Variant Gradient Projection Methods for the Minimization Problems

Abstract

The gradient projection algorithm plays an important role in solving constrained convex minimization problems. In general, the gradient projection algorithm has only weak convergence in infinite-dimensional Hilbert spaces. Recently, H. K. Xu (2011) provided two modified gradient projection algorithms which have strong convergence. Motivated by Xu’s work, in the present paper, we suggest three more simpler variant gradient projection methods so that strong convergence is guaranteed.

1. Introduction

Let 𝐻 be a real Hilbert space and 𝐶 a nonempty closed and convex subset of 𝐻. Let 𝑓∶𝐻→𝑅  be a real-valued convex function. Now we consider the following constrained convex minimization problem: min_𝑥∈𝐶𝑓(𝑥). (1.1) Assume that (1.1) is consistent; that is, it has a solution and we use 𝑆 to denote its solution set. If 𝑓 is Fréchet differentiable, then 𝑥^∗∈𝐶 solves (1.1) if and only if 𝑥^∗∈𝐶 satisfies the following optimality condition: ∇𝑓𝑥^∗,𝑥−𝑥^∗≥0,∀𝑥∈𝐶, (1.2) where ∇𝑓 denotes the gradient of 𝑓. Note that (1.2) can be rewritten as ⟨𝑥^∗−𝑥^∗−∇𝑓𝑥^∗,𝑥−𝑥^∗⟩≥0,∀𝑥∈𝐶. (1.3) This shows that the minimization (1.1) is equivalent to the fixed-point problem: Proj_𝐶𝑥^∗−𝛾∇𝑓𝑥^∗=𝑥^∗, (1.4) where 𝛾>0 is an any constant and Proj_𝐶 is the nearest point projection from 𝐻 onto 𝐶. By using this relationship, the gradient-projection algorithm is usually applied to solve the minimization problem (1.1). This algorithm generates a sequence {𝑥_𝑛} through the recursion: 𝑥_𝑛+1=Proj_𝐶𝑥_𝑛−𝛾_𝑛∇𝑓𝑥_𝑛,𝑛≥0, (1.5) where the initial guess 𝑥₀∈𝐶 is chosen arbitrarily and {𝛾_𝑛} is a sequence of step sizes which may be chosen in different ways. The gradient-projection algorithm (1.5) is a powerful tool for solving constrained convex optimization problems and has well been studied in the case of constant stepsizes 𝛾_𝑛=𝛾 for all 𝑛. The reader can refer to [1–9]. It has recently been applied to solve split feasibility problems which find applications in image reconstructions and the intensity modulated radiation therapy (see [10–17]).

It is known [3] that if 𝑓 has a Lipschitz continuous and strongly monotone gradient, then the sequence {𝑥_𝑛} can be strongly convergent to a minimizer of 𝑓 in 𝐶. If the gradient of 𝑓 is only assumed to be Lipschitz continuous, then {𝑥_𝑛} can only be weakly convergent if 𝐻 is infinite dimensional. This gives naturally rise to a question.

Question 1. How to appropriately modify the gradient projection algorithm so as to have strong convergence?

For this purpose, recently, Xu [18] first introduced the following modification: 𝑥_𝑛+1=𝜃_𝑛ℎ𝑥_𝑛+1−𝜃_𝑛Proj_𝐶𝑥_𝑛−𝛾_𝑛∇𝑓𝑥_𝑛,𝑛≥0, (1.6) where the sequences {𝜃_𝑛}⊂(0,1) and {𝛾_𝑛}⊂(0,∞) satisfy the following conditions: (i)lim_𝑛→∞𝜃_𝑛=0, ∑^∞_𝑛=0𝜃_𝑛=∞ and ∑^∞_𝑛=0|𝜃_𝑛+1−𝜃_𝑛|<∞; (ii)0<liminf_𝑛→∞𝛾_𝑛≤limsup_𝑛→∞𝛾_𝑛<2/𝐿 and ∑^∞_𝑛=0|𝛾_𝑛+1−𝛾_𝑛|<∞. Xu [18] proved that the sequence {𝑥_𝑛} converges strongly to a minimizer of (1.1).

Remark 1.1. Xu's modification (1.6) is a convex combination of the gradient-projection algorithm (1.5) and a self-mapping ℎ(𝑥) which is usually referred as a so-called viscosity item.

In [18], Xu presented another modification as follows: 𝑦_𝑛=Proj_𝐶𝑥_𝑛−𝛾_𝑛∇𝑓𝑥_𝑛𝐶,𝑛≥0,_𝑛=𝑧∈𝐶∶𝑦_𝑛−𝑧≤𝑥_𝑛,𝑄−𝑧_𝑛=𝑧∈𝐶∶𝑥_𝑛−𝑧,𝑥₀−𝑥_𝑛,𝑥≥0_𝑛+1=Proj_{𝐶𝑛∩𝑄𝑛}𝑥₀. (1.7) Consequently, Xu [18] proved that Algorithm (1.7) also converges strongly to 𝑥^∗ which solves the minimization problem (1.1).

Remark 1.2. Equation (1.7) involved in additional projections which couple the gradient projection method (1.5) with the so-called CQ method.

It should be pointed out that Xu's modifications (1.6) and (1.7) are interesting and provide us with a direction for solving (1.1) in infinite-dimensional Hilbert spaces.

Motivated by Xu's work, in the present paper, we suggest three variant gradient projection methods so that strong convergence is guaranteed for solving (1.1) in infinite-dimensional Hilbert spaces. Our motivations are mainly in the two respects.

