 and 𝑗 with∑^𝑡_𝑖=1𝛼_𝑖+∑^𝑟_𝑗=1𝛽_𝑗=1. Let 𝑃_𝑆 denote the projection onto the convex set 𝑆, that is,𝑃_𝑆𝑥=argmin_𝑦∈𝑆𝑥−𝑦. (1.3) There also came out other algorithms for solving MSSFP, such that Xu in [7] and Masad and Reich in [8] introduced strong convergence methods in infinite dimensional Hilbert space, respectively. Censor et al. in [9] presented the perturbed projection and simultaneous subgradient projection algorithm to deal with the limit of accurately computing the orthogonal projection, Censor and Segal proposed string-averaging algorithmic scheme for sparse case in [10] and employed product space formulation to derive and analyze the simultaneous algorithm for MSSFP in [11]. However, the above algorithms use a fixed stepsize related to the largest eigenvalue of the matrix 𝐴^𝑇𝐴, which sometimes affects the convergence speed of the algorithms.

Extrapolated iterative method was first proposed in [12], it is an accelerated method in optimization since Pierra observed that the extrapolation parameter can be much larger than 1 and that the sequence generated by the extrapolated method converges fast. Subsequently, Heinz et al. in [13], proposed a general parallel block-iterative algorithmic framework by introducing extrapolated overrelaxations to solve the affine-convex feasibility problems, the corresponding numerical results also show the fast convergence.

Motivated by the extrapolated method for solving the affine-convex feasibility problems, in this paper, we present an extrapolated iterative method to solve the MSSFP, which includes the algorithm proposed by Censor et al. in [5]. As will be shown our algorithm extends and includes as a special case of the method in [5].

The paper is organized as follows. Section 2 reviews some preliminaries. Section 3 gives an extrapolated algorithm and shows its convergence. Section 4 provides some numerical experiments.

2. Preliminaries

Under normal circumstances, the MSSFP considers both the feasible and the infeasible cases by the use of a proximity function, that is, if the MSSFP problem is consistent then unconstrained minimization of the proximity function yields the value 0; in the inconsistent case, it finds a point which is least violating the feasibility by being “closest” to all sets, as “measured” by the proximity function. The minimization problem is1min2𝑥−𝑃_𝐶(𝑥)²+12𝐴𝑥−𝑃_𝑄(𝐴𝑥)². (2.1)

We know that the projections of a point onto the sets 𝐶 and 𝑄 are difficult to implement, even if each individual sets 𝐶_𝑖 and 𝑄_𝑗 have simple or special structures such that projection onto each of them is easy to implement. In practical applications, the projections onto individual sets 𝐶_𝑖 are more easily calculated than the projection onto the intersection 𝐶. For this purpose, Censor et al. [5] introduced the proximity function 𝑝(𝑥), to measure the distance of a point to all sets. We have𝑝(𝑥)1∶=2^𝑡_𝑖=1𝛼_𝑖𝑃_𝐶𝑖(𝑥)−𝑥²+12^𝑟_𝑗=1𝛽_𝑗𝑃_𝑄𝑗(𝐴𝑥)−𝐴𝑥², (2.2) where 𝛼_𝑖>0,𝛽_𝑗>0, for all 𝑖 and 𝑗 with∑^𝑡_𝑖=1𝛼_𝑖+∑^𝑟_𝑗=1𝛽_𝑗=1. Then,∇𝑝(𝑥)=^𝑡_𝑖=1𝛼_𝑖𝑥^𝑘−𝑃_𝐶𝑖𝑥^𝑘+^𝑟_𝑗=1𝛽_𝑗𝐴^𝑇𝐴𝑥^𝑘−𝑃_𝑄𝑗𝐴𝑥^𝑘, (2.3)

Hence, (1.2) can be rewritten as𝑥^𝑘+1=𝑥^𝑘−𝑠⋅1𝐿⋅∇𝑝𝑥^𝑘. (2.4)

The lemma provides well-known properties of orthogonal projections.

