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Journal of Applied Mathematics
Volume 2012 (2012), Article ID 410137, 10 pages
http://dx.doi.org/10.1155/2012/410137
Research Article

Strong Convergence of a Projected Gradient Method

Department of Mathematics, Tianjin Polytechnic University, Tianjin 300387, China

Received 16 January 2012; Accepted 5 February 2012

Academic Editor: Yeong-Cheng Liou

Copyright © 2012 Shunhou Fan and Yonghong Yao. 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.

Abstract

The projected-gradient method is a powerful tool for solving constrained convex optimization problems and has extensively been studied. In the present paper, a projected-gradient method is presented for solving the minimization problem, and the strong convergence analysis of the suggested gradient projection method is given.

1. Introduction

In the present paper, our main purpose is to solve the following minimization problem:min𝑥𝐶𝑓(𝑥),(1.1) where 𝐶 is a nonempty closed and convex subset of a real Hilbert space 𝐻, 𝑓𝐻𝑅 is a real-valued convex function.

Now it is well known that the projected-gradient method is a powerful tool for solving the above minimization problem and has extensively been studied. See, for instance, [18]. The classic algorithm is the following form of the projected-gradient method:𝑥𝑛+1=𝑃𝐶𝑥𝑛𝑥𝛾𝑓𝑛,𝑛0,(1.2) where 𝛾>0 is an any constant, 𝑃𝐶 is the nearest point projection from 𝐻 onto 𝐶, and 𝑓 denotes the gradient of 𝑓.

It is known [1] that if 𝑓 has a Lipschitz continuous and strongly monotone gradient, then the sequence {𝑥𝑛} generated by (1.2) can be strongly convergent to a minimizer of 𝑓 in 𝐶. If the gradient of 𝑓 is only assumed to be Lipschitz continuous, then {𝑥𝑛} can only be weakly convergent if 𝐻 is infinite dimensional. An interesting problem is how to appropriately modify the projected gradient algorithm so as to have strong convergence? For this purpose, recently, Xu [9] introduced the following algorithm:𝑥𝑛+1=𝜃𝑛𝑥𝑛+1𝜃𝑛𝑃𝐶𝑥𝑛𝛾𝑛𝑥𝑓𝑛,𝑛0.(1.3) Under some additional assumptions, Xu [9] proved that the sequence {𝑥𝑛} converges strongly to a minimizer of (1.1). At the same time, Xu [9] also suggested a regularized method:𝑥𝑛+1=𝑃𝐶𝐼𝛾𝑛𝑓+𝛼𝑛𝐼𝑥𝑛,𝑛0.(1.4) Consequently, Yao et al. [10] proved the strong convergence of the regularized method (1.4) under some weaker conditions.

Motivated by the above works, in this paper we will further construct a new projected gradient method for solving the minimization problem (1.1). It should be pointed out that our method also has strong convergence under some mild conditions.

2. Preliminaries

Let 𝐶 be a nonempty closed convex subset of a real Hilbert space 𝐻. A bounded linear operator 𝐵 is said to be strongly positive on 𝐻 if there exists a constant 𝛼>0 such that𝐵𝑥,𝑥𝛼𝑥2,𝑥𝐻.(2.1) A mapping 𝑇𝐶𝐶 is called nonexpansive if𝑇𝑥𝑇𝑦𝑥𝑦,𝑥,𝑦𝐶.(2.2) A mapping 𝑇𝐶𝐶 is said to be an averaged mapping, if and only if it can be written as the average of the identity 𝐼 and a nonexpansive mapping; that is,𝑇=(1𝛼)𝐼+𝛼𝑅,(2.3) where 𝛼(0,1) is a constant and 𝑅𝐶𝐶 is a nonexpansive mappings. In this case, we call 𝑇 is 𝛼-averaged.

A mapping 𝑇𝐶𝐶 is said to be 𝜈-inverse strongly monotone (𝜈-ism), if and only if𝑥𝑦,𝑇𝑥𝑇𝑦𝜈𝑇𝑥𝑇𝑦2,𝑥,𝑦𝐶.(2.4) The following proposition is well known, which is useful for the next section.

Proposition 2.1 (See [9]). (1) 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. In particular, if 𝑇1 is 𝛼1-averaged and 𝑇2 is 𝛼2-averaged, where 𝛼1,𝛼2(0,1), then the composite 𝑇1𝑇2 is 𝛼-averaged, where 𝛼=𝛼1𝛼2𝛼1𝛼2.
(2) T is 𝜈-ism, then for 𝛾>0, 𝛾𝑇 is (𝜈/𝛾)-ism.

