An Explicit Method for the Split Feasibility Problem with Self-Adaptive Step Sizes

Abstract

An explicit iterative method with self-adaptive step-sizes for solving the split feasibility problem is presented. Strong convergence theorem is provided.

1. Introduction

Since its publication in 1994, the split feasibility problem has been studied by many authors. For some related works, please consult [1–18]. Among them, a more popular algorithm that solves the split feasibility problems is Byrne’s 𝐶𝑄 method [2]: 𝑥_𝑛+1=𝑃_𝐶𝑥_𝑛−𝜏𝐴^∗𝐼−𝑃_𝑄𝐴𝑥_𝑛, (1.1) where 𝐶 and 𝑄 are two closed convex subsets of two real Hilbert spaces 𝐻₁ and 𝐻₂, respectively, and 𝐴∶𝐻₁→𝐻₂ is a bounded linear operator. The 𝐶𝑄 algorithm only involves the computations of the projections 𝑃_𝐶 and 𝑃_𝑄 onto the sets 𝐶 and 𝑄, respectively, and is therefore implementable in the case where 𝑃_𝐶 and 𝑃_𝑄 have closed-form expressions.

Note that 𝐶𝑄 algorithm can be obtained from optimization. If we set 𝑓(𝑥)1∶=2𝐴𝑥−𝑃_𝑄𝐴𝑥², (1.2) then the convex objective 𝑓 is differentiable and has a Lipschitz gradient given by ∇𝑓(𝑥)=𝐴^∗𝐼−𝑃_𝑄𝐴. (1.3) Thus, the 𝐶𝑄 algorithm can be obtained by minimizing the following convex minimization problem min_𝑥∈𝐶𝑓(𝑥). (1.4) We can use a gradient projection algorithm below to solve the split feasibility problem: 𝑥_𝑛+1=𝑃_𝐶𝑥_𝑛−𝜏_𝑛∇𝑓𝑥_𝑛, (1.5) where 𝜏_𝑛, the step size at iteration 𝑛, is chosen in the interval (0,2/𝐿), where 𝐿 is the Lipschitz constant of ∇𝑓.

However, we observe that the determination of the step size 𝜏_𝑛 depends on the operator (matrix) norm ‖𝐴‖ (or the largest eigenvalue of 𝐴^∗𝐴). This means that in order to implement the 𝐶𝑄 algorithm, one has first to compute (or, at least, estimate) the matrix norm of 𝐴, which is in general not an easy work in practice. To overcome the above difficulty, the so-called self-adaptive method which permits step size 𝜏_𝑛 being selected self-adaptively was developed. See, for example, [10, 14, 15, 19–23].

Inspired by the above results and the self-adaptive method, in this paper, we present an explicit iterative method with self-adaptive step sizes for solving the split feasibility problem. Convergence analysis result is given.

2. Preliminaries

Let 𝐻₁ and 𝐻₂ be two real Hilbert spaces and 𝐶 and 𝑄 two closed convex subsets of 𝐻₁ and 𝐻₂, respectively. Let 𝐴∶𝐻₁→𝐻₂ be a bounded linear operator. The split feasibility problem is to find a point 𝑥^∗ such that 𝑥^∗∈𝐶,𝐴𝑥^∗∈𝑄. (2.1) Next, we use Γ to denote the solution set of the split feasibility problem, that is, Γ={𝑥∈𝐶∶𝐴𝑥∈𝑄}.

We know that a point 𝑥^∗∈𝐶 is a stationary point of problem (1.4) if it satisfies ∇𝑓𝑥^∗,𝑥−𝑥^∗≥0,∀𝑥∈𝐶. (2.2) Given 𝑥^∗∈𝐻₁. 𝑥^∗ solves the split feasibility problem if and only if 𝑥^∗ solves the fixed point equation 𝑥^∗=𝑃_𝐶𝑥^∗−𝛾𝐴^∗𝐼−𝑃_𝑄𝐴𝑥^∗. (2.3) Next we adopt the following notation:(i)𝑥_𝑛→𝑥 means that 𝑥_𝑛 converges strongly to 𝑥;(ii)𝑥_𝑛⇀𝑥 means that 𝑥_𝑛 converges weakly to 𝑥;(iii)𝜔_𝑤(𝑥_𝑛)∶={𝑥∶∃𝑥_𝑛𝑗⇀𝑥} is the weak 𝜔-limit set of the sequence {𝑥_𝑛}.

