Journal of Applied Mathematics

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Journal of Applied Mathematics / 2012 / Article
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Numerical and Analytical Methods for Variational Inequalities and Related Problems with Applications

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Research Article | Open Access

Volume 2012 |Article ID 782960 | 14 pages | https://doi.org/10.1155/2012/782960

A Hybrid Gradient-Projection Algorithm for Averaged Mappings in Hilbert Spaces

Ming Tian 1 and Min-Min Li1

1College of Science, Civil Aviation University of China, Tianjin 300300, China

Academic Editor: Hong-Kun Xu
Received18 Apr 2012
Accepted21 Jun 2012
Published25 Jul 2012

Abstract

It is well known that the gradient-projection algorithm (GPA) is very useful in solving constrained convex minimization problems. In this paper, we combine a general iterative method with the gradient-projection algorithm to propose a hybrid gradient-projection algorithm and prove that the sequence generated by the hybrid gradient-projection algorithm converges in norm to a minimizer of constrained convex minimization problems which solves a variational inequality.

1. Introduction

Let 𝐻 be a real Hilbert space and 𝐶 a nonempty closed and convex subset of 𝐻. Consider the following constrained convex minimization problem: minimize𝑥∈𝐶𝑓(𝑥),(1.1) where 𝑓∶𝐶→ℝ is a real-valued convex and continuously Fréchet differentiable function. The gradient ∇𝑓 satisfies the following Lipschitz condition: ‖∇𝑓(𝑥)−∇𝑓(𝑦)‖≤𝐿‖𝑥−𝑦‖,∀𝑥,𝑦∈𝐶,(1.2) where 𝐿>0. Assume that the minimization problem (1.1) is consistent, and let 𝑆 denote its solution set.

It is well known that the gradient-projection algorithm is very useful in dealing with constrained convex minimization problems and has extensively been studied ([1–5] and the references therein). It has recently been applied to solve split feasibility problems [6–10]. Levitin and Polyak [1] consider the following gradient-projection algorithm: 𝑥𝑛+1∶=Proj𝐶𝑥𝑛−𝜆𝑛𝑥∇𝑓𝑛,𝑛≥0.(1.3) Let {𝜆𝑛}∞𝑛=0 satisfy 0<liminf𝑛→∞𝜆𝑛≤limsup𝑛→∞𝜆𝑛<2𝐿.(1.4) It is proved that the sequence {𝑥𝑛} generated by (1.3) converges weakly to a minimizer of (1.1).

Xu proved that under certain appropriate conditions on {𝛼𝑛} and {𝜆𝑛} the sequence {𝑥𝑛} defined by the following relaxed gradient-projection algorithm: 𝑥𝑛+1=1−𝛼𝑛𝑥𝑛+𝛼𝑛Proj𝐶𝑥𝑛−𝜆𝑛𝑥∇𝑓𝑛,𝑛≥0,(1.5) converges weakly to a minimizer of (1.1) [11].

Since the Lipschitz continuity of the gradient of 𝑓 implies that it is indeed inverse strongly monotone (ism) [12, 13], its complement can be an averaged mapping. Recall that a mapping 𝑇 is nonexpansive if and only if it is Lipschitz with Lipschitz constant not more than one, that a mapping is an averaged mapping if and only if it can be expressed as a proper convex combination of the identity mapping and a nonexpansive mapping, and that a mapping 𝑇 is said to be 𝜈-inverse strongly monotone if and only if ⟨𝑥−𝑦,𝑇𝑥−𝑇𝑦⟩≥𝜈‖𝑇𝑥−𝑇𝑦‖2forall𝑥,𝑦∈𝐻, where the number 𝜈>0. Recall also that the composite of finitely many averaged mappings is averaged. That is, if each of the mappings {𝑇𝑖}𝑁𝑖=1 is averaged, then so is the composite 𝑇1⋯𝑇𝑁 [14]. In particular, an averaged mapping is a nonexpansive mapping [15]. As a result, the GPA can be rewritten as the composite of a projection and an averaged mapping which is again an averaged mapping.

Generally speaking, in infinite-dimensional Hilbert spaces, GPA has only weak convergence. Xu [11] provided a modification of GPA so that strong convergence is guaranteed. He considered the following hybrid gradient-projection algorithm: 𝑥𝑛+1=𝜃𝑛ℎ𝑥𝑛+1−𝜃𝑛Proj𝐶𝑥𝑛−𝜆𝑛𝑥∇𝑓𝑛.(1.6)

It is proved that if the sequences {𝜃𝑛} and {𝜆𝑛} satisfy appropriate conditions, the sequence {𝑥𝑛} generated by (1.6) converges in norm to a minimizer of (1.1) which solves the variational inequality 𝑥∗∈𝑆,⟨(𝐼−ℎ)𝑥∗,𝑥−𝑥∗⟩≥0,𝑥∈𝑆.(1.7)

On the other hand, Ming Tian [16] introduced the following general iterative algorithm for solving the variational inequality 𝑥𝑛+1=𝛼𝑛𝑥𝛾𝑓𝑛+𝐼−𝜇𝛼𝑛𝐹𝑇𝑥𝑛,𝑛≥0,(1.8) where 𝐹 is a 𝜅-Lipschitzian and 𝜂-strongly monotone operator with 𝜅>0, 𝜂>0 and 𝑓 is a contraction with coefficient 0<𝛼<1. Then, he proved that if {𝛼𝑛} satisfying appropriate conditions, the {𝑥𝑛} generated by (1.8) converges strongly to the unique solution of variational inequality ⟨(𝜇𝐹−𝛾𝑓)̃𝑥,̃𝑥−𝑧⟩≤0,𝑧∈Fix(𝑇).(1.9)

In this paper, motivated and inspired by the research work in this direction, we will combine the iterative method (1.8) with the gradient-projection algorithm (1.3) and consider the following hybrid gradient-projection algorithm: 𝑥𝑛+1=𝜃𝑛𝑥𝛾ℎ𝑛+𝐼−𝜇𝜃𝑛𝐹Proj𝐶𝑥𝑛−𝜆𝑛𝑥∇𝑓𝑛,𝑛≥0.(1.10)

We will prove that if the sequence {𝜃𝑛} of parameters and the sequence {𝜆𝑛} of parameters satisfy appropriate conditions, then the sequence {𝑥𝑛} generated by (1.10) converges in norm to a minimizer of (1.1) which solves the variational inequality (𝑉𝐼)𝑥∗∈𝑆,⟨(𝜇𝐹−𝛾ℎ)𝑥∗,𝑥−𝑥∗⟩≥0,∀𝑥∈𝑆,(1.11) where 𝑆 is the solution set of the minimization problem (1.1).