Recall that the (nearest point or metric) projection from 𝐻 onto 𝐶, denoted by 𝑃𝐶, assigns, to each 𝑥𝐻, the unique point 𝑃𝐶(𝑥)𝐶 with the property 𝑥𝑃𝐶(𝑥)=inf{𝑥𝑦𝑦𝐶}.(2.5) We use 𝑆 to denote the solution set of (1.1). Assume that (1.1) is consistent, that is, 𝑆. If 𝑓 is Frechet differentiable, then 𝑥𝐶 solves (1.1) if and only if 𝑥𝐶 satisfies the following optimality condition:𝑥𝑓,𝑥𝑥0,𝑥𝐶,(2.6) where 𝑓 denotes the gradient of 𝑓. Observe that (2.6) can be rewritten as the following VI𝑥𝑥𝑥𝑓,𝑥𝑥0,𝑥𝐶.(2.7) (Note that the VI has been extensively studied in the literature, see, for instance [1125].) This shows that the minimization (1.1) is equivalent to the fixed point problem𝑃𝐶𝑥𝑥𝛾𝑓=𝑥,(2.8) where 𝛾>0 is an any constant. This relationship is very important for constructing our method.

Next we adopt the following notation:(i)𝑥𝑛𝑥 means that 𝑥𝑛 converges strongly to 𝑥;(ii)𝑥𝑛𝑥 means that 𝑥𝑛 converges weakly to 𝑥;(iii)Fix(𝑇)={𝑥𝑇𝑥=𝑥} is the fixed points set of 𝑇.

Lemma 2.2 (See [26]). Let {𝑥𝑛} and {𝑦𝑛} be bounded sequences in a Banach space 𝑋 and let {𝛽𝑛} be a sequence in [0,1] with 0<liminf𝑛𝛽𝑛limsup𝑛𝛽𝑛<1.(2.9) Suppose 𝑥𝑛+1=1𝛽𝑛𝑦𝑛+𝛽𝑛𝑥𝑛(2.10) for all 𝑛0 and limsup𝑛𝑦𝑛+1𝑦𝑛𝑥𝑛+1𝑥𝑛0.(2.11) Then, lim𝑛𝑦𝑛𝑥𝑛=0.

Lemma 2.3 (See [27] (demiclosedness principle)). Let 𝐶 be a closed and convex subset of a Hilbert space 𝐻 and let 𝑇𝐶𝐶 be a nonexpansive mapping with Fix(𝑇). If {𝑥𝑛} is a sequence in 𝐶 weakly converging to 𝑥 and if {(𝐼𝑇)𝑥𝑛} converges strongly to 𝑦, then (𝐼𝑇)𝑥=𝑦.(2.12) In particular, if 𝑦=0, then 𝑥Fix(𝑇).

Lemma 2.4 (See [28]). Assume {𝑎𝑛} is a sequence of nonnegative real numbers such that 𝑎𝑛+11𝛾𝑛𝑎𝑛+𝛿𝑛,(2.13) where {𝛾𝑛} is a sequence in (0,1) and {𝛿𝑛} is a sequence such that(1)𝑛=1𝛾𝑛=;(2)limsup𝑛𝛿𝑛/𝛾𝑛0 or 𝑛=1|𝛿𝑛|<.Then, lim𝑛𝑎𝑛=0.

3. Main Results

Let 𝐶 be a closed convex subset of a real Hilbert space 𝐻. Let 𝑓𝐶𝑅 be a real-valued Frechet differentiable convex function with the gradient 𝑓. Let 𝐴𝐶𝐻 be a 𝜌-contraction. Let 𝐵𝐻𝐻 be a self-adjoint, strongly positive bounded linear operator with coefficient 𝛼>0. First, we present our algorithm for solving (1.1). Throughout, we assume 𝑆.

Algorithm 3.1. For given 𝑥0𝐶, compute the sequence {𝑥𝑛} iteratively by 𝑥𝑛+1=𝑃𝐶𝐼+(𝜎𝐴𝐵)𝜃𝑛𝑃𝐶(𝐼𝛾𝑓)𝑥𝑛,𝑛0,(3.1) where 𝜎>0,𝛾>0 are two constants and the real number sequence {𝜃n}[0,1].

Remark 3.2. In (3.1), we use two projections. Now, it is well-known that the advantage of projections, which makes them successful in real-word applications, is computational.

Next, we show the convergence analysis of this Algorithm 3.1.

