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Regularized gradient-projection methods for equilibrium and constrained convex minimization problems
Journal of Inequalities and Applications volume 2013, Article number: 243 (2013)
Abstract
In this article, based on Marino and Xu’s method, an iterative method which combines the regularized gradient-projection algorithm (RGPA) and the averaged mappings approach is proposed for finding a common solution of equilibrium and constrained convex minimization problems. Under suitable conditions, it is proved that the sequences generated by implicit and explicit schemes converge strongly. The results of this paper extend and improve some existing results.
MSC:58E35, 47H09, 65J15.
1 Introduction
Let H be a real Hilbert space with the inner product and the induced norm . Let C be a nonempty, closed and convex subset of H. We need some nonlinear operators which are introduced below.
Let be nonlinear operators.
-
T is nonexpansive if for all .
-
T is Lipschitz continuous if there exists a constant such that for all .
-
A is a strongly positive bounded linear operator if there exists a constant such that for all .
-
is monotone if for all .
-
Given is a number , is η-strongly monotone if for all .
-
Given is a number . is υ-inverse strongly monotone (υ-ism) if for all .
It is known that inverse strongly monotone operators have been studied widely (see [1–3]) and applied to solve practical problems in various fields; for instance, in traffic assignment problems (see [4, 5]).
-
T is firmly nonexpansive if and only if is nonexpansive or, equivalently, for all .
-
is said to be an averaged mapping if , where α is a number in and is nonexpansive. In particular, projections are -averaged mappings.
Averaged mappings have received many investigations, see ([6–10]).
Let ϕ be a bifunction of into ℝ, where ℝ is the set of real numbers. The equilibrium problem for is to find such that
The set of solutions of (1.1) is denoted by . Given a mapping , let for all . Then if and only if for all , i.e., z is a solution of the variational inequality. Numerous problems in physics, optimization and economics reduce to finding a solution of (1.1). Some methods have been proposed to solve the equilibrium problem; see, for instance, [11–15].
In 2000, Moudafi [16] introduced the viscosity approximation method for nonexpansive mappings. Let h be a contraction on H, starting with an arbitrary initial , define a sequence recursively by
where is a sequence in . Xu [17] proved that under certain conditions on , the sequence generated by (1.2) converges strongly to the unique solution of the variational inequality
We use to denote the set of fixed points of the mapping T; that is, .
In 2006, Marino and Xu [18] introduced a general iterative method for nonexpansive mappings. Let h be a contraction on H with a coefficient , and let A be a strongly positive bounded linear operator on H with a constant . Starting with an arbitrary initial guess , define a sequence recursively by
where is a sequence in , and is a constant. It is proved that the sequence converges strongly to the unique solution of the variational inequality
For finding the common solution of and a fixed point problem, Takahashi and Takahashi [11] introduced the following iterative scheme by the viscosity approximation method in a Hilbert space: and
for all , where and satisfy some appropriate conditions. Further, they proved and converge strongly to , where .
On the other hand, let be a convex function, and consider the following minimization problem:
Assume that the constrained convex minimization problem (1.5) is solvable, and let U denote the set of solutions of (1.5). Then the gradient-projection algorithm (GPA) generates a sequence according to the recursive formula
where the parameters are real positive numbers, and is the metric projection from H onto C. It is known that the convergence of the sequence depends on the behavior of the gradient ∇f. If the gradient ∇f is only assumed to be inverse strongly monotone, then can only be convergent weakly to a minimizer of (1.5). If the gradient ∇f is Lipschitz continuous and strongly monotone, then the sequence can be convergent strongly to a minimizer of (1.5) provided the parameters satisfy appropriate conditions.
As everyone knows, Xu [9] gave an averaged mapping approach to the gradient-projection method, and he constructed a counterexample which shows that the sequence generated by the gradient-projection method does not converge strongly in the infinite-dimensional space. Moreover, he presented two modifications of the gradient-projection method which are shown to have strong convergence.
In 2011, motivated by Xu, Ceng [19] proposed the following iterative algorithm:
where is an l-Lipschitzian mapping with a constant , and is a k-Lipschitzian and η-strongly monotone operator with constants . Let , , and . Let and satisfy , . Under suitable conditions, it is proved that the sequence generated by (1.6) converges strongly to a minimizer of (1.5).
In 2012, Tian and Liu [20] introduced the following iterative method in a Hilbert space: and
where , , and , and , , satisfy appropriate conditions. Further, they proved the sequence converges strongly to a point , which solves the variational inequality
It is the first time that the equilibrium and constrained convex minimization problems have been solved.
Since, in general, the minimization problem (1.5) has more than one solution, so regularization is needed. Now we consider the following regularized minimization problem:
where is the regularization parameter, f is a convex function with a -ism continuous gradient ∇f. Then the regularized GPA generates a sequence by the following recursive formula:
where the parameter , γ is a constant with , and is the metric projection from H onto C. We all know that the sequence generated by algorithm (1.8) converges weakly to a minimizer of (1.5) in the setting of infinite-dimensional spaces (see [21]).
In this paper, motivated and inspired by the above results, we introduce a new iterative method: and
for finding a element of , where , , , . Under appropriate conditions, it is proved that the sequence generated by (1.9) converges strongly to a point , which solves the variational inequality
2 Preliminaries
In this section we introduce some useful properties and lemmas which will be used in the proofs for the main results in the next section.