2. Preliminaries

This section collects some lemmas which will be used in the proofs for the main results in the next section. Some of them are known; others are not hard to derive.

Throughout this paper, we write 𝑥𝑛⇀𝑥 to indicate that the sequence {𝑥𝑛} converges weakly to 𝑥, 𝑥𝑛→𝑥 implies that {𝑥𝑛} converges strongly to 𝑥. 𝜔𝑤(𝑥𝑛)∶={𝑥∶∃𝑥𝑛𝑗⇀𝑥} is the weak 𝜔-limit set of the sequence {𝑥𝑛}∞𝑛=1.

Lemma 2.1 (see [17]). Assume that {𝑎𝑛}∞𝑛=0 is a sequence of nonnegative real numbers such that 𝑎𝑛+1≤1−𝛾𝑛𝑎𝑛+𝛾𝑛𝛿𝑛+𝛽𝑛,𝑛≥0,(2.1) where {𝛾𝑛}∞𝑛=0 and {𝛽𝑛}∞𝑛=0 are sequences in [0,1] and {𝛿𝑛}∞𝑛=0 is a sequence in ℝ such that(i)∑∞𝑛=0𝛾𝑛=∞; (ii)either limsup𝑛→∞𝛿𝑛≤0 or ∑∞𝑛=0𝛾𝑛|𝛿𝑛|<∞;(iii)∑∞𝑛=0𝛽𝑛<∞. Then lim𝑛→∞𝑎𝑛=0.

Lemma 2.2 (see [18]). Let 𝐶 be a closed and convex subset of a Hilbert space 𝐻, and let 𝑇∶𝐶→𝐶 be a nonexpansive mapping with Fix𝑇≠∅. If {𝑥𝑛}∞𝑛=1 is a sequence in 𝐶 weakly converging to 𝑥 and if {(𝐼−𝑇)𝑥𝑛}∞𝑛=1 converges strongly to y, then (𝐼−T)𝑥=𝑦.

Lemma 2.3. Let 𝐻 be a Hilbert space, and let 𝐶 be a nonempty closed and convex subset of 𝐻. ℎ∶𝐶→𝐶 a contraction with coefficient 0<𝜌<1, and 𝐹∶𝐶→𝐶 a 𝜅-Lipschitzian continuous operator and 𝜂-strongly monotone operator with 𝜅,𝜂>0. Then, for 0<𝛾<𝜇𝜂/𝜌, ⟨𝑥−𝑦,(𝜇𝐹−𝛾ℎ)𝑥−(𝜇𝐹−𝛾ℎ)𝑦⟩≥(𝜇𝜂−𝛾𝜌)‖𝑥−𝑦‖2,∀𝑥,𝑦∈𝐶.(2.2) That is, 𝜇𝐹−𝛾ℎ is strongly monotone with coefficient 𝜇𝜂−𝛾𝜌.

Lemma 2.4. Let 𝐶 be a closed subset of a real Hilbert space 𝐻, given 𝑥∈𝐻 and 𝑦∈𝐶. Then, 𝑦=𝑃𝐶𝑥 if and only if there holds the inequality ⟨𝑥−𝑦,𝑦−𝑧⟩≥0,∀𝑧∈𝐶.(2.3)