Theorem 3.3. Assume that the gradient 𝑓 is 𝐿-Lipschitzian and 𝜎𝜌<𝛼. Let {𝑥𝑛} be a sequence generated by (3.1), where 𝛾(0,2/𝐿) is a constant and the sequence {𝜃𝑛} satisfies the conditions: (i) lim𝑛𝜃𝑛=0 and (ii) 𝑛=0𝜃𝑛=. Then {𝑥𝑛} converges to a minimizer ̃𝑥 of (1.1) which solves the following variational inequality: ̃𝑥S𝑠𝑢𝑐𝑡𝑎𝑡𝜎A(̃𝑥)B(̃𝑥),𝑥̃𝑥0,𝑥𝑆.(3.2)

By Algorithm 3.1 involved in the projection, we will use the properties of the metric projection for proving Theorem 3.3. For convenience, we list the properties of the projection as follows.

Proposition 3.4. It is well known that the metric projection 𝑃𝐶 of 𝐻 onto 𝐶 has the following basic properties:(i)𝑃𝐶(𝑥)𝑃𝐶(𝑦)𝑥𝑦, for all 𝑥,𝑦𝐻;(ii)𝑥𝑦,𝑃𝐶(𝑥)𝑃𝐶(𝑦)𝑃𝐶(𝑥)𝑃𝐶(𝑦)2, for every 𝑥,𝑦𝐻;(iii)𝑥𝑃𝐶(𝑥),𝑦𝑃𝐶(𝑥)0, for all 𝑥𝐻, 𝑦𝐶.