Reason 1. The solution of the minimization problem (1.1) is not always unique, so that there may be many solutions to the problem. In that case, a special solution (e.g., the minimum norm solution) must be found from among candidate solutions. The minimum norm problem is motivated by the following least squares solution to the constrained linear inverse problem: 𝐵𝑥=𝑏,𝑥∈Ω, (1.8) where Ω is a nonempty closed convex subset of a real Hilbert space 𝐻, 𝐵 is a bounded linear operator from 𝐻 to another real Hilbert space 𝐻₁, 𝐵^∗ is the adjoint of 𝐵, and 𝑏 is a given point in 𝐻₁. The least-squares solution to (1.8) is the least-norm minimizer of the minimization problem: min_𝑥∈Ω‖𝐵𝑥−𝑏‖. (1.9) For some related works, please see Solodov and Svaiter [19], Goebel and Kirk [20], and Martinez-Yanes and Xu [21].

Reason 2. Projection methods are used extensively in a variety of methods in optimization theory. Apart from theoretical interest, the main advantage of projection methods, which makes them successful in real-world applications, is computational (see [22–31]). In this respect, (1.7) is particularly useful. But we observe that (1.7) involves two half-spaces 𝐶_𝑛 and 𝑄_𝑛. If the sets 𝐶_𝑛 and 𝑄_𝑛 are simple enough, then 𝑃_𝐶𝑛 and 𝑃_𝑄𝑛 are easily executed. But 𝐶_𝑛∩𝑄_𝑛 may be complicate, so that the projection 𝑃_{𝐶𝑛∩𝑄𝑛} is not easily executed. This might seriously affect the efficiency of the method. Hence, it is interesting that one can relax 𝐶_𝑛 or 𝑄_𝑛 from (1.7).

In the present paper, we suggest the following three methods: 𝑥_𝑛+1=Proj_𝐶𝜃_𝑛ℎ(𝑥)+1−𝜃_𝑛𝑥_𝑛−𝛾_𝑛∇𝑓𝑥_𝑛𝑥,𝑛≥0,_𝑛+1=Proj_𝐶𝐼−𝛾∇𝑓Proj_𝐶1−𝜃_𝑛𝑥_𝑛𝑦,𝑛≥0,_𝑛=Proj_𝐶𝑥_𝑛−𝛾_𝑛∇𝑓𝑥_𝑛𝐶,𝑛≥0,_𝑛=𝑧∈𝐶_𝑛−1∶𝑦_𝑛−𝑧≤𝑥_𝑛,𝑥−𝑧_𝑛+1=Proj_𝐶𝑛𝑥₀. (1.10)(1.11) We will show that (1.10) can be used to find the minimum norm solution of the minimization problem (1.1), and (1.11) which is only involved in 𝐶_𝑛 also has strong convergence.

2. Preliminaries

Let 𝐶 be a nonempty closed convex subset of a real Hilbert space 𝐻. A mapping 𝑇∶𝐶→𝐶 is called nonexpansive if 𝑇𝑥−𝑇𝑦≤𝑥−𝑦,∀𝑥,𝑦∈𝐶. (2.1) Recall that the (nearest point or metric) projection from 𝐻 onto 𝐶, denoted Proj_𝐶, assigns, to each 𝑥∈𝐻, the unique point Proj_𝐶(𝑥)∈𝐶 with the property 𝑥−Proj_𝐶(𝑥)=inf𝑥−𝑦∶𝑦∈𝐶. (2.2) It is well known that the metric projection Proj_𝐶 of 𝐻 onto 𝐶 has the following basic properties: (i)‖Proj_𝐶(𝑥)−Proj_𝐶(𝑦)‖≤‖𝑥−𝑦‖, for all 𝑥,𝑦∈𝐻;(ii)⟨𝑥−𝑦,Proj_𝐶(𝑥)−Proj_𝐶(𝑦)⟩≥‖Proj_𝐶(𝑥)−Proj_𝐶(𝑦)‖², for every 𝑥,𝑦∈𝐻;(iii)⟨𝑥−Proj_𝐶(𝑥),𝑦−Proj_𝐶(𝑥)⟩≤0, for all 𝑥∈𝐻, 𝑦∈𝐶. 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}.

Lemma 2.1 (see [32] (Demiclosedness Principle)). Let 𝐶 be a closed and convex subset of a Hilbert space 𝐻, and let 𝑇∶𝐶→𝐶 be a nonexpansive mapping with Fix(𝑇)≠∅. If {𝑥_𝑛} is a sequence in 𝐶 weakly converging to 𝑥 and if {(𝐼−𝑇)𝑥_𝑛} converges strongly to 𝑦, then (𝐼−𝑇)𝑥=𝑦. (2.3)
In particular, if 𝑦=0, then 𝑥∈Fix(𝑇).

Lemma 2.2 (see [33]). Let 𝐶 be a closed convex subset of 𝐻. Let {𝑥_𝑛} be a sequence in 𝐻 and 𝑥₀∈𝐻. If {𝑥_𝑛} is such that 𝜔_𝑤(𝑥_𝑛)⊂𝐶 and satisfies the condition 𝑥_𝑛−𝑥₀≤𝑥₀−Proj_𝐶𝑥₀,∀𝑛≥0, (2.4) then 𝑥_𝑛→Proj_𝐶(𝑥₀).