Recall that a function 𝑓∶𝐻→ℝ is called convex if 𝑓𝜆𝑥+(1−𝜆)𝑦≤𝜆𝑓(𝑥)+(1−𝜆)𝑓𝑦, (2.4) for all 𝜆∈(0,1) and ∀𝑥, 𝑦∈𝐻. It is known that a differentiable function 𝑓 is convex if and only if there holds the relation: 𝑓(𝑧)≥𝑓(𝑥)+∇𝑓(𝑥),𝑧−𝑥, (2.5) for all 𝑧∈𝐻. Recall that an element 𝑔∈𝐻 is said to be a subgradient of 𝑓∶𝐻→ℝ at 𝑥 if 𝑓(𝑧)≥𝑓(𝑥)+𝑔,𝑧−𝑥, (2.6) for all 𝑧∈𝐻. If the function 𝑓∶𝐻→ℝ has at least one subgradient at 𝑥 is said to be sub-differentiable at 𝑥. The set of subgradients of 𝑓 at the point 𝑥 is called the subdifferential of 𝑓 at 𝑥, and is denoted by 𝜕𝑓(𝑥). A function 𝑓 is called sub-differentiable if it is subdifferentiable at all 𝑥∈𝐻. If 𝑓 is convex and differentiable, then its gradient and subgradient coincide. A function 𝑓∶𝐻→ℝ is said to be weakly lower semi continuous (w-lsc) at 𝑥 if 𝑥_𝑛⇀𝑥 implies 𝑓(𝑥)≤liminf_𝑛→∞𝑓𝑥_𝑛. (2.7)𝑓 is said to be w-lsc on 𝐻 if it is w-lsc at every point 𝑥∈𝐻.

A mapping 𝑇∶𝐶→𝐶 is called nonexpansive if 𝑇𝑥−𝑇𝑦≤𝑥−𝑦, (2.8) for all 𝑥,𝑦∈𝐶.

Recall that the (nearest point or metric) projection from 𝐻 onto 𝐶, denoted 𝑃_𝐶, assigns, to each 𝑥∈𝐻, the unique point 𝑃_𝐶(𝑥)∈𝐶 with the property 𝑥−𝑃_𝐶(𝑥)=inf𝑥−𝑦∶𝑦∈𝐶. (2.9) It is well known that the metric projection 𝑃_𝐶 of 𝐻 onto 𝐶 has the following basic properties:(a)‖𝑃_𝐶(𝑥)−𝑃_𝐶(𝑦)‖≤‖𝑥−𝑦‖ for all 𝑥,𝑦∈𝐻;(b)⟨𝑥−𝑦,𝑃_𝐶(𝑥)−𝑃_𝐶(𝑦)⟩≥‖𝑃_𝐶(𝑥)−𝑃_𝐶(𝑦)‖² for every 𝑥,𝑦∈𝐻;(c)⟨𝑥−𝑃_𝐶(𝑥),𝑦−𝑃_𝐶(𝑥)⟩≤0 for all 𝑥∈𝐻, 𝑦∈𝐶.

Lemma 2.1 (see [24]). Assume that {𝑎_𝑛} is a sequence of nonnegative real numbers such that 𝑎_𝑛+1≤1−𝛾_𝑛𝑎_𝑛+𝛿_𝑛, (2.10) where {𝛾_𝑛} is a sequence in (0,1) and {𝛿_𝑛} is a sequence such that(1)∑^∞_𝑛=1𝛾_𝑛=∞; (2)limsup_𝑛→∞(𝛿_𝑛/𝛾_𝑛)≤0 or ∑^∞_𝑛=1|𝛿_𝑛|<∞.
Then lim_𝑛→∞𝑎_𝑛=0.

Lemma 2.2 (see [25]). Let (𝑠_𝑛) be a sequence of real numbers that does not decrease at infinity, in the sense that there exists a subsequence (𝑠_𝑛𝑖) of (𝑠_𝑛) such that 𝑠_𝑛𝑖≤𝑠_𝑛𝑖+1 for all 𝑖≥0. For every 𝑛≥𝑛₀, define an integer sequence (𝜏(𝑛)) as 𝜏(𝑛)=max𝑘≤𝑛∶𝑠_𝑛𝑖<𝑠_𝑛𝑖+1. (2.11) Then 𝜏(𝑛)→∞ as 𝑛→∞ and for all 𝑛≥𝑛₀max𝑠_𝜏(𝑛),𝑠_𝑛≤𝑠_𝜏(𝑛)+1. (2.12)

3. Main Results

In this section, we will introduce our algorithm and prove our main results.