Lemma 2.1 (see [14]). Let 𝑆 be a nonempty closed convex subset of ℜ^𝑁, for any 𝑥,𝑦∈ℜ^𝑁 and any 𝑧∈𝑆, the following properties hold:(1)𝑥∈𝑆⇔𝑃_𝑆(𝑥)=𝑥, (2)⟨𝑥−𝑃_𝑆(𝑥),𝑧−𝑃_𝑆(𝑥)⟩≤0, (3)‖𝑃_𝑆(𝑥)−𝑧‖²≤‖𝑥−𝑧‖²−‖𝑃_𝑆(𝑥)−𝑥‖², (4)‖𝑃_𝑆(𝑥)−𝑃_𝑆(𝑦)‖≤‖𝑥−𝑦‖.

3. The Extrapolated Projection Algorithm and Its Convergence

The following is our extrapolated projection algorithm.

Algorithm 3.1. For an arbitrary initial point 𝑥⁰,{𝑥^𝑘}_𝑘≥0 is generated by the iteration 𝑥^𝑘+1=𝑥^𝑘1+𝑠max𝐿,𝜆_𝑘^𝑡_𝑖=1𝛼_𝑖𝑃_𝐶𝑖𝑥^𝑘−𝑥^𝑘+^𝑟_𝑗=1𝛽_𝑗𝐴^𝑇𝑃_𝑄𝑗𝐴𝑥^𝑘−𝐴𝑥^𝑘, (3.1) where 𝑠 is a positive scalar such that 0<𝑠<2, 𝛼_𝑖>0,𝛽_𝑗>0 for all 𝑖 and 𝑗 with ∑^𝑡_𝑖=1𝛼_𝑖+∑^𝑟_𝑗=1𝛽_𝑗=1, 𝐿=∑^𝑡_𝑖=1𝛼_𝑖+𝜌(𝐴^𝑇𝐴)∑^𝑟_𝑗=1𝛽_𝑗 and 𝜌(𝐴^𝑇𝐴) being the spectral radius of 𝐴^𝑇𝐴, 𝜆_𝑘=∑^𝑡_𝑖=1𝛼_𝑖𝑃_𝐶𝑖𝑥^𝑘−𝑥^𝑘2+∑^𝑟_𝑗=1𝛽_𝑗𝑃_𝑄𝑗𝐴𝑥^𝑘−𝐴𝑥^𝑘2∑^𝑡_𝑖=1𝛼_𝑖𝑃_𝐶𝑖𝑥^𝑘−𝑥^𝑘+∑^𝑟_𝑗=1𝛽_𝑗𝐴^𝑇𝑃_𝑄𝑗𝐴𝑥^𝑘−𝐴𝑥^𝑘2. (3.2)
Evidently, (3.1) happens to be (1.2), when 1/𝐿>𝜆_𝑘.

Now we prove the convergence of the Algorithm 3.1.

Theorem 3.2. Assume that the set of the solution of the multiple-sets split feasibility problem (MSSFP) is nonempty. Then, any sequence {𝑥^𝑘}^∞_𝑘=0 generated by Algorithm 3.1 converges to a solution of MSSFP (1.1).