Lemma 2.3 (see [34]). Assume {𝑎_𝑛} is a sequence of nonnegative real numbers such that 𝑎_𝑛+1≤1−𝛾_𝑛𝑎_𝑛+𝛿_𝑛, (2.5) 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 2.4 (see [35]). Let {𝑥_𝑛} and {𝑦_𝑛} be bounded sequences in a Banach space 𝑋, and let {𝛽_𝑛} be a sequence in [0,1] with 0<liminf_𝑛→∞𝛽_𝑛≤limsup_𝑛→∞𝛽_𝑛<1. (2.6) Suppose 𝑥_𝑛+1=1−𝛽_𝑛𝑦_𝑛+𝛽_𝑛𝑥_𝑛, (2.7) for all 𝑛≥0, and limsup_𝑛→∞𝑦_𝑛+1−𝑦_𝑛−𝑥_𝑛+1−𝑥_𝑛≤0. (2.8) Then, lim_𝑛→∞‖𝑦_𝑛−𝑥_𝑛‖=0.

3. Main Results

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

Theorem 3.1. Let 𝐶 be a closed convex subset of a real Hilbert space 𝐻. Let 𝑓∶𝐶→𝑅 be a real-valued Fréchet differentiable convex function. Assume that the solution set 𝑆 of (1.1) is nonempty. Assume that the gradient ∇𝑓 is 𝐿-Lipschitzian. Let ℎ∶𝐶→𝐻 be a 𝜌-contraction with 𝜌∈[0,1). Let {𝑥_𝑛} be a sequence generated by the following hybrid gradient projection algorithm: 𝑥_𝑛+1=Proj_𝐶𝜃_𝑛ℎ𝑥_𝑛+1−𝜃_𝑛𝑥_𝑛−𝛾_𝑛∇𝑓𝑥_𝑛,𝑛≥0, (3.1) where the sequences {𝜃_𝑛}⊂(0,1) and {𝛾_𝑛}⊂(0,∞) satisfy the following conditions: (i)lim_𝑛→∞𝜃_𝑛=0, ∑^∞_𝑛=0𝜃_𝑛=∞ and ∑^∞_𝑛=0|𝜃_𝑛+1−𝜃_𝑛|<∞; (ii)0<liminf_𝑛→∞𝛾_𝑛≤limsup_𝑛→∞𝛾_𝑛<2/𝐿 and ∑^∞_𝑛=0|𝛾_𝑛+1−𝛾_𝑛|<∞. Then the sequence {𝑥_𝑛} generated by (3.1) converges to a minimizer 𝑥 of (1.1) which is the unique solution of the following variational inequality: 𝑥∈𝑆,⟨(𝐼−ℎ)𝑥,𝑥−𝑥⟩≥0,𝑥∈𝑆. (3.2)

