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Journal of Applied Mathematics
Volume 2012 (2012), Article ID 410137, 10 pages
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.
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.
In the present paper, our main purpose is to solve the following minimization problem: where is a nonempty closed and convex subset of a real Hilbert space , 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, [1–8]. The classic algorithm is the following form of the projected-gradient method: where is an any constant, is the nearest point projection from onto , and denotes the gradient of .
It is known  that if has a Lipschitz continuous and strongly monotone gradient, then the sequence generated by (1.2) can be strongly convergent to a minimizer of in . If the gradient of is only assumed to be Lipschitz continuous, then can only be weakly convergent if 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  introduced the following algorithm: Under some additional assumptions, Xu  proved that the sequence converges strongly to a minimizer of (1.1). At the same time, Xu  also suggested a regularized method: Consequently, Yao et al.  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.
Let be a nonempty closed convex subset of a real Hilbert space . A bounded linear operator is said to be strongly positive on if there exists a constant such that A mapping is called nonexpansive if A mapping is said to be an averaged mapping, if and only if it can be written as the average of the identity and a nonexpansive mapping; that is, where is a constant and is a nonexpansive mappings. In this case, we call is -averaged.
A mapping is said to be -inverse strongly monotone (-ism), if and only if The following proposition is well known, which is useful for the next section.
Proposition 2.1 (See ). (1) The composite of finitely many averaged mappings is averaged. That is, if each of the mappings is averaged, then so is the composite . In particular, if is -averaged and is -averaged, where , then the composite is -averaged, where .
(2) T is -ism, then for , is ()-ism.
Recall that the (nearest point or metric) projection from onto , denoted by , assigns, to each , the unique point with the property We use to denote the solution set of (1.1). Assume that (1.1) is consistent, that is, . If is Frechet differentiable, then solves (1.1) if and only if satisfies the following optimality condition: where denotes the gradient of . Observe that (2.6) can be rewritten as the following VI (Note that the VI has been extensively studied in the literature, see, for instance [11–25].) This shows that the minimization (1.1) is equivalent to the fixed point problem where is an any constant. This relationship is very important for constructing our method.
Next we adopt the following notation:(i) means that converges strongly to ;(ii) means that converges weakly to ;(iii) is the fixed points set of .
Lemma 2.2 (See ). Let and be bounded sequences in a Banach space and let be a sequence in with Suppose for all and Then, .
Lemma 2.3 (See  (demiclosedness principle)). Let be a closed and convex subset of a Hilbert space and let be a nonexpansive mapping with . If is a sequence in weakly converging to and if converges strongly to , then In particular, if , then .
Lemma 2.4 (See ). Assume is a sequence of nonnegative real numbers such that where is a sequence in and is a sequence such that(1);(2) or .Then, .
3. Main Results
Let be a closed convex subset of a real Hilbert space . Let be a real-valued Frechet differentiable convex function with the gradient . Let be a -contraction. Let be a self-adjoint, strongly positive bounded linear operator with coefficient . First, we present our algorithm for solving (1.1). Throughout, we assume .
Algorithm 3.1. For given , compute the sequence iteratively by where are two constants and the real number sequence .
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 is -Lipschitzian and . Let be a sequence generated by (3.1), where is a constant and the sequence satisfies the conditions: (i) and (ii) . Then converges to a minimizer of (1.1) which solves the following variational inequality:
Proposition 3.4. It is well known that the metric projection of onto has the following basic properties:(i), for all ;(ii), for every ;(iii), for all , .
The Proof of Theorem 3.3
Let . First, from (2.8), we note that . By (3.1), we have Thus, by induction, we obtain
Note that the Lipschitz condition implies that the gradient is -inverse strongly monotone (ism), which then implies that is -ism. So, is -averaged. Now since the projection is -averaged, we see that is -averaged. Hence we have that where are nonexpansive and . Then we can rewrite (3.1) as where Set for all . Since is bounded, we deduce , , and are all bounded. Hence, there exists a constant such that Thus, It follows that This together with Lemma 2.2 implies that So, Since we deduce Next we prove where is the unique solution of VI (3.2).
Indeed, we can choose a subsequence of such that Since is bounded, there exists a subsequence of which converges weakly to a point . Without loss of generality, we may assume that converges weakly to . Since , is nonexpansive. Thus, from (3.14) and Lemma 2.3, we have . Therefore, Finally, we show . By using the property of the projection , we have It follows that It is obvious that . Then we can apply Lemma 2.4 to the last inequality to conclude that . The proof is completed.
Algorithm 3.5. For given , compute the sequence iteratively by where are two constants and the real number sequence .
From Theorem 3.3, we have the following result.
Theorem 3.6. Assume that the gradient is -Lipschitzian and . Let be a sequence generated by (3.20), where is a constant and the sequences satisfies the conditions: (i) and (ii) . Then converges to a minimizer of (1.1) which is the minimum norm element in .
Proof. As a consequence of Theorem 3.3, we obtain that the sequence generated by (3.20) converges strongly to which satisfies This implies Thus, That is, is the minimum norm element in . This completes the proof.
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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