Proof. Let ℎ_𝑘=max{1/L,𝜆_𝑘} and take a point 𝑧∈𝐶 with 𝐴𝑧∈𝑄.
Step 1. First we show that ‖𝑥^𝑘−𝑧‖≤‖𝑥^𝑘+1−𝑧‖. From (3.1), we have 𝑥^𝑘+1−𝑧²=𝑥^𝑘+𝑠ℎ_𝑘^𝑡_𝑖=1𝛼_𝑖𝑃_𝐶𝑖𝑥^𝑘−𝑥^𝑘+^𝑟_𝑗=1𝛽_𝑗𝐴^𝑇𝑃_𝑄𝑗𝐴𝑥^𝑘−𝐴𝑥^𝑘−𝑧²=𝑥^𝑘−𝑧²+𝑠²ℎ²_𝑘^𝑡_𝑖=1𝛼_𝑖𝑃_𝐶𝑖𝑥^𝑘−𝑥^𝑘+^𝑟_𝑗=1𝛽_𝑗𝐴^𝑇𝑃_𝑄𝑗𝐴𝑥^𝑘−𝐴𝑥^𝑘2+2𝑠ℎ_𝑘𝑥^𝑘−𝑧,^𝑡_𝑖=1𝛼_𝑖𝑃_𝐶𝑖𝑥^𝑘−𝑥^𝑘+2𝑠ℎ_𝑘𝑥^𝑘−𝑧,^𝑟_𝑗=1𝛽_𝑗𝐴^𝑇𝑃_𝑄𝑗𝐴𝑥^𝑘−𝐴𝑥^𝑘. (3.3) Observe that 𝑥^𝑘−𝑧,^𝑡_𝑖=1𝛼_𝑖𝑃_𝐶𝑖𝑥^𝑘−𝑥^𝑘=^𝑡_𝑖=1𝛼_𝑖𝑥^𝑘−𝑧,𝑃_𝐶𝑖𝑥^𝑘−𝑥^𝑘=^𝑡_𝑖=1𝛼_𝑖𝑥^𝑘−𝑃_𝐶𝑖𝑥^𝑘,𝑃_𝐶𝑖𝑥^𝑘−𝑥^𝑘+^𝑡_𝑖=1𝛼_𝑖𝑃_𝐶𝑖𝑥^𝑘−𝑧,𝑃_𝐶𝑖𝑥^𝑘−𝑥^𝑘. (3.4) By the property (2) in Lemma 2.1, we get ^𝑡_𝑖=1𝛼_𝑖𝑃_𝐶𝑖𝑥^𝑘−𝑧,𝑃_𝐶𝑖𝑥^𝑘−𝑥^𝑘≤0. (3.5) Therefore, 𝑥^𝑘−𝑧,^𝑡_𝑖=1𝛼_𝑖𝑃_𝐶𝑖𝑥^𝑘−𝑥^𝑘≤−^𝑡_𝑖=1𝛼_𝑖𝑃_𝐶𝑖𝑥^𝑘−𝑥^𝑘2. (3.6) Similarly, we have 𝑥^𝑘−𝑧,^𝑟_𝑗=1𝛽_𝑗𝐴^𝑇𝑃_𝑄𝑗𝐴𝑥^𝑘−𝐴𝑥^𝑘=^𝑟_𝑗=1𝛽_𝑗𝑥^𝑘−𝑧,𝐴^𝑇𝑃_𝑄𝑗𝐴𝑥^𝑘−𝐴𝑥^𝑘=^𝑟_𝑗=1𝛽_𝑗𝐴𝑥^𝑘−𝐴𝑧,𝑃_𝑄𝑗𝐴𝑥^𝑘−𝐴𝑥^𝑘. (3.7) Since 𝐴𝑧∈𝑄, using again property (2) in Lemma 2.1, we obtain that 𝑥^𝑘−𝑧,^𝑟_𝑗=1𝛽_𝑗𝐴^𝑇𝑃_𝑄𝑗𝐴𝑥^𝑘−𝐴𝑥^𝑘≤−^𝑟_𝑗=1𝛽_𝑗𝑃_𝑄𝑗𝐴𝑥^𝑘−𝐴𝑥^𝑘2. (3.8) Substituting (3.6) and (3.8) into (3.3), we get the following: 𝑥^𝑘+1−𝑧²≤𝑥^𝑘−𝑧²+𝑠²ℎ²_𝑘^𝑡_𝑖=1𝛼_𝑖𝑃_𝐶𝑖𝑥^𝑘−𝑥^𝑘+^𝑟_𝑗=1𝛽_𝑗𝐴^𝑇𝑃_𝑄𝑗𝐴𝑥^𝑘−𝐴𝑥^𝑘2−2𝑠ℎ_𝑘^𝑡_𝑖=1𝛼_𝑖𝑃_𝐶𝑖𝑥^𝑘−𝑥^𝑘2−2𝑠ℎ_𝑘^𝑟_𝑗=1𝛽_𝑗𝑃_𝑄𝑗𝐴𝑥^𝑘−𝐴𝑥^𝑘2. (3.9) Assume at 𝑘 th step, 1/𝐿>𝜆_𝑘, then algorithms (3.1) and (2.4) coincide. Since ∇𝑝(𝑥) has a Lipschitz constant 𝐿, and ∇𝑝(𝑥) is 1/𝐿-ism (inverse-strongly monotone), that is, (𝑥)𝑦,(𝑥)𝑦≥1∇𝑝−∇𝑝≤𝐿𝑥−𝑦∇𝑝−∇𝑝,𝑥−𝑦𝐿∇𝑝(𝑥)−∇𝑝𝑦², (3.10) see [5]; then, from the proof of Theorem  2.1 in [2], we get that the sequence {𝑥^𝑘} generated by (1.2) satisfies 𝑥^𝑘+1−𝑧²≤𝑥^𝑘−𝑧²()∑−𝑠2−𝑠^𝑡𝑖=1𝛼_𝑖𝑃_𝐶𝑖𝑥^𝑘−𝑥^𝑘+∑^𝑟_𝑗=1𝛽_𝑗𝐴^𝑇𝑃_𝑄𝑗𝐴𝑥^𝑘−𝐴𝑥^𝑘2𝐿². (3.11) Similarly, assume at 𝑘th step, 1/𝐿≤𝜆_𝑘, that is ℎ_𝑘=𝜆_𝑘, then replacing 1/𝐿 with 𝜆_𝑘 in (3.9), we get the following: 𝑥^𝑘+1−𝑧²≤𝑥^𝑘−𝑧²()∑−𝑠2−𝑠^𝑡𝑖=1𝛼_𝑖𝑃_𝐶𝑖𝑥^𝑘−𝑥^𝑘2+∑^𝑟_𝑗=1𝛽_𝑗𝑃_𝑄𝑗𝐴𝑥^𝑘−𝐴𝑥^𝑘22∑^𝑡_𝑖=1𝛼_𝑖𝑃_𝐶𝑖𝑥^𝑘−𝑥^𝑘+∑^𝑟_𝑗=1𝛽_𝑗𝐴^𝑇𝑃_𝑄𝑗𝐴𝑥^𝑘−𝐴𝑥^𝑘2. (3.12) Combing (3.11) with (3.12) 𝑥^𝑘+1−𝑧²≤𝑥^𝑘−𝑧²()∑−𝑠2−𝑠max^𝑡𝑖=1𝛼_𝑖𝑃_𝐶𝑖𝑥^𝑘−𝑥^𝑘+∑^𝑟_𝑗=1𝛽_𝑗𝐴^𝑇𝑃_𝑄𝑗𝐴𝑥^𝑘−𝐴𝑥^𝑘2𝐿²,∑^𝑡_𝑖=1𝛼_𝑖𝑃_𝐶𝑖𝑥^𝑘−𝑥^𝑘2+∑^𝑟_𝑗=1𝛽_𝑗𝑃_𝑄𝑗𝐴𝑥^𝑘−𝐴𝑥^𝑘22∑^𝑡_𝑖=1𝛼_𝑖𝑃_𝐶𝑖𝑥^𝑘−𝑥^𝑘+∑^𝑟_𝑗=1𝛽_𝑗𝐴^𝑇𝑃_𝑄𝑗𝐴𝑥^𝑘−𝐴𝑥^𝑘2. (3.13) Since 𝑠∈(0,2), we have 𝑥^𝑘+1−𝑧≤𝑥^𝑘−𝑧 (3.14) for all 𝑧∈𝐶 such that 𝐴𝑧∈𝑄. Evidently, both {𝑥^𝑘} and {‖𝑥^𝑘−𝑧‖} are bounded.Step 2. Secondly we show that lim_𝑘→∞𝑥^𝑘=𝑥^∗ with 𝑥^∗∈𝐶 and 𝐴𝑥^∗∈𝑄.