Proof. Take any 𝑥^∗∈𝑆. Since 𝑥^∗∈𝐶 solves the minimization problem (1.1) if and only if 𝑥^∗ solves the fixed-point equation, 𝑥^∗=Proj_𝐶(𝐼−𝛾∇𝑓)𝑥^∗ for any fixed positive number 𝛾. So, we have 𝑥^∗=Proj_𝐶(𝐼−𝛾_𝑛∇𝑓)𝑥^∗ for all 𝑛≥0. It can be rewritten as 𝑥^∗=Proj_𝐶𝜃_𝑛𝑥^∗+1−𝜃_𝑛𝑥^∗−𝛾_𝑛1−𝜃_𝑛∇𝑓𝑥^∗,𝑛≥0. (3.3) From condition (ii) 0<liminf_𝑛→∞𝛾_𝑛≤limsup_𝑛→∞𝛾_𝑛<2/𝐿, there exist two constants 𝑎 and 𝑏 such that 0<𝑎≤𝛾_𝑛≤𝑏<2/𝐿 for sufficiently large 𝑛; without loss of generality, we can assume 0<𝑎≤𝛾_𝑛≤𝑏<2/𝐿 for all 𝑛. Since lim_𝑛→∞𝜃_𝑛=0, without loss of generality, we can assume that 0<𝜃_𝑛<1−𝑏𝐿/2 for all 𝑛≥0. So, 0<liminf_𝑛→∞(𝛾_𝑛/(1−𝜃_𝑛))≤limsup_𝑛→∞(𝛾_𝑛/(1−𝜃_𝑛))<2/𝐿. Hence, 𝐼−(𝛾_𝑛/(1−𝜃_𝑛))∇𝑓 is nonexpansive.
From (3.1), we get 𝑥_𝑛+1−𝑥^∗=Proj_𝐶𝜃_𝑛ℎ𝑥_𝑛+1−𝜃_𝑛𝑥_𝑛−𝛾_𝑛∇𝑓𝑥_𝑛−Proj_𝐶𝐼−𝛾_𝑛∇𝑓𝑥^∗=Proj_𝐶𝜃_𝑛ℎ𝑥_𝑛+1−𝜃_𝑛𝑥_𝑛−𝛾_𝑛1−𝜃_𝑛∇𝑓𝑥_𝑛−Proj_𝐶𝜃_𝑛𝑥^∗+1−𝜃_𝑛𝑥^∗−𝛾_𝑛1−𝜃_𝑛∇𝑓𝑥^∗≤𝜃_𝑛ℎ𝑥_𝑛−𝑥^∗+1−𝜃_𝑛×𝛾𝐼−^𝑛1−𝜃_𝑛∇𝑓𝑥_𝑛−𝛾𝐼−^𝑛1−𝜃_𝑛∇𝑓𝑥^∗≤𝜃_𝑛ℎ𝑥_𝑛−ℎ𝑥^∗+𝜃_𝑛ℎ𝑥^∗−𝑥^∗+1−𝜃_𝑛𝑥_𝑛−𝑥^∗≤𝜃_𝑛𝜌𝑥_𝑛−𝑥^∗+𝜃_𝑛ℎ𝑥^∗−𝑥^∗+1−𝜃_𝑛𝑥_𝑛−𝑥^∗=1−1−𝜌𝜃_𝑛𝑥_𝑛−𝑥^∗+𝜃_𝑛ℎ𝑥^∗−𝑥^∗≤max𝑥_𝑛−𝑥^∗,1ℎ𝑥1−𝜌^∗−𝑥^∗. (3.4) Thus, we deduce by induction that 𝑥_𝑛−𝑥^∗≤max𝑥₀−𝑥^∗,1ℎ𝑥1−𝜌^∗−𝑥^∗. (3.5) This indicates that the sequence {𝑥_𝑛} is bounded and so are the sequences {ℎ(𝑥_𝑛)} and {∇𝑓(𝑥_𝑛)}. Then, we can chose a constant 𝑀>0 such that sup_𝑛≥0ℎ𝑥_𝑛+𝑥_𝑛+∇𝑓𝑥_𝑛≤𝑀. (3.6) Next, we estimate ‖𝑥_𝑛+1−𝑥_𝑛‖. By (3.1), we have 𝑥_𝑛+1−𝑥_𝑛=Proj_𝐶𝜃_𝑛ℎ𝑥_𝑛+1−𝜃_𝑛𝑥_𝑛−𝛾_𝑛∇𝑓𝑥_𝑛−Proj_𝐶𝜃_𝑛−1ℎ𝑥_𝑛−1+1−𝜃_𝑛−1𝑥_𝑛−1−𝛾_𝑛−1∇𝑓𝑥_𝑛−1≤1−𝜃_𝑛𝑥_𝑛−𝛾_𝑛∇𝑓𝑥_𝑛−1−𝜃_𝑛−1𝑥_𝑛−1−𝛾_𝑛−1∇𝑓𝑥_𝑛−1+𝜃_𝑛ℎ𝑥_𝑛−𝜃_𝑛−1ℎ𝑥_𝑛−1=1−𝜃_𝑛𝛾𝐼−^𝑛1−𝜃_𝑛∇𝑓𝑥_𝑛−1−𝜃_𝑛𝛾𝐼−^𝑛1−𝜃_𝑛∇𝑓𝑥_𝑛−1+𝜃_𝑛−1−𝜃_𝑛𝑥_𝑛−1+𝛾_𝑛−1−𝛾_𝑛∇𝑓𝑥_𝑛−1+𝜃_𝑛ℎ𝑥_𝑛−ℎ𝑥_𝑛−1+𝜃_𝑛−𝜃_𝑛−1ℎ𝑥_𝑛−1≤1−𝜃_𝑛𝛾𝐼−^𝑛1−𝜃_𝑛∇𝑓𝑥_𝑛−𝛾𝐼−^𝑛1−𝜃_𝑛∇𝑓𝑥_𝑛−1+𝜃_𝑛−𝜃_𝑛−1𝑥_𝑛−1+𝛾_𝑛−𝛾_𝑛−1∇𝑓𝑥_𝑛−1+𝜃_𝑛ℎ𝑥_𝑛−ℎ𝑥_𝑛−1+𝜃_𝑛−𝜃_𝑛−1ℎ𝑥_𝑛−1≤1−1−𝜌𝜃_𝑛𝑥_𝑛−𝑥_𝑛−1+𝜃_𝑛−𝜃_𝑛−1ℎ𝑥_𝑛−1+𝑥_𝑛−1+𝛾_𝑛−𝛾_𝑛−1∇𝑓𝑥_𝑛−1≤1−1−𝜌𝜃_𝑛𝑥_𝑛−𝑥_𝑛−1+𝛾_𝑛−𝛾_𝑛−1+𝜃_𝑛−𝜃_𝑛−1𝑀. (3.7) Then, we can combine the last inequality and Lemma 2.3 to conclude that lim_𝑛→∞𝑥_𝑛+1−𝑥_𝑛=0. (3.8) Now we show that the weak limit set 𝜔_𝑤(𝑥_𝑛)⊂𝑆. Choose