The Proof of Theorem 3.3
Let 𝑥𝑆. First, from (2.8), we note that 𝑃𝐶(𝐼𝛾𝑓)𝑥=𝑥. By (3.1), we have 𝑥𝑛+1𝑥=𝑃𝐶𝐼+(𝜎𝐴𝐵)𝜃𝑛𝑃𝐶(𝐼𝛾f)𝑥𝑛𝑥𝑃𝐶𝐼+(𝜎𝐴𝐵)𝜃𝑛𝑃𝐶(𝐼𝛾𝑓)𝑥𝑛𝑃𝐶𝐼+(𝜎𝐴𝐵)𝜃𝑛𝑃𝐶(𝐼𝛾𝑓)𝑥+𝑃𝐶𝐼+(𝜎𝐴𝐵)𝜃𝑛𝑥𝑥1(𝛼𝜎𝜌)𝜃𝑛𝑥𝑛𝑥+𝜃𝑛𝑥𝜎𝐴𝑥𝐵=1(𝛼𝜎𝜌)𝜃𝑛𝑥𝑛𝑥+(𝛼𝜎𝜌)𝜃𝑛𝑥𝜎𝐴𝑥𝐵𝑥𝛼𝜎𝜌max𝑛𝑥,𝑥𝜎𝐴𝑥𝐵.𝛼𝜎𝜌(3.3) Thus, by induction, we obtain 𝑥𝑛𝑥𝑥max0𝑥,𝑥𝜎𝐴𝑥𝐵𝛼𝜎𝜌.(3.4)
Note that the Lipschitz condition implies that the gradient 𝑓 is (1/𝐿)-inverse strongly monotone (ism), which then implies that 𝛾𝑓 is (1/𝛾𝐿)-ism. So, 𝐼𝛾𝑓 is (𝛾𝐿/2)-averaged. Now since the projection 𝑃𝐶 is (1/2)-averaged, we see that 𝑃𝐶(𝐼𝛾𝑓) is ((2+𝛾𝐿)/4)-averaged. Hence we have that 𝑃𝐶=121𝐼+2𝑅𝑃𝐶(𝐼𝛾𝑓)=2𝛾𝐿4𝐼+2+𝛾𝐿4𝑇=(1𝛽)𝐼+𝛽𝑇,(3.5) where 𝑅,𝑇 are nonexpansive and 𝛽=(2+𝛾𝐿)/4(0,1). Then we can rewrite (3.1) as 𝑥𝑛+1=121𝐼+2𝑅𝐼+(𝜎𝐴𝐵)𝜃𝑛(1𝛽)𝑥𝑛+𝛽𝑇𝑥𝑛=1𝛽2𝑥𝑛+𝛽2𝑇𝑥𝑛+𝜃𝑛2𝑅(𝜎𝐴𝐵)+2𝐼+(𝜎𝐴𝐵)𝜃𝑛(1𝛽)𝑥𝑛+𝛽𝑇𝑥𝑛=1𝛽2𝑥𝑛+1+𝛽2𝑦𝑛,(3.6) where 𝑦𝑛=2𝜃1+𝛽𝑛2𝑅(𝜎𝐴𝐵)+2𝐼+(𝜎𝐴𝐵)𝜃𝑛(1𝛽)𝑥𝑛+𝛽𝑇𝑥𝑛+𝛽1+𝛽𝑇𝑥𝑛.(3.7) Set 𝑧𝑛=(1𝛽)𝑥𝑛+𝛽𝑇𝑥𝑛 for all 𝑛. Since {𝑥𝑛} is bounded, we deduce {𝐴(𝑥𝑛)}, {𝐵(𝑥𝑛)}, and {𝑇𝑥𝑛} are all bounded. Hence, there exists a constant 𝑀>0 such that sup𝑛(𝜎𝐴𝐵)𝑧𝑛𝑀.(3.8) Thus, 𝑦𝑛+1𝑦𝑛2𝜃1+𝛽𝑛+12(𝜎𝐴𝐵)𝑧𝑛+1𝜃𝑛2(𝜎𝐴𝐵)𝑧𝑛+𝛽1+𝛽𝑇𝑥𝑛+1𝑇𝑥𝑛+1𝑅𝜃1+𝛽𝑛+1𝜎𝐴+𝐼𝜃𝑛+1𝐵𝑧𝑛+1𝑅𝐼+(𝜎𝐴𝐵)𝜃𝑛𝑧𝑛1𝜃1+𝛽𝑛+1+𝜃𝑛𝛽𝑀+𝑥1+𝛽𝑛+1𝑥𝑛+1𝑧1+𝛽𝑛+1𝑧𝑛+1𝜃1+𝛽𝑛+1(𝜎𝐴𝐵)𝑧𝑛+1𝜃𝑛(𝜎𝐴𝐵)𝑧𝑛2𝜃1+𝛽𝑛+1+𝜃𝑛𝑥𝑀+𝑛+1𝑥𝑛.(3.9) It follows that limsup𝑛𝑦𝑛+1𝑦𝑛𝑥𝑛+1𝑥𝑛0.(3.10) This together with Lemma 2.2 implies that lim𝑛𝑦𝑛𝑥𝑛=0.(3.11) So, lim𝑛𝑥𝑛+1𝑥𝑛=lim𝑛1+𝛽2𝑦𝑛𝑥𝑛=0.(3.12) Since 𝑥𝑛𝑃𝐶(𝐼𝛾𝑓)𝑥𝑛𝑥𝑛𝑥𝑛+1+𝑥𝑛+1𝑃𝐶(𝐼𝛾𝑓)𝑥𝑛=𝑥𝑛𝑥𝑛+1+𝑃𝐶𝐼+(𝜎𝐴𝐵)𝜃𝑛𝑃𝐶(𝐼𝛾𝑓)𝑃𝐶(𝐼𝛾𝑓)𝑥𝑛𝑥𝑛𝑥𝑛+1+𝜃𝑛(𝜎𝐴𝐵)𝑃𝐶(𝐼𝛾𝑓)𝑥𝑛,(3.13) we deduce lim𝑛𝑥𝑛𝑃𝐶(𝐼𝛾𝑓)𝑥𝑛=0.(3.14) Next we prove limsup𝑘𝑥𝜎𝐴𝑥𝐵,𝑥𝑛𝑥0,(3.15) where 𝑥 is the unique solution of VI (3.2).
Indeed, we can choose a subsequence {𝑥𝑛𝑖} of {𝑥𝑛} such that limsup𝑛𝑥𝜎𝐴𝑥𝐵,𝑥𝑛𝑥=lim𝑖𝑥𝜎𝐴𝑥𝐵,𝑥𝑛𝑖𝑥.(3.16) Since {𝑥𝑛𝑖} is bounded, there exists a subsequence of {𝑥𝑛𝑖} which converges weakly to a point ̃𝑥. Without loss of generality, we may assume that {𝑥𝑛𝑖} converges weakly to ̃𝑥. Since 𝛾(0,2/𝐿), 𝑃𝐶(𝐼𝛾𝑓) is nonexpansive. Thus, from (3.14) and Lemma 2.3, we have 𝑥𝑛𝑖̃𝑥Fix(𝑃𝐶(𝐼𝛾𝑓))=𝑆. Therefore, limsup𝑛𝑥𝜎𝐴𝑥𝐵,𝑥𝑛𝑥=lim𝑖𝑥𝜎𝐴𝑥𝐵,𝑥𝑛𝑖𝑥=𝑥𝜎𝐴𝑥𝐵,̃𝑥𝑥0.(3.17) Finally, we show 𝑥𝑛̃𝑥. By using the property of the projection 𝑃𝐶, we have 𝑥𝑛+1̃𝑥2=𝑃𝐶𝐼+(𝜎𝐴𝐵)𝜃𝑛𝑃𝐶(𝐼𝛾𝑓)𝑥𝑛𝑃𝐶(̃𝑥)2𝐼+(𝜎𝐴𝐵)𝜃𝑛𝑃𝐶(𝐼𝛾𝑓)𝑥𝑛̃𝑥,𝑥𝑛+1=̃𝑥𝐼+(𝜎𝐴𝐵)𝜃𝑛𝑃𝐶(𝐼𝛾𝑓)𝑥𝑛̃𝑥,𝑥𝑛+1̃𝑥+𝜃𝑛𝜎𝐴(̃𝑥)𝐵(̃𝑥),𝑥𝑛+1̃𝑥𝐼+(𝜎𝐴𝐵)𝜃𝑛𝑃𝐶(𝐼𝛾𝑓)𝑥𝑛𝑃𝐶𝑥(𝐼𝛾𝑓)̃𝑥𝑛+1̃𝑥+𝜃𝑛𝜎𝐴(̃𝑥)𝐵(̃𝑥),𝑥𝑛+1̃𝑥1(𝛼𝜎𝜌)𝜃𝑛𝑥𝑛𝑥̃𝑥𝑛+1̃𝑥+𝜃𝑛𝜎𝐴(̃𝑥)𝐵(̃𝑥),𝑥𝑛+1̃𝑥1(𝛼𝜎𝜌)𝜃𝑛2𝑥𝑛̃𝑥2+12𝑥𝑛+1̃𝑥2+𝜃𝑛𝜎𝐴(̃𝑥)𝐵(̃𝑥),𝑥𝑛+1.̃𝑥(3.18) It follows that 𝑥𝑛+1̃𝑥21(𝛼𝜎𝜌)𝜃𝑛𝑥𝑛̃𝑥2+2𝜃𝑛𝜎𝐴(̃𝑥)𝐵(̃𝑥),𝑥𝑛+1=̃𝑥1(𝛼𝜎𝜌)𝜃𝑛𝑥𝑛̃𝑥2+(𝛼𝜎𝜌)𝜃𝑛2𝛼𝜎𝜌𝜎𝐴(̃𝑥)𝐵(̃𝑥),𝑥𝑛+1.̃𝑥(3.19) It is obvious that limsup𝑛((2/(𝛼𝜎𝜌))𝜎𝐴(̃𝑥)𝐵(̃𝑥),𝑥𝑛+1̃𝑥0). Then we can apply Lemma 2.4 to the last inequality to conclude that 𝑥𝑛̃𝑥. The proof is completed.