As shown in Step 1, the sequence {‖𝑥^𝑘−𝑧‖} is monotonically decreasing and bounded, and there exists the limit lim_𝑘→∞𝑥^𝑘−𝑧=𝑑. (3.15) Since the case ℎ_𝑘=1/𝐿 is already treated in [5], we only need to consider the subsequence {𝑥^𝑘𝑝}_{𝜆𝑘𝑝>1/𝐿}. Hence, we need to show that the subsequence {𝑥^𝑘𝑝} converges to 𝑥^∗ with 𝑥^∗∈𝐶 and 𝐴𝑥^∗∈𝑄. From (3.12) and (3.15), and replacing 𝑥^𝑘 by 𝑥^𝑘𝑝, we have lim_𝑝→∞∑^𝑡_𝑖=1𝛼_𝑖𝑃_𝐶𝑖𝑥^𝑘𝑝−𝑥^𝑘𝑝2+∑^𝑟_𝑗=1𝛽_𝑗𝑃_𝑄𝑗𝐴𝑥^𝑘𝑝−𝐴𝑥^𝑘𝑝22∑^𝑡_𝑖=1𝛼_𝑖𝑃_𝐶𝑖𝑥^𝑘𝑝−𝑥^𝑘𝑝+∑^𝑟_𝑗=1𝛽_𝑗𝐴^𝑇𝑃_𝑄𝑗𝐴𝑥^𝑘𝑝−𝐴𝑥^𝑘𝑝2=0 (3.16) From (3) in Lemma 2.1, we know that ‖𝑃_𝐶𝑖(𝑥^𝑘𝑝)−𝑥^𝑘𝑝)‖≤‖𝑥^𝑘𝑝−𝑧‖and ‖𝑃_𝑄𝑗(𝐴𝑥^𝑘𝑝)−𝐴𝑥^𝑘𝑝)‖≤‖𝐴𝑥^𝑘𝑝−𝐴𝑧‖, then, we may assume that there exists a constant 𝑀 such that ^𝑡_𝑖=1𝛼_𝑖𝑃_𝐶𝑖𝑥^𝑘𝑝−𝑥^𝑘𝑝+^𝑟_𝑗=1𝛽_𝑗𝐴^𝑇𝑃_𝑄𝑗𝐴𝑥^𝑘𝑝−𝐴𝑥^𝑘𝑝2≤𝑀. (3.17) Therefore, ∑^𝑡_𝑖=1𝛼_𝑖𝑃_𝐶𝑖(𝑥^𝑘𝑝)−𝑥^𝑘𝑝2+∑^𝑟_𝑗=1𝛽_𝑗(𝑃_𝑄𝑗(𝐴𝑥^𝑘𝑝)−𝐴𝑥^𝑘𝑝)²²∑^𝑡_𝑖=1𝛼_𝑖𝑃_𝐶𝑖𝑥^𝑘𝑝−𝑥^𝑘𝑝+∑^𝑟_𝑗=1𝛽_𝑗𝑃_𝑄𝑗𝐴𝑥^𝑘𝑝−𝐴𝑥^𝑘𝑝2≥1𝑀^𝑡_𝑖=1𝛼_𝑖𝑃_𝐶𝑖𝑥^𝑘𝑝−𝑥^𝑘𝑝2+^𝑟_𝑗=1𝛽_𝑗(𝑃_𝑄𝑗(𝐴𝑥^𝑘𝑝)−𝐴𝑥^𝑘𝑝)²²≥0. (3.18) Taking limits as 𝑝→∞ in (3.18) and considering (3.16) lead to lim^𝑡_{𝑝→∞𝑖=1}𝛼_𝑖𝑃_𝐶𝑖𝑥^𝑘𝑝−𝑥^𝑘𝑝2+^𝑟_𝑗=1𝛽_𝑗𝑃_𝑄𝑗𝐴𝑥^𝑘𝑝−𝐴𝑥^𝑘𝑝2=0, (3.19) this implies that lim_𝑝→∞𝑃_𝐶𝑖𝑥^𝑘𝑝−𝑥^𝑘𝑝=0,𝑖=1,…,𝑡,lim_𝑝→∞𝑃_𝑄𝑗𝐴𝑥^𝑘𝑝−𝐴𝑥^𝑘𝑝=0,𝑗=1,…,𝑟. (3.20) Since the sequence {𝑥^𝑘𝑝} is bounded, there exists a subsequence {𝑥^𝑘𝑝_𝑙} of {𝑥^𝑘𝑝} which converges to a point 𝑏, and a corresponding subsequence {𝐴𝑥^𝑘𝑝_𝑙} of {𝐴𝑥^𝑘𝑝} which converges to a point 𝐴𝑏. Therefore, from (3.20), it is easy to get that 𝑏∈𝐶 and 𝐴𝑏∈𝑄.