any 𝑥∈𝜔_𝑤(𝑥_𝑛). Then there must exist a subsequence {𝑥_𝑛𝑗} of {𝑥_𝑛} such that 𝑥_𝑛𝑗⇀𝑥. At the same time, the real number sequence {𝛾_𝑛𝑗} is bounded. Thus, there exists a subsequence {𝛾_𝑛𝑗𝑖} of {𝛾_𝑛𝑗} which converges to 𝛾. Without loss of generality, we may assume that 𝛾_𝑛𝑗→𝛾. Note that 0<liminf_𝑛→∞𝛾_𝑛≤limsup_𝑛→∞𝛾_𝑛<2/𝐿. So, 𝛾∈(0,2/𝐿). That is, 𝛾_𝑛𝑗→𝛾∈(0,2/𝐿) as 𝑗→∞. Next, we only need to show that 𝑥∈𝑆. First, from (3.8) we have that ‖𝑥_𝑛𝑗+1−𝑥_𝑛𝑗‖→0. Then, we have 𝑥_𝑛𝑗−Proj_𝐶𝐼−𝛾∇𝑓𝑥_𝑛𝑗≤𝑥_𝑛𝑗−𝑥_𝑛𝑗+1+𝑥_𝑛𝑗+1−Proj_𝐶𝐼−𝛾_𝑛𝑗∇𝑓𝑥_𝑛𝑗+Proj_𝐶𝐼−𝛾_𝑛𝑗∇𝑓𝑥_𝑛𝑗−Proj_𝐶𝐼−𝛾∇𝑓𝑥_𝑛𝑗=Proj_𝐶𝜃_𝑛𝑗ℎ𝑥_𝑛𝑗+1−𝜃_𝑛𝑗𝑥_𝑛𝑗−𝛾_𝑛𝑗∇𝑓𝑥_𝑛𝑗−Proj_𝐶𝐼−𝛾_𝑛𝑗∇𝑓𝑥_𝑛𝑗+Proj_𝐶𝐼−𝛾_𝑛𝑗∇𝑓𝑥_𝑛𝑗−Proj_𝐶𝐼−𝛾∇𝑓𝑥_𝑛𝑗+𝑥_𝑛𝑗−𝑥_𝑛𝑗+1≤𝜃_𝑛𝑗ℎ𝑥_𝑛𝑗+𝑥_𝑛𝑗+𝛾_𝑛𝑗−𝛾∇𝑓𝑥_𝑛𝑗+𝑥_𝑛𝑗−𝑥_𝑛𝑗+1⟶0. (3.9) Since 𝛾∈(0,2/𝐿), Proj_𝐶(𝐼−𝛾∇𝑓) is nonexpansive. It then follows from Lemma 2.1 (demi-closedness principle) that 𝑥∈Fix(Proj_𝐶(𝐼−𝛾∇𝑓)). Hence, 𝑥∈𝑆 because of 𝑆=Fix(Proj_𝐶(𝐼−𝛾∇𝑓)). So, 𝜔_𝑤(𝑥_𝑛)⊂𝑆.
Finally, we prove that 𝑥_𝑛→𝑥, where 𝑥 is the unique solution of the VI (3.2). First, we show that limsup_𝑛→∞⟨(𝐼−ℎ)𝑥,𝑥_𝑛−𝑥⟩≥0. Observe that there exists a subsequence {𝑥_𝑛𝑗} of {𝑥_𝑛} satisfying limsup_𝑛→∞⟨(𝐼−ℎ)𝑥,𝑥_𝑛−𝑥⟩=lim_𝑗→∞⟨(𝐼−ℎ)𝑥,𝑥_𝑛𝑗−𝑥⟩. (3.10) Since {𝑥_𝑛𝑗} is bounded, there exists a subsequence {𝑥_𝑛𝑗𝑖} of {𝑥_𝑛𝑗} such that 𝑥_𝑛𝑗𝑖⇀𝑥. Without loss of generality, we assume that 𝑥_𝑛𝑗⇀𝑥. Then, we obtain limsup_𝑛→∞⟨(𝐼−ℎ)𝑥,𝑥_𝑛−𝑥⟩=lim_𝑗→∞⟨(𝐼−ℎ)𝑥,𝑥_𝑛𝑗−𝑥⟩=⟨(𝐼−ℎ)𝑥,𝑥−𝑥⟩≥0. (3.11) By using the property (ii) of Proj_𝐶, we have 𝑥_𝑛+1−𝑥²=Proj_𝐶𝜃_𝑛ℎ𝑥_𝑛+1−𝜃_𝑛𝑥_𝑛−𝛾_𝑛∇𝑓𝑥_𝑛−Proj_𝐶𝜃_𝑛𝑥+1−𝜃_𝑛𝑥−𝛾_𝑛∇𝑓(𝑥)²≤⟨𝜃_𝑛ℎ𝑥_𝑛−𝑥+1−𝜃_𝑛𝛾𝐼−^𝑛1−𝜃_𝑛∇𝑓𝑥_𝑛−𝛾𝐼−^𝑛1−𝜃_𝑛∇𝑓𝑥,𝑥_𝑛+1−𝑥⟩≤𝜃_𝑛⟨ℎ𝑥_𝑛−ℎ(𝑥),𝑥_𝑛+1−𝑥⟩+𝜃_𝑛⟨ℎ(𝑥)−𝑥,𝑥_𝑛+1+−𝑥⟩1−𝜃_𝑛𝛾𝐼−^𝑛1−𝜃_𝑛∇𝑓𝑥_𝑛−𝛾𝐼−^𝑛1−𝜃_𝑛∇𝑓𝑥𝑥_𝑛+1−𝑥≤𝜃_𝑛𝜌𝑥_𝑛−𝑥𝑥_𝑛+1−𝑥+𝜃_𝑛⟨ℎ(𝑥)−𝑥,𝑥_𝑛+1−𝑥⟩+1−𝜃_𝑛𝑥_𝑛−𝑥𝑥_𝑛+1=𝜃−𝑥1−1−𝜌_𝑛𝑥_𝑛−𝑥𝑥_𝑛+1−𝑥+𝜃_𝑛⟨ℎ𝑥^∗−𝑥,𝑥_𝑛+1≤𝜃−𝑥⟩1−1−𝜌_𝑛2𝑥_𝑛−𝑥²+12𝑥_𝑛+1−𝑥²+𝜃_𝑛ℎ(𝑥)−𝑥,𝑥_𝑛+1−𝑥. (3.12) It follows that 𝑥_𝑛+1−𝑥²≤1−1−𝜌𝜃_𝑛𝑥_𝑛−𝑥²+1−𝜌𝜃_𝑛2ℎ()1−𝜌𝑥−𝑥,𝑥_𝑛+1−𝑥. (3.13) From Lemma 2.3, (3.11), and (3.13), we deduce that 𝑥_𝑛→𝑥. This completes the proof.