3. Main Results

Let 𝐻 be a real Hilbert space, and let 𝐶 be a nonempty closed and convex subset of 𝐻 such that 𝐶±𝐶⊂𝐶. Assume that the minimization problem (1.1) is consistent, and let 𝑆 denote its solution set. Assume that the gradient ∇𝑓 satisfies the Lipschitz condition (1.2). Since 𝑆 is a closed convex subset, the nearest point projection from 𝐻 onto 𝑆 is well defined. Recall also that a contraction on 𝐶 is a self-mapping of 𝐶 such that ‖ℎ(𝑥)−ℎ(𝑦)‖≤𝜌‖𝑥−𝑦‖,forall𝑥,𝑦∈𝐶, where 𝜌∈[0,1) is a constant. Let 𝐹 be a 𝜅-Lipschitzian and 𝜂-strongly monotone operator on 𝐶 with 𝜅,𝜂>0. Denote by Π the collection of all contractions on 𝐶, namely, Π={ℎ∶ℎisacontractionon𝐶}.(3.1) Now given ℎ∈Π with 0<𝜌<1, 𝑠∈(0,1). Let 0<𝜇<2𝜂/𝜅2,   0<𝛾<𝜇(𝜂−(𝜇𝜅2)/2)/𝜌=𝜏/𝜌. Assume that 𝜆𝑠 with respect to 𝑠 is continuous and, in addition, 𝜆𝑠∈[𝑎,𝑏]⊂(0,2/𝐿). Consider a mapping 𝑋𝑠 on 𝐶 defined by 𝑋𝑠(𝑥)=𝑠𝛾ℎ(𝑥)+(𝐼−𝑠𝜇𝐹)Proj𝐶𝐼−𝜆𝑠∇𝑓(𝑥),𝑥∈𝐶.(3.2) It is easy to see that 𝑋𝑠 is a contraction. Setting 𝑉𝑠∶=Proj𝐶(𝐼−𝜆𝑠∇𝑓). It is obvious that 𝑉𝑠 is a nonexpansive mapping. We can rewrite 𝑋𝑠(𝑥) as 𝑋𝑠(𝑥)=𝑠𝛾ℎ(𝑥)+(𝐼−𝑠𝜇𝐹)𝑉𝑠(𝑥).(3.3) First observe that for 𝑠∈(0,1), we can get ‖‖(𝐼−𝑠𝜇𝐹)𝑉𝑠(𝑥)−(𝐼−s𝜇𝐹)𝑉𝑠‖‖(𝑦)2=‖‖𝑉𝑠(𝑥)−𝑉𝑠(𝑦)−𝑠𝜇𝐹𝑉𝑠(𝑥)−𝐹𝑉𝑠‖‖(𝑦)2=‖‖𝑉𝑠(𝑥)−𝑉𝑠‖‖(𝑦)2−2𝑠𝜇⟨𝑉𝑠(𝑥)−𝑉𝑠(𝑦),𝐹𝑉𝑠(𝑥)−𝐹𝑉𝑠(𝑦)⟩+𝑠2𝜇2‖‖𝐹𝑉𝑠(𝑥)−𝐹𝑉𝑠‖‖(𝑦)2≤‖𝑥−𝑦‖2‖‖𝑉−2𝑠𝜇𝜂𝑠(𝑥)−𝑉𝑠‖‖(𝑦)2+𝑠2𝜇2𝜅2‖‖𝑉𝑠(𝑥)−𝑉𝑠‖‖(𝑦)2≤1−𝑠𝜇2𝜂−𝑠𝜇𝜅2‖𝑥−𝑦‖2≤1−𝑠𝜇2𝜂−𝑠𝜇𝜅222‖𝑥−𝑦‖2≤1−𝑠𝜇𝜂−𝜇𝜅222‖𝑥−𝑦‖2=(1−𝑠𝜏)2‖𝑥−𝑦‖2.(3.4) Indeed, we have ‖‖𝑋𝑠(𝑥)−𝑋𝑠‖‖=‖‖(𝑦)𝑠𝛾ℎ(𝑥)+(𝐼−𝑠𝜇𝐹)𝑉𝑠(𝑥)−𝑠𝛾ℎ(𝑦)−(𝐼−𝑠𝜇𝐹)𝑉𝑠‖‖‖‖(𝑦)≤𝑠𝛾‖ℎ(𝑥)−ℎ(𝑦)‖+(𝐼−𝑠𝜇𝐹)𝑉𝑠(𝑥)−(𝐼−𝑠𝜇𝐹)𝑉𝑠‖‖(𝑦)≤𝑠𝛾𝜌‖𝑥−𝑦‖+(1−𝑠𝜏)‖𝑥−𝑦‖=(1−𝑠(𝜏−𝛾𝜌))‖𝑥−𝑦‖.(3.5) Hence, 𝑋𝑠 has a unique fixed point, denoted 𝑥𝑠, which uniquely solves the fixed-point equation 𝑥𝑠𝑥=𝑠𝛾ℎ𝑠+(𝐼−𝑠𝜇𝐹)𝑉𝑠𝑥𝑠.(3.6) The next proposition summarizes the properties of {𝑥𝑠}.

Proposition 3.1. Let 𝑥𝑠 be defined by (3.6).(i){𝑥𝑠}isboundedfor𝑠∈(0,(1/𝜏)). (ii)lim𝑠→0‖𝑥𝑠−Proj𝐶(𝐼−𝜆𝑠∇𝑓)(𝑥𝑠)‖=0. (iii)𝑥𝑠definesacontinuouscurvefrom(0,1/𝜏)into𝐻.