To obtain the result that the sequence {𝑥^𝑘𝑝} itself is convergent to a point 𝑏∈𝐶 with 𝐴𝑏∈𝑄, it is now sufficient to show that the subsequence of {𝑥^𝑘𝑝} converges to the same point 𝑏 and the corresponding subsequence of {𝐴𝑥^𝑘𝑝} is convergent to 𝐴𝑏. Let us suppose that there exists a subsequence {𝑥^𝑘𝑝_𝑙′} of {𝑥^𝑘𝑝} that is convergent to point 𝑏^′, as above, 𝑏^′∈𝐶 and 𝐴𝑏^′∈𝑄. For 𝑙∈𝑍⁺, we obtain 𝑥^𝑘𝑝_𝑙−𝑏^′2−𝑥^𝑘𝑝_𝑙−𝑏²=𝑥^𝑘𝑝_𝑙−𝑏+𝑏−𝑏^′,𝑥^𝑘𝑝_𝑙−𝑏+𝑏−𝑏^′−𝑥^𝑘𝑝_𝑙−𝑏², (3.21) which, after calculating the inner product, leads to 𝑥^𝑘𝑝_𝑙−𝑏^′2−𝑥^𝑘𝑝_𝑙−𝑏²=2𝑥^𝑘𝑝_𝑙−𝑏,𝑏−𝑏^′+𝑏^′−𝑏². (3.22) Similarly, for 𝑙^′∈𝑍⁺, it is easily to obtain that 𝑥^𝑘𝑝_𝑙′−𝑏²−𝑥^𝑘𝑝_𝑙′−𝑏^′2=2𝑥^𝑘𝑝_𝑙′−𝑏^′,𝑏^′−𝑏+𝑏−𝑏^′2. (3.23) As remarked, the sequences {‖𝑥^𝑘𝑝−𝑏‖}^+∞_𝑝=1 and {‖𝑥^𝑘𝑝−𝑏^′‖}^+∞_𝑝=1 are convergent to 𝑑(𝑏) and 𝑑(𝑏^′). In particular, we get the following: lim_𝑝→+∞𝑥^𝑘𝑝−𝑏−𝑥^𝑘𝑝−𝑏^′=𝑑(𝑏)−𝑑𝑏^′. (3.24) Taking the limits in (3.22) and (3.23), for 𝑙→+∞ and for 𝑙^′→+∞, we deduce that 𝑑𝑏^′2−𝑑(𝑏)²=0+𝑏−𝑏^′2,𝑑(𝑏)²−𝑑𝑏^′2=0+𝑏^′−𝑏², (3.25) from above we conclude that 𝑏=𝑏^′. Similarly, 