From Theorem 3.1, we obtain immediately the following theorem.

Theorem 3.2. Let 𝐶 be a closed convex subset of a real Hilbert space 𝐻. Let 𝑓∶𝐶→𝑅 be a real-valued Fréchet differentiable convex function. Assume 𝑆≠∅. Assume that the gradient ∇𝑓 is 𝐿-Lipschitzian. Let {𝑥_𝑛} be a sequence generated by the following hybrid gradient projection algorithm: 𝑥_𝑛+1=Proj_𝐶1−𝜃_𝑛𝑥_𝑛−𝛾_𝑛∇𝑓𝑥_𝑛,𝑛≥0, (3.14) where the sequences {𝜃_𝑛}⊂(0,1) and {𝛾_𝑛}⊂(0,∞) satisfy the following conditions: (i)lim_𝑛→∞𝜃_𝑛=0, ∑^∞_𝑛=0𝜃_𝑛=∞ and ∑^∞_𝑛=0|𝜃_𝑛+1−𝜃_𝑛|<∞; (ii)0<liminf_𝑛→∞𝛾_𝑛≤limsup_𝑛→∞𝛾_𝑛<2/𝐿 and ∑^∞_𝑛=0|𝛾_𝑛+1−𝛾_𝑛|<∞. Then the sequence {𝑥_𝑛} generated by (3.14) converges to a minimizer 𝑥 of (1.1) which is the minimum norm element in 𝑆.

Proof. In Theorem 3.1, we note that ℎ is a non-self mapping from 𝐶 to the whole space 𝐻. Hence, if we chose ℎ(𝑥)≡0 for all 𝑥∈𝐶, then Algorithm (3.1) reduces to (3.14). And sequence 𝑥_𝑛 converges strongly to 𝑥=Proj_𝑆(0) which is obviously the minimum norm element in 𝑆. The proof is completed.

Next, we suggest another simple algorithm for dropping the assumption ∑^∞_𝑛=0|𝜃_𝑛+1−𝜃_𝑛|<∞.

Theorem 3.3. Let 𝐶 be a closed convex subset of a real Hilbert space 𝐻. Let 𝑓∶𝐶→𝑅 be a real-valued Fréchet differentiable convex function. Assume 𝑆≠∅. Assume that the gradient ∇𝑓 is 𝐿-Lipschitzian. Let {𝑥_𝑛} be a sequence generated by the following hybrid gradient projection algorithm: x_n+1=Proj_𝐶I−𝛾∇𝑓Proj_𝐶1−𝜃_𝑛x_𝑛,𝑛≥0, (3.15) where 𝛾∈(0,2/𝐿) is a constant and the sequences {𝜃_𝑛}⊂(0,1) satisfy the following conditions: (1)lim_𝑛→∞𝜃_𝑛=0; (2)∑^∞_𝑛=0𝜃_𝑛=∞. Then the sequence {𝑥_𝑛} generated by (3.15) converges to a minimizer 𝑥 of (1.1) which is the minimum norm element in 𝑆.