Proof. (i) Take a 𝑥∈𝑆, then we have ‖‖𝑥𝑠−𝑥‖‖=‖‖𝑥𝑠𝛾ℎ𝑠+(𝐼−𝑠𝜇𝐹)Proj𝐶𝐼−𝜆𝑠𝑥∇𝑓𝑠−𝑥‖‖=‖‖(𝐼−𝑠𝜇𝐹)Proj𝐶𝐼−𝜆𝑠𝑥∇𝑓𝑠−(𝐼−𝑠𝜇𝐹)Proj𝐶𝐼−𝜆𝑠∇𝑓𝑥𝑥+𝑠𝛾ℎ𝑠−𝜇𝐹Proj𝐶𝐼−𝜆𝑠∇𝑓𝑥‖‖‖‖𝑥≤(1−𝑠𝜏)𝑠−𝑥‖‖‖‖𝑥+𝑠𝛾ℎ𝑠−𝜇𝐹𝑥‖‖≤‖‖𝑥(1−𝑠𝜏)𝑠−𝑥‖‖‖‖𝑥+𝑠𝛾𝜌𝑠−𝑥‖‖‖‖+𝑠𝛾ℎ𝑥−𝜇𝐹𝑥‖‖.(3.7) It follows that ‖‖𝑥𝑠−𝑥‖‖≤‖‖𝛾ℎ𝑥−𝜇𝐹𝑥‖‖.𝜏−𝛾𝜌(3.8) Hence, {𝑥𝑠} is bounded.
(ii) By the definition of {𝑥𝑠}, we have ‖‖𝑥𝑠−Proj𝐶𝐼−𝜆𝑠𝑥∇𝑓𝑠‖‖=‖‖𝑥𝑠𝛾ℎ𝑠+(𝐼−𝑠𝜇𝐹)Proj𝐶𝐼−𝜆𝑠𝑥∇𝑓𝑠−Proj𝐶𝐼−𝜆𝑠𝑥∇𝑓𝑠‖‖‖‖𝑥=𝑠𝛾ℎ𝑠−𝜇𝐹Proj𝐶𝐼−𝜆𝑠𝑥∇𝑓s‖‖⟶0,(3.9){𝑥𝑠} is bounded, so are {ℎ(𝑥𝑠)} and {𝐹Proj𝐶(𝐼−𝜆𝑠∇𝑓)(𝑥𝑠)}.
(iii) Take 𝑠, 𝑠0∈(0,1/𝜏), and we have ‖‖𝑥𝑠−𝑥𝑠0‖‖=‖‖𝑥𝑠𝛾ℎ𝑠+(𝐼−𝑠𝜇𝐹)Proj𝐶𝐼−𝜆𝑠𝑥∇𝑓𝑠−𝑠0𝑥𝛾ℎ𝑠0−𝐼−𝑠0𝜇𝐹Proj𝐶𝐼−𝜆𝑠0𝑥∇𝑓𝑠0‖‖≤‖‖𝑠−𝑠0𝑥𝛾ℎ𝑠+𝑠0𝛾ℎ𝑥𝑠𝑥−ℎ𝑠0‖‖+‖‖𝐼−𝑠0𝜇𝐹Proj𝐶𝐼−𝜆𝑠0𝑥∇𝑓𝑠−𝐼−𝑠0𝜇𝐹Proj𝐶𝐼−𝜆𝑠0𝑥∇𝑓𝑠0‖‖+‖‖(𝐼−𝑠𝜇𝐹)Proj𝐶𝐼−𝜆𝑠𝑥∇𝑓𝑠−𝐼−𝑠0𝜇𝐹Proj𝐶𝐼−𝜆𝑠0𝑥∇𝑓𝑠‖‖≤‖‖𝑠−𝑠0𝑥𝛾ℎ𝑠+𝑠0𝛾ℎ𝑥𝑠𝑥−ℎ𝑠0‖‖+‖‖𝐼−𝑠0𝜇𝐹Proj𝐶𝐼−𝜆𝑠0𝑥∇𝑓𝑠−𝐼−𝑠0𝜇𝐹Proj𝐶𝐼−𝜆𝑠0𝑥∇𝑓𝑠0‖‖+‖‖(𝐼−𝑠𝜇𝐹)Proj𝐶𝐼−𝜆𝑠𝑥∇𝑓𝑠−(𝐼−𝑠𝜇𝐹)Proj𝐶𝐼−𝜆𝑠0𝑥∇𝑓𝑠‖‖+‖‖(𝐼−𝑠𝜇𝐹)Proj𝐶𝐼−𝜆𝑠0𝑥∇𝑓𝑠−𝐼−𝑠0𝜇𝐹Proj𝐶𝐼−𝜆𝑠0𝑥∇𝑓𝑠‖‖≤||𝑠−𝑠0||𝛾‖‖ℎ𝑥𝑠‖‖+𝑠0‖‖𝑥𝛾𝜌𝑠−𝑥𝑠0‖‖+1−𝑠0𝜏‖‖𝑥𝑠−𝑥𝑠0‖‖+||𝜆𝑠−𝜆𝑠0||‖‖𝑥∇𝑓𝑠‖‖+‖‖𝑠𝜇𝐹Proj𝐶𝐼−𝜆𝑠0𝑥∇𝑓𝑠−𝑠0𝜇𝐹Proj𝐶𝐼−𝜆𝑠0𝑥∇𝑓𝑠‖‖=||𝑠−𝑠0||𝛾‖‖ℎ𝑥𝑠‖‖+𝑠0‖‖𝑥𝛾𝜌𝑠−𝑥𝑠0‖‖+1−𝑠0𝜏‖‖𝑥𝑠−𝑥𝑠0‖‖+||𝜆𝑠−𝜆𝑠0||‖‖𝑥∇𝑓𝑠‖‖+||𝑠−𝑠0||‖‖𝜇𝐹Proj𝐶𝐼−𝜆𝑠0𝑥∇𝑓𝑠‖‖=𝛾‖‖ℎ𝑥𝑠‖‖‖‖+𝜇𝐹Proj𝐶𝐼−𝜆𝑠0𝑥∇𝑓𝑠‖‖||𝑠−𝑠0||+𝑠0‖‖𝑥𝛾𝜌𝑠−𝑥𝑠0‖‖+1−𝑠0𝜏‖‖𝑥𝑠−𝑥𝑠0‖‖+||𝜆𝑠−𝜆𝑠0||‖‖𝑥∇𝑓𝑠‖‖.(3.10) Therefore, ‖‖𝑥𝑠−𝑥𝑠0‖‖≤𝛾‖‖ℎ𝑥𝑠‖‖‖‖+𝜇𝐹Proj𝐶𝐼−𝜆𝑠0𝑥∇𝑓𝑠‖‖𝑠0||(𝜏−𝛾𝜌)𝑠−𝑠0||+‖‖𝑥∇𝑓𝑠‖‖𝑠0||𝜆(𝜏−𝛾𝜌)𝑠−𝜆𝑠0||.(3.11) Therefore, 𝑥𝑠→𝑥𝑠0 as 𝑠→𝑠0. This means 𝑥𝑠 is continuous.

Our main result in the following shows that {𝑥𝑠} converges in norm to a minimizer of (1.1) which solves some variational inequality.

Theorem 3.2. Assume that {𝑥𝑠} is defined by (3.6), then 𝑥𝑠 converges in norm as 𝑠→0 to a minimizer of (1.1) which solves the variational inequality ⟨(𝜇𝐹−𝛾ℎ)𝑥∗,̃𝑥−𝑥∗⟩≥0,∀̃𝑥∈𝑆.(3.12) Equivalently, we have Proj𝑠(𝐼−(𝜇𝐹−𝛾ℎ))𝑥∗=𝑥∗.