Let 𝐶 and 𝑄 be nonempty closed convex subsets of real Hilbert spaces 𝐻₁ and 𝐻₂, respectively. Let 𝐴∶𝐻₁→𝐻₂ be a bounded linear operator. In the sequel, we assume that the split feasibility problem is consistent, that is Γ≠∅.

Algorithm 3.1. For 𝑢∈𝐶 and given 𝑥₀∈𝐶, let the sequence {𝑥_𝑛+1} defined by 𝑦_𝑛=𝛼_𝑛𝑢+1−𝛼_𝑛𝑥_𝑛,𝑥_𝑛+1=𝑃_𝐶𝑦_𝑛−𝜏_𝑛𝑓𝑦_𝑛∇𝑓𝑦_𝑛∇𝑓𝑦_𝑛²,𝑛≥0, (3.1) where {𝛼_𝑛}⊂(0,1) and {𝜏_𝑛}⊂(0,2).

Remark 3.2. In the sequel, we may assume that ∇𝑓(𝑦_𝑛)≠0 for all 𝑛. Note that this fact can be guaranteed if the sequence {𝑦_𝑛} is infinite; that is, Algorithm 3.1 does not terminate in a finite number of iterations.

Theorem 3.3. Assume that the following conditions are satisfied:(i)lim_𝑛→∞𝛼_𝑛=0 and ∑^∞_𝑛=1𝛼_𝑛=∞;(ii)inf_𝑛𝜏_𝑛(2−𝜏_𝑛)>0.
Then {𝑥_𝑛} defined by (3.1) converges strongly to 𝑃_Γ(𝑢).