Proof. Claim 1. The sequence {𝑥_𝑛} is bounded.
Take 𝑥^∗∈𝑆. Then we have 𝑥_𝑛+1−𝑥^∗=Proj_𝐶𝐼−𝛾∇𝑓Proj_𝐶1−𝜃_𝑛𝑥_𝑛−Proj_𝐶𝐼−𝛾∇𝑓𝑥^∗≤1−𝜃_𝑛𝑥_𝑛−𝑥^∗≤1−𝜃_𝑛𝑥_𝑛−𝑥^∗+𝜃_𝑛𝑥^∗≤max𝑥_𝑛−𝑥^∗,𝑥^∗. (3.16) By induction, 𝑥_𝑛−𝑥^∗≤max𝑥₀−𝑥^∗,𝑥^∗. (3.17)
Claim 2. ‖𝑥_𝑛−Proj_𝐶(𝐼−𝛾∇𝑓)𝑥_𝑛‖→0 and 𝜔_𝑤(𝑥_𝑛)⊂𝑆.
By the similar argument as that in [18, page 366], we can write Proj_𝐶𝐼−𝛾∇𝑓=1−𝛽𝐼+𝛽𝑇, (3.18) where 𝑇 is nonexpansive and 𝛽=(2+𝛾𝐿)/4⊂(0,1). Then we can rewrite (3.15) as 𝑥_𝑛+1=1−𝛽𝐼+𝛽𝑇Proj_𝐶1−𝜃_𝑛𝑥_𝑛=1−𝛽𝑥_𝑛+𝛽𝑇Proj_𝐶1−𝜃_𝑛𝑥_𝑛+1−𝛽Proj_𝐶1−𝜃_𝑛𝑥_𝑛−𝑥_𝑛=1−𝛽𝑥_𝑛+𝛽𝑦_𝑛, (3.19) where 𝑦_𝑛=𝑇Proj_𝐶1−𝜃_𝑛𝑥_𝑛+1−𝛽𝛽Proj_𝐶1−𝜃_𝑛𝑥_𝑛−𝑥_𝑛. (3.20) It follows that 𝑦_𝑛+1−𝑦_𝑛=𝑇Proj_𝐶1−𝜃_𝑛+1𝑥_𝑛+1+1−𝛽𝛽Proj_𝐶1−𝜃_𝑛+1𝑥_𝑛+1−𝑥_𝑛+1−𝑇Proj_𝐶1−𝜃_𝑛𝑥_𝑛−1−𝛽𝛽Proj_𝐶1−𝜃_𝑛𝑥_𝑛−𝑥_𝑛≤𝑇Proj_𝐶1−𝜃_𝑛+1𝑥_𝑛+1−𝑇Proj_𝐶1−𝜃_𝑛𝑥_𝑛+1−𝛽𝛽Proj_𝐶1−𝜃_𝑛+1𝑥_𝑛+1−𝑥_𝑛+1+1−𝛽𝛽Proj_𝐶1−𝜃_𝑛𝑥_𝑛−𝑥_𝑛≤𝑥_𝑛+1−𝑥_𝑛+𝜃_𝑛+1𝛽𝑥_𝑛+1+𝜃_𝑛𝛽𝑥_𝑛. (3.21) So, limsup_𝑛→∞𝑦_𝑛+1−𝑦_𝑛−𝑥_𝑛+1−𝑥_𝑛≤0. (3.22) This together with Lemma 2.4 implies that lim_𝑛→∞𝑦_𝑛−𝑥_𝑛=0. (3.23) Thus, lim_𝑛→∞𝑥_𝑛+1−𝑥_𝑛=lim_𝑛→∞1−𝛽𝑦_𝑛−𝑥_𝑛=0. (3.24) Note that 𝑥_𝑛−Proj_𝐶𝐼−𝛾∇𝑓𝑥_𝑛≤𝑥_𝑛−𝑥_𝑛+1+𝑥_𝑛+1−Proj_𝐶𝐼−𝛾∇𝑓𝑥_𝑛=𝑥_𝑛−𝑥_𝑛+1+Proj_𝐶𝐼−𝛾∇𝑓Proj_𝐶1−𝜃_𝑛𝑥_𝑛−Proj_𝐶𝐼−𝛾∇𝑓𝑥_𝑛≤𝑥_𝑛−𝑥_𝑛+1+𝜃_𝑛𝑥_𝑛. (3.25) Therefore, lim_𝑛→∞𝑥_𝑛−Proj_𝐶𝐼−𝛾∇𝑓𝑥_𝑛=0. (3.26) Now repeating the proof of Theorem 3.1, we conclude that 𝜔_𝑤(𝑥_𝑛)⊂𝑆.
Claim 3. limsup_𝑛→∞⟨𝑥,𝑥_𝑛−𝑥⟩≥0 where 𝑥 is the minimum norm element in 𝑆.
Observe that there exists a subsequence {𝑥_𝑛𝑗} of {𝑥_𝑛} satisfying limsup_𝑛→∞𝑥,𝑥_𝑛−𝑥=lim_𝑗→∞𝑥,𝑥_𝑛𝑗−𝑥. (3.27) Since {𝑥_𝑛𝑗} is bounded, there exists a subsequence {𝑥_𝑛𝑗𝑖} of {𝑥_𝑛𝑗} such that 𝑥_𝑛𝑗𝑖⇀𝑥∈𝑆. Without loss of generality, we assume that 𝑥_𝑛𝑗⇀𝑥∈𝑆. Then, we obtain limsup_𝑛→∞𝑥,𝑥_𝑛−𝑥=lim_𝑗→∞𝑥,𝑥_𝑛𝑗−𝑥=⟨𝑥,𝑥−𝑥⟩≥0. (3.28)
Claim 4. 𝑥_𝑛→𝑥. From (3.15), we have 𝑥_𝑛+1−𝑥²=Proj_𝐶𝐼−𝛾∇𝑓Proj_𝐶1−𝜃_𝑛𝑥_𝑛−Proj_𝐶𝐼−𝛾∇𝑓𝑥²≤1−𝜃_𝑛𝑥_𝑛−𝑥²=1−𝜃_𝑛𝑥_𝑛−𝑥−𝜃_𝑛𝑥²=1−𝜃_𝑛²𝑥_𝑛−𝑥²−2𝜃_𝑛1−𝜃_𝑛⟨𝑥,𝑥_𝑛−𝑥⟩+𝜃²_𝑛‖𝑥‖²≤1−𝜃_𝑛𝑥_𝑛−𝑥²+𝜃_𝑛−21−𝜃_𝑛𝑥,𝑥_𝑛−𝑥+𝜃_𝑛‖𝑥‖². (3.29) It is obvious that limsup_𝑛→∞(−2(1−𝜃_𝑛)⟨𝑥,𝑥_𝑛−𝑥⟩+𝜃_𝑛‖𝑥‖²)≤0. Then we can apply Lemma 2.3 to the last inequality to conclude that 𝑥_𝑛→𝑥. The proof is completed.

Next, we suggest another algorithm with the additional projections applied to the gradient projection algorithm. We show that this algorithm has strong convergence.

Theorem 3.4. Let 𝐶 be a closed convex subset of a real Hilbert space 𝐻. Let 𝑓∶𝐶→𝑅 be a real-valued Fréchet differentiable convex function. Assume 𝑆≠∅. Assume that the gradient ∇𝑓 is 𝐿-Lipschitzian. Let 𝑥₀∈𝐻. For 𝐶₁=𝐶 and 𝑥₁=Proj_𝐶1𝑥₀, define a sequence {𝑥_𝑛} of 𝐶 as follows: 𝑦_𝑛=Proj_𝐶𝑥_𝑛−𝛾_𝑛∇𝑓𝑥_𝑛𝐶,𝑛≥0,_𝑛=𝑧∈𝐶_𝑛−1∶𝑦_𝑛−𝑧≤𝑥_𝑛,𝑥−𝑧_𝑛+1=Proj_𝐶𝑛𝑥₀. (3.30) where the sequence {𝛾_𝑛}⊂(0,∞) satisfies the condition 0<liminf_𝑛→∞𝛾_𝑛≤limsup_𝑛→∞𝛾_𝑛<2/𝐿. Then the sequence {𝑥_𝑛} generated by (3.30) converges to 𝑥=Proj_𝑆(𝑥₀).