Proof. It is easy to see that the uniqueness of a solution of the variational inequality (3.12). By Lemma 2.3, 𝜇𝐹−𝛾ℎ is strongly monotone, so the variational inequality (3.12) has only one solution. Let 𝑥∗∈𝑆 denote the unique solution of (3.12).
To prove that 𝑥𝑠→𝑥∗(𝑠→0), we write, for a given ̃𝑥∈𝑆, 𝑥𝑠𝑥−̃𝑥=𝑠𝛾ℎ𝑠+(𝐼−𝑠𝜇𝐹)Proj𝐶𝐼−𝜆𝑠𝑥∇𝑓𝑠𝑥−̃𝑥=𝑠𝛾ℎ𝑠−𝜇𝐹̃𝑥+(𝐼−𝑠𝜇𝐹)Proj𝐶𝐼−𝜆𝑠𝑥∇𝑓𝑠−(𝐼−𝑠𝜇𝐹)Proj𝐶𝐼−𝜆𝑠∇𝑓(̃𝑥).(3.13) It follows that ‖‖𝑥𝑠‖‖−̃𝑥2𝑥=𝑠𝛾ℎ𝑠−𝜇𝐹̃𝑥,𝑥𝑠+−̃𝑥(𝐼−𝑠𝜇𝐹)Proj𝐶𝐼−𝜆𝑠𝑥∇𝑓𝑠−(𝐼−𝑠𝜇𝐹)Proj𝐶𝐼−𝜆𝑠∇𝑓(̃𝑥),𝑥𝑠‖‖𝑥−̃𝑥≤(1−𝑠𝜏)𝑠‖‖−̃𝑥2𝑥+𝑠𝛾ℎ𝑠−𝜇𝐹̃𝑥,𝑥𝑠.−̃𝑥(3.14) Hence, ‖‖𝑥𝑠‖‖−̃𝑥2≤1𝜏𝑥𝛾ℎ𝑠−𝜇𝐹̃𝑥,𝑥𝑠≤1−̃𝑥𝜏‖‖𝑥𝛾𝜌𝑠‖‖−̃𝑥2+⟨𝛾ℎ(̃𝑥)−𝜇𝐹̃𝑥,𝑥𝑠.−̃𝑥⟩(3.15) To derive that ‖‖𝑥𝑠‖‖−̃𝑥2≤1𝜏−𝛾𝜌⟨𝛾ℎ(̃𝑥)−𝜇𝐹̃𝑥,𝑥𝑠−̃𝑥⟩.(3.16) Since {𝑥𝑠} is bounded as 𝑠→0, we see that if {𝑠𝑛} is a sequence in (0,1) such that 𝑠𝑛→0 and 𝑥𝑠𝑛⇀𝑥, then by (3.16), 𝑥𝑠𝑛→𝑥. We may further assume that 𝜆𝑠𝑛→𝜆∈[0,2/𝐿] due to condition (1.4). Notice that Proj𝐶(𝐼−𝜆∇𝑓) is nonexpansive. It turns out that ‖‖𝑥𝑠𝑛−Proj𝐶(𝐼−𝜆∇𝑓)𝑥𝑠𝑛‖‖≤‖‖𝑥𝑠𝑛−Proj𝐶𝐼−𝜆𝑠𝑛𝑥∇𝑓𝑠𝑛‖‖+‖‖Proj𝐶𝐼−𝜆𝑠𝑛𝑥∇𝑓𝑠𝑛−Proj𝐶(𝐼−𝜆∇𝑓)𝑥𝑠𝑛‖‖≤‖‖𝑥𝑠𝑛−Proj𝐶𝐼−𝜆𝑠𝑛𝑥∇𝑓𝑠𝑛‖‖+‖‖𝜆−𝜆𝑠𝑛𝑥∇𝑓𝑠𝑛‖‖=‖‖𝑥𝑠𝑛−Proj𝐶𝐼−𝜆𝑠𝑛𝑥∇𝑓𝑠𝑛‖‖+||𝜆−𝜆𝑠𝑛||‖‖𝑥∇𝑓𝑠𝑛‖‖.(3.17) From the boundedness of {𝑥𝑠} and lim𝑠→0‖Proj𝐶(𝐼−𝜆𝑠∇𝑓)𝑥𝑠−𝑥𝑠‖=0, we conclude that lim𝑛→∞‖‖𝑥𝑠𝑛−Proj𝐶(𝐼−𝜆∇𝑓)𝑥𝑠𝑛‖‖=0.(3.18) Since 𝑥𝑠𝑛⇀𝑥, by Lemma 2.2, we obtain 𝑥=Proj𝐶(𝐼−𝜆∇𝑓)𝑥.(3.19) This shows that 𝑥∈𝑆.
We next prove that 𝑥 is a solution of the variational inequality (3.12). Since 𝑥𝑠𝑥=𝑠𝛾ℎ𝑠+(𝐼−𝑠𝜇𝐹)Proj𝐶𝐼−𝜆𝑠𝑥∇𝑓𝑠,(3.20) we can derive that 𝑥(𝜇𝐹−𝛾ℎ)𝑠1=−𝑠𝐼−Proj𝐶𝐼−𝜆𝑠𝑥∇𝑓𝑠𝐹𝑥+𝜇𝑠−𝐹Proj𝐶𝐼−𝜆𝑠𝑥∇𝑓𝑠.(3.21) Therefore, for ̃𝑥∈𝑆, 𝑥(𝜇𝐹−𝛾ℎ)𝑠,𝑥𝑠1−̃𝑥=−𝑠𝐼−Proj𝐶𝐼−𝜆𝑠𝑥∇𝑓𝑠−𝐼−Proj𝐶𝐼−𝜆𝑠∇𝑓(̃𝑥),𝑥𝑠𝐹𝑥−̃𝑥+𝜇𝑠−𝐹Proj𝐶𝐼−𝜆𝑠𝑥∇𝑓𝑠,𝑥𝑠𝐹𝑥−̃𝑥≤𝜇𝑠−𝐹Proj𝐶𝐼−𝜆𝑠𝑥∇𝑓𝑠,𝑥𝑠.−̃𝑥(3.22) Since Proj𝐶(𝐼−𝜆𝑠∇𝑓) is nonexpansive, we obtain that 𝐼−Proj𝐶(𝐼−𝜆𝑠∇𝑓) is monotone, that is, 𝐼−Proj𝐶𝐼−𝜆𝑠𝑥∇𝑓𝑠−𝐼−Proj𝐶𝐼−𝜆𝑠∇𝑓(̃𝑥),𝑥𝑠−̃𝑥≥0.(3.23) Taking the limit through 𝑠=𝑠𝑛→0 ensures that 𝑥 is a solution to (3.12). That is to say (𝜇𝐹−𝛾ℎ)𝑥,𝑥−̃𝑥≤0.(3.24) Hence 𝑥=𝑥∗ by uniqueness. Therefore, 𝑥𝑠→𝑥∗ as 𝑠→0. The variational inequality (3.12) can be written as ⟨(𝐼−𝜇𝐹+𝛾ℎ)𝑥∗−𝑥∗,̃𝑥−𝑥∗⟩≤0,∀̃𝑥∈𝑆.(3.25) So, by Lemma 2.4, it is equivalent to the fixed-point equation 𝑃𝑆(𝐼−𝜇𝐹+𝛾ℎ)𝑥∗=𝑥∗.(3.26)

Taking 𝐹=𝐴, 𝜇=1 in Theorem 3.2, we get the following

Corollary 3.3. We have that {𝑥𝑠} converges in norm as 𝑠→0 to a minimizer of (1.1) which solves the variational inequality ⟨(𝐴−𝛾ℎ)𝑥∗,̃𝑥−𝑥∗⟩≥0,∀̃𝑥∈𝑆.(3.27) Equivalently, we have Proj𝑠(𝐼−(𝐴−𝛾ℎ))𝑥∗=𝑥∗.