𝐴𝑏=𝐴𝑏^′. Hence, lim_𝑝→∞𝑥^𝑘𝑝=𝑏 with 𝑏∈𝐶 and 𝐴𝑏∈𝑄, that is, lim_𝑝→∞‖𝑥^𝑘𝑝−𝑏‖=0 with 𝑏∈𝐶 and 𝐴𝑏∈𝑄. Replace 𝑏 with 𝑥^∗, it can be written as lim_𝑝→∞‖𝑥^𝑘𝑝−𝑥^∗‖=0 with 𝑥^∗∈𝐶 and 𝐴𝑥^∗∈𝑄. And by reason of the monotonicity and boundness of the sequence {‖𝑥^𝑘−𝑥^∗‖}, we get the result.
Here we shortly explain the rational for the choice of the parameter 𝜆_𝑘 in Algorithm 3.1. In fact, if 𝑠=1, (3.9) can be rewritten as 𝑥^𝑘−𝑧²−𝑥^𝑘+1−𝑧²≥2ℎ_𝑘^𝑡_𝑖=1𝛼_𝑖𝑃_𝐶𝑖(𝑥^𝑘)−𝑥^𝑘2+2ℎ_𝑘^𝑟_𝑗=1𝛽_𝑗𝑃_𝑄𝑗(𝐴𝑥^𝑘)−𝐴𝑥^𝑘2−ℎ²_𝑘^𝑡_𝑖=1𝛼_𝑖𝑃_𝐶𝑖𝑥^𝑘−𝑥^𝑘−^𝑟_𝑗=1𝛽_𝑗𝐴^𝑇𝑃_𝑄𝑗𝐴𝑥^𝑘−𝐴𝑥^𝑘2. (3.26) Evidently, when ℎ_𝑘=∑^𝑡_𝑖=1𝛼_𝑖𝑃_𝐶𝑖𝑥^𝑘−𝑥^𝑘2+∑^𝑟_𝑗=1𝛽_𝑗𝑃_𝑄𝑗(𝐴𝑥^𝑘)−𝐴𝑥^𝑘2∑^𝑡_𝑖=1𝛼_𝑖𝑃_𝐶𝑖𝑥^𝑘−𝑥^𝑘−∑^𝑟_𝑗=1𝛽_𝑗𝐴^𝑇𝑃_𝑄𝑗𝐴𝑥^𝑘−𝐴𝑥^𝑘2, (3.27) the maximal value of the right hand side expression of (3.26) is obtained. Hence, if 𝑠=1, for the case 𝜆_𝑘>1/𝐿, the factor 𝜆_𝑘 can be considered as the “best” possible value which assures that 𝑥^𝑘+1 as the “closest” point to the set of solution of MSSFP along the direction ∑^𝑡_𝑖=1𝛼_𝑖(𝑃_𝐶𝑖(𝑥^𝑘)−𝑥^𝑘)−∑^𝑟_𝑗=1𝛽_𝑗𝐴^𝑇(𝑃_𝑄𝑗(𝐴𝑥^𝑘)−𝐴𝑥^𝑘). Therefore, to some extent, the extrapolated factor 𝜆_𝑘 plays an important role for the accelerated convergence for Algorithm 3.1.