Proof. It is obvious that 𝐶_𝑛 is convex. For any 𝑥^∗∈𝑆, we have 𝑦_𝑛−𝑥^∗=Proj_𝐶𝑥_𝑛−𝛾_𝑛∇𝑓𝑥_𝑛−Proj_𝐶𝑥^∗−𝛾_𝑛∇𝑓𝑥^∗≤𝑥_𝑛−𝑥^∗. (3.31) This implies that 𝑥^∗∈𝐶_𝑛. Hence, 𝑆⊂𝐶_𝑛. From 𝑥_𝑛+1=Proj_𝐶𝑛𝑥₀, we have ⟨𝑥₀−𝑥_𝑛+1,𝑥_𝑛+1−𝑦⟩≥0,∀𝑦∈𝐶_𝑛. (3.32) Since 𝑆⊂𝐶_𝑛, we have ⟨𝑥₀−𝑥_𝑛+1,𝑥_𝑛+1−𝑢⟩≥0,∀𝑢∈𝑆. (3.33) So, for 𝑢∈𝑆, we have 0≤⟨𝑥₀−𝑥_𝑛+1,𝑥_𝑛+1−𝑢⟩=⟨𝑥₀−𝑥_𝑛+1,𝑥_𝑛+1−𝑥₀+𝑥₀𝑥−𝑢⟩=−₀−𝑥_𝑛+1²+⟨𝑥₀−𝑥_𝑛+1,𝑥₀𝑥−𝑢⟩≤−₀−𝑥_𝑛+1²+𝑥₀−𝑥_𝑛+1𝑥₀−𝑢. (3.34) Hence, 𝑥₀−𝑥_𝑛+1≤𝑥₀−𝑢,∀𝑢∈𝑆. (3.35) This implies that {𝑥_𝑛} is bounded.
From 𝑥_𝑛=Proj_𝐶𝑛−1𝑥₀ and 𝑥_𝑛+1=Proj_𝐶𝑛𝑥₀∈𝐶_𝑛⊂𝐶_𝑛−1, we have ⟨𝑥₀−𝑥_𝑛,𝑥_𝑛−𝑥_𝑛+1⟩≥0. (3.36) Hence, 0≤⟨𝑥₀−𝑥_𝑛,𝑥_𝑛−𝑥_𝑛+1⟩=⟨𝑥₀−𝑥_𝑛,𝑥_𝑛−𝑥₀+𝑥₀−𝑥_𝑛+1⟩=−𝑥₀−𝑥_𝑛²+⟨𝑥₀−𝑥_𝑛,𝑥₀−𝑥_𝑛+1⟩≤−𝑥₀−𝑥_𝑛²+𝑥₀−𝑥_𝑛𝑥₀−𝑥_𝑛+1, (3.37) and therefore 𝑥₀−𝑥_𝑛≤𝑥₀−𝑥_𝑛+1. (3.38) This implies that lim_𝑛→∞𝑥_𝑛−𝑥₀exists. (3.39)
From (3.36) and (3.39), we obtain 𝑥_𝑛+1−𝑥_𝑛²=(𝑥_𝑛+1−𝑥₀)−(𝑥_𝑛−𝑥₀)²=𝑥_𝑛+1−𝑥₀²−𝑥_𝑛−𝑥₀²−2⟨𝑥_𝑛+1−𝑥_𝑛,𝑥_𝑛−𝑥₀⟩≤𝑥_𝑛+1−𝑥₀²−𝑥_𝑛−𝑥₀²⟶0. (3.40) By the fact 𝑥_𝑛+1∈𝐶_𝑛, we get 𝑦_𝑛−𝑥_𝑛+1≤𝑥_𝑛−𝑥_𝑛+1⟶0. (3.41) Therefore, from (3.40) and (3.41), we deduce lim_𝑛→∞𝑥_𝑛−𝑦_𝑛=lim_𝑛→∞𝑥_𝑛−Proj_𝐶𝐼−𝛾_𝑛∇𝑓𝑥_𝑛=0. (3.42) Now (3.42) and Lemma 2.1 guarantee that every weak limit point of {𝑥_𝑛} is a fixed point of Proj_𝐶(𝐼−𝛾_𝑛∇𝑓). That is, 𝜔_𝑤(𝑥_𝑛)⊂Fix(Proj_𝐶(𝐼−𝛾_𝑛∇𝑓))=𝑆. At the same time, if we choose 𝑢=Proj_𝑆(𝑥₀) in (3.35), we have 𝑥₀−𝑥_𝑛+1≤𝑥₀−Proj_𝑆𝑥₀. (3.43) This fact and Lemma 2.2 ensure the strong convergence of {𝑥_𝑛} to Proj_𝑆(𝑥₀). This completes the proof.

Now we give some remarks on our variant gradient projection methods.

Remark 3.5. Under the same control parameters, the gradient projection methods (3.1) and (1.6) are all strong convergent. However, (3.1) seems to have more advantage than (1.6) as ℎ is a non-self-mapping.

Remark 3.6. The gradient projection method (3.14) is similar to (1.5) by using (1−𝜃_𝑛)𝑥_𝑛 instead of 𝑥_𝑛. But (3.1) has strong convergence, and especially (3.1) converges strongly to the minimum norm element of 𝑆.

Remark 3.7. The advantage of the gradient projection method (3.15) is that it has strong convergence under some weaker assumptions on parameter 𝜃_𝑛.

Remark 3.8. The gradient projection method (3.30) is simpler than (1.7).

Acknowledgments

Y. Yao was supported in part by NSFC 11071279 and NSFC 71161001-G0105. Y.-C. Liou was supported in part by NSC 100-2221-E-230-012. C.-F. Wen was supported in part by NSC 100-2115-M-037-001.