Taking 𝐹=𝐼, 𝜇=1, 𝛾=1 in Theorem 3.2, we get the following.

Corollary 3.4. Let 𝑧𝑠∈𝐻 be the unique fixed point of the contraction 𝑧↦𝑠ℎ(𝑧)+(1−𝑠)Proj𝐶(𝐼−𝜆𝑠∇𝑓)(𝑧). Then, {𝑧𝑠} converges in norm as 𝑠→0 to the unique solution of the variational inequality ⟨(𝐼−ℎ)𝑥∗,̃𝑥−𝑥∗⟩≥0,∀̃𝑥∈𝑆.(3.28)

Finally, we consider the following hybrid gradient-projection algorithm, 𝑥0𝑥∈𝐶arbitrarily,𝑛+1=𝜃𝑛𝑥𝛾ℎ𝑛+𝐼−𝜇𝜃𝑛𝐹Proj𝐶𝑥𝑛−𝜆𝑛𝑥∇𝑓𝑛,∀𝑛≥0.(3.29) Assume that the sequence {𝜆𝑛}∞𝑛=0 satisfies the condition (1.4) and, in addition, that the following conditions are satisfied for {𝜆𝑛}∞𝑛=0 and {𝜃𝑛}∞𝑛=0⊂[0,1]:(i)𝜃𝑛→0; (ii)∑∞𝑛=0𝜃𝑛=∞; (iii)∑∞𝑛=0|𝜃𝑛+1−𝜃𝑛|<∞; (iv)∑∞𝑛=0|𝜆𝑛+1−𝜆𝑛|<∞.

Theorem 3.5. Assume that the minimization problem (1.1) is consistent and the gradient ∇𝑓 satisfies the Lipschitz condition (1.2). Let {𝑥𝑛} be generated by algorithm (3.29) with the sequences {𝜃𝑛} and {𝜆𝑛} satisfying the above conditions. Then, the sequence {𝑥𝑛} converges in norm to 𝑥∗ that is obtained in Theorem 3.2.