4. Numerical Experiments

In the numerical results listed in the following table CPU time in seconds. We denote that 𝑒₀=(0,0,…,0)∈ℜ^𝑁 and 𝑒₁=(1,1,…,1)∈ℜ^𝑁. “Algorithm (1.2)” in the tables denotes the projection algorithm developed by Censor et al., in [5] as (1.2). “Algorithm 3.1” in the tables denotes Algorithm 3.1.

Now we give the following examples to test the efficiency of the above algorithm.

Example 4.1. In this example, we considered the multiples-set split feasibility problem, where ,𝐶𝐴=2−1323125212021−22−10−35₁=𝑥∈ℜ⁵∣𝑥₁+𝑥₂,𝐶≤0.25₂=𝑥∈ℜ⁵∣𝑥₂+𝑥₃,𝐶≤0.25₃=𝑥∈ℜ⁵∣𝑥₃+𝑥₄,𝐶≤0.25₄=𝑥∈ℜ⁵∣𝑥₄+𝑥₅,𝐶≤0.25₅=𝑥∈ℜ⁵∣𝑥₁+𝑥₅≤0.25, (4.1) and 𝑄={𝑥∈ℜ⁴|𝑥≤𝑑} with 𝑑=(1,1,1,1). Consider the following three cases:Case I. 𝑥⁰=(1,−1,1,−1,1),Case II. 𝑥⁰=(1,1,1,1,1),Case III. 𝑥⁰=(10,0,10,0,10). The number of iterative step needed for Algorithm (1.2) and Algorithm 3.1, and the corresponding solutions of this example are shown in Table 1.

Table 1 

The numerical results of Example 4.1.

Example 4.2. In this example, we consider a multiple-set split feasibility where 𝐴=(𝑎_𝑖𝑗)_𝑁×𝑁∈ℜ^𝑁×𝑁, and 𝑎_𝑖𝑗∈(0,10) are generated randomly 𝐶_𝑖=𝑥∈ℜ^𝑁∣𝑥−𝑖𝑒₁²≤(38+2𝑖)²𝑄,𝑖=1,…,𝑡,_𝑗=𝑥∈ℜ^𝑁∣24≤𝑥_𝑗≤26,𝑗=1,…,𝑟(𝑁). (4.2) Take initial point 𝑥⁰=𝑒₀∈ℜ^𝑁, we test the algorithms with different values of 𝑠,𝑡,𝑟, and 𝑁, respectively, in different dimensional Euclidean space.The number of iterative step needed for Algorithm (1.2) and Algorithm 3.1 is displayed in Table 2.

Table 2 

The numerical results of Example 4.2.

Example 4.3. In this example, we considered the multiples-set split feasibility problem, where ,𝐶𝐴=100100100100100000000000000000₁=𝑥∈ℜ⁵∣𝑥₁+𝑥₂;𝐶≤0.25₂=𝑥∈ℜ⁵∣𝑥₂+𝑥₃,𝐶≤0.25₃=𝑥∈ℜ⁵∣𝑥₃+𝑥₄,𝐶≤0.25₄=𝑥∈ℜ⁵∣𝑥₄+𝑥₅,𝐶≤0.25₅=𝑥∈ℜ⁵∣𝑥₁+𝑥₅≤0.25, (4.3) and 𝑄={𝑥∈ℜ⁴|𝑥≤𝑑} with 𝑑=(1,1,1,1). Consider the following three cases:Case I. 𝑥⁰=(1,−1,1,−1,1), Case II. 𝑥⁰=(1,1,1,1,1), Case III. 𝑥⁰=(10,0,10,0,10). The number of iterative step needed for Algorithm (1.2) and Algorithm 3.1, and the corresponding solutions of this example are shown in Table 3.
In all numerical experiments, we take the weights as 𝛼_𝑖=𝛽_𝑗=1/(𝑟+𝑡),𝑖=1,…,𝑡, 𝑗=1,…,𝑟. The stopping criterion is 𝑝(𝑥)<𝜀=10⁻⁴.

Table 3 

The numerical results of Example 4.3.

From these preliminary numerical results, we can see that the method is efficient, while the computational burden is not too large using the extrapolated technique.

Acknowledgments

This work was supported by National Science Foundation of China (under Grant 111712210), Shanghai Municipal Committee of Science and Technology (under Grant 10550500800), Shanghai Municipal Government (under Grant S30501), and the Innovation Fund Project for Graduate Student of Shanghai (under Grant JWCXSL1001).