Proof. Let 𝜈∈Γ. It follows that ∇𝑓(𝜈)=0 for all 𝜈∈Γ. From (2.5), we deduce that 𝑓𝑦_𝑛=𝑓𝑦_𝑛−𝑓(𝜈)≤∇𝑓𝑦_𝑛,𝑦_𝑛−𝜈. (3.2) Thus, by (3.1) and (3.2), we have 𝑥_𝑛+1−𝜈²=𝑃_𝐶𝑦_𝑛−𝜏_𝑛𝑓𝑦_𝑛∇𝑓𝑦_𝑛∇𝑓𝑦_𝑛²−𝜈²≤𝑦_𝑛−𝜏_𝑛𝑓𝑦_𝑛∇𝑓𝑦_𝑛∇𝑓𝑦_𝑛²−𝜈²=𝑦_𝑛−𝜈²−2𝜏_𝑛𝑓𝑦_𝑛∇𝑓𝑦_𝑛²∇𝑓𝑦_𝑛,𝑦_𝑛−𝜈+𝜏²_𝑛𝑓²𝑦_𝑛∇𝑓𝑦_𝑛²≤𝑦_𝑛−𝜈²−2𝜏_𝑛𝑓²𝑦_𝑛∇𝑓𝑦_𝑛²+𝜏²_𝑛𝑓²𝑦_𝑛∇𝑓𝑦_𝑛²=𝑦_𝑛−𝜈²−𝜏_𝑛2−𝜏_𝑛𝑓²𝑦_𝑛∇𝑓𝑦_𝑛²,𝑦_𝑛−𝜈²=𝛼_𝑛(𝑢−𝜈)+1−𝛼_𝑛𝑥_𝑛−𝜈²≤𝛼_𝑛‖𝑢−𝜈‖²+1−𝛼_𝑛𝑥_𝑛−𝜈². (3.3) It follows that 𝑥_𝑛+1−𝜈²≤𝛼_𝑛‖𝑢−𝜈‖²+1−𝛼_𝑛𝑥_𝑛−𝜈²−𝜏_𝑛2−𝜏_𝑛𝑓²𝑦_𝑛∇𝑓𝑦_𝑛²≤𝛼_𝑛‖𝑢−𝜈‖²+1−𝛼_𝑛𝑥_𝑛−𝜈²≤max‖𝑢−𝜈‖²,𝑥_𝑛−𝜈². (3.4) By induction, we deduce 𝑥_𝑛+1−𝜈≤max‖𝑢−𝜈‖,𝑥₀−𝜈. (3.5) Hence, {𝑥_𝑛} is bounded.
At the same time, we note that 𝑦_𝑛−𝜈²=𝛼_𝑛(𝑢−𝜈)+1−𝛼_𝑛𝑥_𝑛−𝜈²≤1−𝛼_𝑛𝑥_𝑛−𝜈²+2𝛼_𝑛𝑢−𝜈,𝑦_𝑛−𝜈. (3.6) Therefore, 𝑥_𝑛+1−𝜈²≤1−𝛼_𝑛𝑥_𝑛−𝜈²+2𝛼_𝑛𝑢−𝜈,𝑦_𝑛−𝜈−𝜏_𝑛2−𝜏_𝑛𝑓²𝑦_𝑛∇𝑓𝑦_𝑛²=1−𝛼_𝑛𝑥_𝑛−𝜈²+2𝛼_𝑛𝑢−𝜈,𝛼_𝑛(𝑢−𝜈)+1−𝛼_𝑛𝑥_𝑛−𝜈−𝜏_𝑛2−𝜏_𝑛𝑓²𝑦_𝑛∇𝑓𝑦_𝑛²=1−𝛼_𝑛𝑥_𝑛−𝜈²+2𝛼²_𝑛‖𝑢−𝜈‖²+2𝛼_𝑛1−𝛼_𝑛𝑢−𝜈,𝑥_𝑛−𝜈−𝜏_𝑛2−𝜏_𝑛𝑓²𝑦_𝑛∇𝑓𝑦_𝑛². (3.7) It follows that 𝑥_𝑛+1−𝜈²−𝑥_𝑛−𝜈²+𝛼_𝑛𝑥_𝑛−𝜈²−2𝛼_𝑛‖𝑢−𝜈‖²+2𝛼_𝑛𝑢−𝜈,𝑥_𝑛−𝜈+𝜏_𝑛2−𝜏_𝑛𝑓²𝑦_𝑛∇𝑓𝑦_𝑛²≤2𝛼_𝑛𝑢−𝜈,𝑥_𝑛−𝜈. (3.8) Next, we will prove that 𝑥_𝑛→𝜈. Set 𝜔_𝑛=‖𝑥_𝑛−𝜈‖² for all 𝑛≥0. Since 𝛼_𝑛→0 and inf_𝑛𝜏_𝑛(2−𝜏_𝑛)>0, we may assume without loss of generality that 𝜏_𝑛(2−𝜏_𝑛)≥𝜎 for some 𝜎>0. Thus, we can rewrite (3.8) as 𝜔_𝑛+1−𝜔_𝑛+𝛼_𝑛𝑈_𝑛+𝜎𝑓²𝑦_𝑛∇𝑓𝑦_𝑛²≤2𝛼_𝑛𝑢−𝜈,𝑥_𝑛−𝜈, (3.9) where 𝑈_𝑛=‖𝑥_𝑛−𝜈‖²−2𝛼_𝑛‖𝑢−𝜈‖²+2𝛼_𝑛⟨𝑢−𝜈,𝑥_𝑛−𝜈⟩.
Now, we consider two possible cases.
Case  1. Assume that {𝜔_𝑛} is eventually decreasing; that is, there exists 𝑁>0 such that {𝜔_𝑛} is decreasing for 𝑛≥𝑁. In this case, {𝜔_𝑛} must be convergent and from (3.9) it follows that 0≤𝜎𝑓²𝑦_𝑛∇𝑓𝑦_𝑛²≤𝜔_𝑛−𝜔_𝑛+1−𝛼_𝑛𝑈_𝑛+2𝛼_𝑛‖𝑢−𝜈‖𝑥_𝑛−𝜈≤𝜔_𝑛−𝜔_𝑛+1+𝑀𝛼_𝑛, (3.10) where 𝑀>0 is a constant such that sup_𝑛{2‖𝑢−𝜈‖‖𝑥_𝑛−𝜈‖+‖𝑈_𝑛‖}≤𝑀. Letting 𝑛→∞ in (3.10), we get lim_𝑛→∞𝑓𝑦_𝑛=0. (3.11) Since {𝑦_𝑛} is bounded, there exists a subsequence {𝑦_𝑛𝑘} of {𝑦_𝑛} converging weakly to 𝑥∈𝐶. Since, 