Proof. (1) The sequence {𝑥𝑛}∞𝑛=0 is bounded. Setting 𝑉𝑛∶=Proj𝐶𝐼−𝜆𝑛∇𝑓.(3.30) Indeed, we have, for 𝑥∈𝑆, ‖‖𝑥𝑛+1−𝑥‖‖=‖‖𝜃𝑛𝑥𝛾ℎ𝑛+𝐼−𝜇𝜃𝑛𝐹𝑉𝑛𝑥𝑛−𝑥‖‖=‖‖𝜃𝑛𝑥𝛾ℎ𝑛−𝜇𝐹𝑥+𝐼−𝜇𝜃𝑛𝐹𝑉𝑛𝑥𝑛−𝐼−𝜇𝜃𝑛𝐹𝑉𝑛𝑥‖‖≤1−𝜃𝑛𝜏‖‖𝑥𝑛−𝑥‖‖+𝜃𝑛‖‖𝑥𝜌𝛾𝑛−𝑥‖‖+𝜃𝑛‖‖𝛾ℎ𝑥−𝜇𝐹𝑥‖‖=1−𝜃𝑛(‖‖𝑥𝜏−𝛾𝜌)𝑛−𝑥‖‖+𝜃𝑛‖‖𝛾ℎ𝑥−𝜇𝐹𝑥‖‖‖‖𝑥≤max𝑛−𝑥‖‖,1‖‖𝜏−𝛾𝜌𝛾ℎ𝑥−𝜇𝐹𝑥‖‖,∀𝑛≥0.(3.31) By induction, ‖‖𝑥𝑛−𝑥‖‖‖‖𝑥≤max0−𝑥‖‖,‖‖𝛾ℎ𝑥−𝜇𝐹𝑥‖‖𝜏−𝛾𝜌.(3.32) In particular, {𝑥𝑛}∞𝑛=0 is bounded.
(2) We prove that ‖𝑥𝑛+1−𝑥𝑛‖→0 as 𝑛→∞. Let 𝑀 be a constant such that 𝑀>maxsup𝑛≥0𝛾‖‖ℎ𝑥𝑛‖‖,sup𝜅,𝑛≥0𝜇‖‖𝐹𝑉𝜅𝑥𝑛‖‖,sup𝑛≥0‖‖𝑥∇𝑓𝑛‖‖.(3.33) We compute ‖‖𝑥𝑛+1−𝑥𝑛‖‖=‖‖𝜃𝑛𝑥𝛾ℎ𝑛+𝐼−𝜇𝜃𝑛𝐹𝑉𝑛𝑥𝑛−𝜃𝑛−1𝑥𝛾ℎ𝑛−1−𝐼−𝜇𝜃𝑛−1𝐹𝑉𝑛−1𝑥𝑛−1‖‖=‖‖𝜃𝑛𝛾ℎ𝑥𝑛𝑥−ℎ𝑛−1𝜃+𝛾𝑛−𝜃𝑛−1ℎ𝑥𝑛−1+𝐼−𝜇𝜃𝑛𝐹𝑉𝑛𝑥𝑛−𝐼−𝜇𝜃𝑛𝐹𝑉𝑛𝑥𝑛−1+𝐼−𝜇𝜃𝑛𝐹𝑉𝑛𝑥𝑛−1−𝐼−𝜇𝜃𝑛−1𝐹𝑉𝑛−1𝑥𝑛−1‖‖=‖‖𝜃𝑛𝛾ℎ𝑥𝑛𝑥−ℎ𝑛−1𝜃+𝛾𝑛−𝜃𝑛−1ℎ𝑥𝑛−1+𝐼−𝜇𝜃𝑛𝐹𝑉𝑛𝑥𝑛−𝐼−𝜇𝜃𝑛𝐹𝑉𝑛𝑥𝑛−1+𝐼−𝜇𝜃𝑛𝐹𝑉𝑛𝑥𝑛−1−𝐼−𝜇𝜃𝑛𝐹𝑉𝑛−1𝑥𝑛−1+𝐼−𝜇𝜃𝑛𝐹𝑉𝑛−1𝑥𝑛−1−𝐼−𝜇𝜃𝑛−1𝐹𝑉𝑛−1𝑥𝑛−1‖‖≤𝜃𝑛‖‖𝑥𝛾𝜌𝑛−𝑥𝑛−1‖‖||𝜃+𝛾𝑛−𝜃𝑛−1||‖‖ℎ𝑥𝑛−1‖‖+1−𝜃𝑛𝜏‖‖𝑥𝑛−𝑥𝑛−1‖‖+‖‖𝑉𝑛𝑥𝑛−1−𝑉𝑛−1𝑥𝑛−1‖‖||𝜃+𝜇𝑛−𝜃𝑛−1||‖‖𝐹𝑉𝑛−1𝑥𝑛−1‖‖≤𝜃𝑛‖‖𝑥𝛾𝜌𝑛−𝑥𝑛−1‖‖||𝜃+𝑀𝑛−𝜃𝑛−1||+1−𝜃𝑛𝜏‖‖𝑥𝑛−𝑥𝑛−1‖‖+‖‖𝑉𝑛𝑥𝑛−1−V𝑛−1𝑥𝑛−1‖‖||𝜃+𝑀𝑛−𝜃𝑛−1||=1−𝜃𝑛‖‖𝑥(𝜏−𝛾𝜌)𝑛−𝑥𝑛−1‖‖||𝜃+2𝑀𝑛−𝜃𝑛−1||+‖‖𝑉𝑛𝑥𝑛−1−𝑉𝑛−1𝑥𝑛−1‖‖,‖‖𝑉(3.34)𝑛𝑥𝑛−1−𝑉𝑛−1𝑥𝑛−1‖‖=‖‖Proj𝐶𝐼−𝜆𝑛𝑥∇𝑓𝑛−1−Proj𝐶𝐼−𝜆𝑛−1𝑥∇𝑓𝑛−1‖‖≤‖‖𝐼−𝜆𝑛𝑥∇𝑓𝑛−1−𝐼−𝜆𝑛−1𝑥∇𝑓𝑛−1‖‖=||𝜆𝑛−𝜆𝑛−1||‖‖𝑥∇𝑓𝑛−1‖‖||𝜆≤𝑀𝑛−𝜆𝑛−1||.(3.35) Combining (3.34) and (3.35), we can obtain ‖‖𝑥𝑛+1−𝑥𝑛‖‖≤1−(𝜏−𝛾𝜌)𝜃𝑛‖‖𝑥𝑛−𝑥𝑛−1‖‖||𝜃+2𝑀𝑛−𝜃𝑛−1||+||𝜆𝑛−𝜆𝑛−1||.(3.36) Apply Lemma 2.1 to (3.36) to conclude that ‖𝑥𝑛+1−𝑥𝑛‖→0 as 𝑛→∞.
(3) We prove that 𝜔𝑤(𝑥𝑛)⊂𝑆. Let ̂𝑥∈𝜔𝑤(𝑥𝑛), and assume that 𝑥𝑛𝑗⇀̂𝑥 for some subsequence {𝑥𝑛𝑗}∞𝑗=1 of {𝑥𝑛}∞𝑛=0. We may further assume that 𝜆𝑛𝑗→𝜆∈[0,2/𝐿] due to condition (1.4). Set 𝑉∶=Proj𝐶(𝐼−𝜆∇𝑓). Notice that 𝑉 is nonexpansive and Fix𝑉=𝑆. It turns out that ‖‖𝑥𝑛𝑗−𝑉𝑥𝑛𝑗‖‖≤‖‖𝑥𝑛𝑗−𝑉𝑛𝑗𝑥𝑛𝑗‖‖+‖‖𝑉𝑛𝑗𝑥𝑛𝑗−𝑉𝑥𝑛𝑗‖‖≤‖‖𝑥𝑛𝑗−𝑥𝑛𝑗+1‖‖+‖‖𝑥𝑛𝑗+1−𝑉𝑛𝑗𝑥𝑛𝑗‖‖+‖‖𝑉𝑛𝑗𝑥𝑛𝑗−𝑉𝑥𝑛𝑗‖‖≤‖‖𝑥𝑛𝑗−𝑥𝑛𝑗+1‖‖+𝜃𝑛𝑗‖‖𝑥𝛾ℎ𝑛𝑗−𝜇𝐹𝑉𝑛𝑗𝑥𝑛𝑗‖‖+‖‖Proj𝐶𝐼−𝜆𝑛𝑗𝑥∇𝑓𝑛𝑗−Proj𝐶(𝐼−𝜆∇𝑓)𝑥𝑛𝑗‖‖≤‖‖𝑥𝑛𝑗−𝑥𝑛𝑗+1‖‖+𝜃𝑛𝑗‖‖𝑥𝛾ℎ𝑛𝑗−𝜇𝐹𝑉𝑛𝑗𝑥𝑛𝑗‖‖+|||𝜆−𝜆𝑛𝑗|||‖‖𝑥∇𝑓𝑛𝑗‖‖≤‖‖𝑥𝑛𝑗−𝑥𝑛𝑗+1‖‖𝜃+2𝑀𝑛𝑗+|||𝜆−𝜆𝑛𝑗|||⟶0as𝑗⟶∞.(3.37) So Lemma 2.2 guarantees that 𝜔𝑤(𝑥𝑛)⊂Fix𝑉=𝑆.
(4) We prove that 𝑥𝑛→𝑥∗ as 𝑛→∞, where 𝑥∗ is the unique solution of the 𝑉𝐼 (3.12). First observe that there is some ̂𝑥∈𝜔𝑤(𝑥𝑛)⊂𝑆 Such that limsup𝑛→∞⟨(𝜇𝐹−𝛾ℎ)𝑥∗,𝑥𝑛−𝑥∗⟩=⟨(𝜇𝐹−𝛾ℎ)𝑥∗,̂𝑥−𝑥∗⟩≥0.(3.38)
We now compute ‖‖𝑥𝑛+1−𝑥∗‖‖2=‖‖𝜃𝑛𝑥𝛾ℎ𝑛+𝐼−𝜇𝜃𝑛𝐹Proj𝐶𝐼−𝜆𝑛𝑥∇𝑓𝑛−𝑥∗‖‖2=‖‖𝜃𝑛𝑥𝛾ℎ𝑛−𝜇𝐹𝑥∗+𝐼−𝜇𝜃𝑛𝐹𝑉𝑛𝑥𝑛−𝐼−𝜇𝜃𝑛𝐹𝑉𝑛𝑥∗‖‖2=‖‖𝜃𝑛𝛾ℎ𝑥𝑛𝑥−ℎ∗+𝐼−𝜇𝜃𝑛𝐹𝑉𝑛𝑥𝑛−(𝐼−𝜇𝜃𝑛𝐹)𝑉𝑛𝑥∗+𝜃𝑛𝑥𝛾ℎ∗−𝜇𝐹𝑥∗‖‖2≤‖‖𝜃𝑛𝛾ℎ𝑥𝑛𝑥−ℎ∗+𝐼−𝜇𝜃𝑛𝐹𝑉𝑛𝑥𝑛−𝐼−𝜇𝜃𝑛𝐹𝑉𝑛𝑥∗‖‖2+2𝜃𝑛(𝛾ℎ−𝜇𝐹)𝑥∗,𝑥𝑛+1−𝑥∗=‖‖𝜃𝑛𝛾ℎ𝑥𝑛𝑥−ℎ∗‖‖2+‖‖𝐼−𝜇𝜃𝑛𝐹𝑉𝑛𝑥𝑛−𝐼−𝜇𝜃𝑛𝐹𝑉𝑛𝑥∗‖‖2+2𝜃𝑛𝑥𝛾⟨ℎ𝑛𝑥−ℎ∗,𝐼−𝜇𝜃𝑛𝐹𝑉𝑛𝑥𝑛−𝐼−𝜇𝜃𝑛𝐹𝑉𝑛𝑥∗⟩+2𝜃𝑛⟨(𝛾ℎ−𝜇𝐹)𝑥∗,𝑥𝑛+1−𝑥∗⟩≤𝜃2𝑛𝛾2𝜌2‖‖𝑥𝑛−𝑥∗‖‖2+1−𝜃𝑛𝜏2‖‖𝑥𝑛−𝑥∗‖‖2+2𝜃𝑛𝛾𝜌1−𝜃𝑛𝜏‖‖𝑥𝑛−𝑥∗‖‖2+2𝜃𝑛(𝛾ℎ−𝜇𝐹)𝑥∗,𝑥𝑛+1−𝑥∗=𝜃2𝑛𝛾2𝜌2+1−𝜃𝑛𝜏2+2𝜃𝑛𝛾𝜌1−𝜃𝑛𝜏‖‖𝑥𝑛−𝑥∗‖‖2+2𝜃𝑛(𝛾ℎ−𝜇𝐹)𝑥∗,𝑥𝑛+1−𝑥∗≤𝜃𝑛𝛾2𝜌2+1−2𝜃𝑛𝜏+𝜃𝑛𝜏2+2𝜃𝑛‖‖𝑥𝛾𝜌𝑛−𝑥∗‖‖2+2𝜃𝑛(𝛾ℎ−𝜇𝐹)𝑥∗,𝑥𝑛+1−𝑥∗=1−𝜃𝑛2𝜏−𝛾2𝜌2−𝜏2‖‖𝑥−2𝛾𝜌𝑛−𝑥∗‖‖2+2𝜃𝑛(𝛾ℎ−𝜇𝐹)𝑥∗,𝑥𝑛+1−𝑥∗.(3.39) Applying Lemma 2.1 to the inequality (3.39), together with (3.38), we get ‖𝑥𝑛−𝑥∗‖→0 as 𝑛→∞.