In (3.1), if we take 𝐴=0 and 𝐵=𝐼, then (3.1) reduces to the following.

Algorithm 3.5. For given 𝑥0𝐶, compute the sequence {𝑥𝑛} iteratively by 𝑥𝑛+1=𝑃𝐶1𝜃𝑛𝑃𝐶(𝐼𝛾𝑓)𝑥𝑛,𝑛0,(3.20) where 𝜎>0,𝛾>0 are two constants and the real number sequence {𝜃𝑛}[0,1].

From Theorem 3.3, we have the following result.

Theorem 3.6. Assume that the gradient 𝑓 is 𝐿-Lipschitzian and 𝜎𝜌<𝛼. Let {𝑥𝑛} be a sequence generated by (3.20), where 𝛾(0,2/𝐿) is a constant and the sequences {𝜃𝑛} satisfies the conditions: (i) lim𝑛𝜃𝑛=0 and (ii) 𝑛=0𝜃𝑛=. Then {𝑥𝑛} converges to a minimizer ̃𝑥 of (1.1) which is the minimum norm element in 𝑆.

Proof. As a consequence of Theorem 3.3, we obtain that the sequence {𝑥𝑛} generated by (3.20) converges strongly to ̃𝑥 which satisfies ̃𝑥𝑆suchthat̃𝑥,𝑥̃𝑥0,𝑥𝑆.(3.21) This implies ̃𝑥2𝑥,̃𝑥𝑥̃𝑥,𝑥𝑆.(3.22) Thus, ̃𝑥𝑥,𝑥𝑆.(3.23) That is, ̃𝑥 is the minimum norm element in 𝑆. This completes the proof.

Acknowledgment

Y. Yao was supported in part by Colleges and Universities Science and Technology Development Foundation (20091003) of Tianjin, NSFC 11071279 and NSFC 71161001-G0105.

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