𝑥_𝑛−𝑦_𝑛→0, we also have {𝑥_𝑛𝑘} of {𝑥_𝑛} converging weakly to 𝑥∈𝐶. From the weak lower semicontinuity of 𝑓, we have 0≤𝑓(𝑥)≤liminf_𝑘→∞𝑓𝑦_𝑛𝑘=lim_𝑛→∞𝑓𝑦_𝑛=0. (3.12) Hence, 𝑓(𝑥)=0; that is, 𝐴𝑥∈𝑄. This indicates that 𝜔_𝑤𝑦_𝑛=𝜔_𝑤𝑥_𝑛⊂Γ. (3.13) Furthermore, by using the property of the projection (𝑐), we deduce limsup_𝑛→∞𝑢−𝜈,𝑥_𝑛𝑥−𝜈=max_{∈𝜔𝑤(𝑥𝑛)}𝑢−𝑃_Γ(𝑢),𝑥−𝑃_Γ(𝑢)≤0. (3.14) From (3.8), we obtain 𝜔_𝑛+1≤1−𝛼_𝑛𝜔_𝑛+𝛼_𝑛2𝛼_𝑛‖𝑢−𝜈‖²+21−𝛼_𝑛𝑢−𝜈,𝑥_𝑛−𝜈. (3.15) This together with Lemma 2.1 imply that 𝜔_𝑛→0.
Case  2. Assume that {𝜔_𝑛} is not eventually decreasing. That is, there exists an integer 𝑛₀ such that 𝜔_𝑛0≤𝜔_𝑛0+1. Thus, we can define an integer sequence {𝜏_𝑛} for all 𝑛≥𝑛₀ as follows: 𝜏(𝑛)=max𝑘∈ℕ∣𝑛₀≤𝑘≤𝑛,𝜔_𝑘≤𝜔_𝑘+1. (3.16) Clearly, 𝜏(𝑛) is a nondecreasing sequence such that 𝜏(𝑛)→+∞ as 𝑛→∞ and 𝜔_𝜏(𝑛)≤𝜔_𝜏(𝑛)+1, (3.17) for all 𝑛≥𝑛₀. In this case, we derive from (3.10) that 𝜎𝑓²𝑦_𝜏(𝑛)∇𝑓𝑦_𝜏(𝑛)²≤𝑀𝛼_𝜏(𝑛)→0. (3.18) It follows that lim_𝑛→∞𝑓𝑦_𝜏(𝑛)=0. (3.19) This implies that every weak cluster point of {𝑦_𝜏(𝑛)} is in the solution set Γ; that is, 𝜔_𝑤(𝑦_𝜏(𝑛))⊂Γ. So, 𝜔_𝑤(𝑥_𝜏(𝑛))⊂Γ. On the other hand, we note that 𝑦_𝜏(𝑛)−𝑥_𝜏(𝑛)=𝛼_𝜏(𝑛)𝑢−𝑥_𝜏(𝑛)𝑥→0,_𝜏(𝑛)+1−𝑦_𝜏(𝑛)≤𝜏_𝜏(𝑛)𝑓𝑦_𝜏(𝑛)∇𝑓𝑦_𝜏(𝑛)→0. (3.20) From which we can deduce that limsup_𝑛→∞𝑢−𝜈,𝑥_𝜏(𝑛)𝑥−𝜈=max_{∈𝜔𝑤(𝑥𝜏(𝑛))}𝑢−𝑃_Γ(𝑢),𝑥−𝑃_Γ(𝑢)≤0. (3.21) Since 𝜔_𝜏(𝑛)≤𝜔_𝜏(𝑛)+1, we have from (3.9) that 𝜔_𝜏(𝑛)≤1−2𝛼_𝜏(𝑛)𝑢−𝜈,𝑥_𝜏(𝑛)−𝜈+2𝛼_𝜏(𝑛)‖𝑢−𝜈‖². (3.22) Combining (3.21) and (3.22) yields limsup_𝑛→∞𝜔_𝜏(𝑛)≤0, (3.23) and hence lim_𝑛→∞𝜔_𝜏(𝑛)=0. (3.24) From (3.15), we have limsup_𝑛→∞𝜔_𝜏(𝑛)+1≤limsup_𝑛→∞𝜔_𝜏(𝑛). (3.25) Thus, lim_𝑛→∞𝜔_𝜏(𝑛)+1=0. (3.26) From Lemma 2.2, we have 0≤𝜔_𝑛≤max𝜔_𝜏(𝑛),𝜔_𝜏(𝑛)+1. (3.27) Therefore, 𝜔_𝑛→0. That is, 𝑥_𝑛→𝜈. This completes the proof.

Acknowledgment

The author was supported in part by the Youth Foundation of Taizhou University (2011QN11).