Corollary 3.6 (see [11]). Let {𝑥𝑛} be generated by the following algorithm: 𝑥𝑛+1=𝜃𝑛ℎ𝑥𝑛+1−𝜃𝑛Proj𝐶𝑥𝑛−𝜆𝑛𝑥∇𝑓𝑛,∀𝑛≥0.(3.40) Assume that the sequence {𝜆𝑛}∞𝑛=0 satisfies the conditions (1.4) and (iv) and that {𝜃𝑛}⊂[0,1] satisfies the conditions (i)–(iii). Then {𝑥𝑛} converges in norm to 𝑥∗ obtained in Corollary 3.4.

Corollary 3.7. Let {𝑥𝑛} be generated by the following algorithm: 𝑥𝑛+1=𝜃𝑛𝑥𝛾ℎ𝑛+𝐼−𝜃𝑛𝐴Proj𝐶𝑥𝑛−𝜆𝑛𝑥∇𝑓𝑛,∀𝑛≥0.(3.41) Assume that the sequences {𝜃𝑛} and {𝜆𝑛} satisfy the conditions contained in Theorem 3.5, then {𝑥𝑛} converges in norm to 𝑥∗ obtained in Corollary 3.3.

Acknowledgments

Ming Tian is Supported in part by The Fundamental Research Funds for the Central Universities (the Special Fund of Science in Civil Aviation University of China: No. ZXH2012 K001) and by the Science Research Foundation of Civil Aviation University of China (No. 2012KYM03).

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Copyright © 2012 Ming Tian and Min-Min Li. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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