Some properties of averaged mappings are gathered in the proposition below.
Let the operators be given:
-
(i)
If for some and if S is averaged and V is nonexpansive, then T is averaged.
-
(ii)
The composition 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 α-averaged, where .
-
(iii)
If the mappings are averaged and have a common fixed point, then
Here the notation denotes the set of fixed points of the mapping T; that is, .
The following proposition gathers some results on the relationship between averaged mappings and inverse strongly monotone operators.
Let be given. We have:
-
(i)
T is nonexpansive if and only if the complement is ()-ism;
-
(ii)
If T is υ-ism, then for , γT is ()-ism;
-
(iii)
T is averaged if and only if the complement is υ-ism for some ; indeed, for , T is α-averaged if and only if is ()-ism.
Lemma 2.1 [9]
Assume that is a sequence of nonnegative real numbers such that
where and are sequences in and is a sequence in ℝ such that
-
(i)
;
-
(ii)
either or ;
-
(iii)
.
Then .
The so-called demiclosed principle for nonexpansive mappings will often be used.
Lemma 2.2 (Demiclosed principle [23])
Let C be a closed and convex subset of a Hilbert space H and let be a nonexpansive mapping with . If is a sequence in C weakly converging to x and if converges strongly to y, then . In particular, if , then .
Lemma 2.3 [18]
Let H be a Hilbert space, let C be a closed and convex subset of H, let be a Lipschitzian operator with a coefficient , and let be a strongly positive bounded linear operator with a coefficient . Then, for ,
That is, is strongly monotone with a coefficient .
Recall the metric (nearest point) projection from a real Hilbert space H to a closed and convex subset C of H is defined as follows: given , is the unique point in C with the property
is characterized as follows.
Lemma 2.4 Let C be a closed and convex subset of a real Hilbert space H. Given and . Then if and only if the following inequality holds:
Lemma 2.5 [18]
Assume that A is a strongly positive bounded linear operator on a Hilbert space H with a coefficient and . Then .
For solving the equilibrium problem for a bifunction , let us assume that ϕ satisfies the following conditions:
(A1) for all ;
(A2) ϕ is monotone, i.e., for all ;
(A3) for each , ;
(A4) for each , is convex and lower semicontinuous.
Lemma 2.6 [24]
Let C be a nonempty, closed and convex subset of H and let ϕ be a bifunction of into ℝ satisfying (A1)-(A4). Let and . Then there exists such that
Lemma 2.7 [13]
Assume that satisfies (A1)-(A4). For and , define a mapping as follows:
Then the following hold:
-
(1)
is single-valued;
-
(2)
is firmly nonexpansive, i.e., for any ;
-
(3)
;
-
(4)
is closed and convex.
We adopt the following notation:
-
means that strongly;
-
means that weakly.
3 Main results
Recall that throughout this paper we always denote U as the solution set of the constrained convex minimization problem (1.5), and denote as the solution set of the equilibrium problem (1.1).
Let H be a real Hilbert space and C be a nonempty closed convex subset of H. Let be Lipschitzian with a constant , and be a strongly positive bounded linear operator with a coefficient , and . Suppose that ∇f is -ism continuous. Let be a mapping defined as in Lemma 2.7. We now consider the following mapping on H defined by
where , , and , . It is easy to see that is a contraction. Indeed, by Lemma 2.5 and Lemma 2.7, we have for each
Since , it follows that is a contraction. Therefore, by the Banach contraction principle, has a unique fixed point such that
Note that indeed depends on V as well, but we will suppress this dependence of on V for simplicity of notation throughout the rest of this paper.
The following theorem summarizes the properties of the sequence .
Theorem 3.1 Let C be a nonempty, closed and convex subset of a Hilbert space H. Let ϕ be a bifunction from satisfying (A1)-(A4), and let be a real-valued convex function, and assume that the gradient ∇f is -ism with a constant . Assume that . Let be Lipschitzian with a constant , and let be a strongly positive bounded linear operator with a coefficient , and . Let the sequences and be generated by
where , , and . Let , and satisfy the following conditions:
-
(i)
, ;
-
(ii)
, ;
-
(iii)
, .
Then the sequence converges strongly to a point , which solves the variational inequality
Equivalently, we have .
Proof It is well known that solves the minimization problem (1.5) if and only if for each fixed , solves the fixed-point equation
It is clear that , i.e., .
First, we assume that . By Lemma 2.5, we obtain . Let , then from , we have
for all . Thus, we have from (3.3)
It follows that
For , note that
and
where and .
Then we get
Since and , there exists such that
where .
It follows from (3.4) and (3.5) that
Since , there exists a real number such that , and
Hence is bounded and we also obtain that is bounded.
Next, we show that .
Indeed, for any , by Lemma 2.7, we have
This implies that
Then from (3.5), we derive that
It follows from (3.6) that
Since both and are bounded and , , it follows that .
We claim that . Indeed,
Since and , we obtain that
Thus,
and
we have and .
Since is bounded, without loss of generality, we can assume that . Next, we show that .
By (3.5), we have
So, by Lemma 2.2, we get .
Next, we show that . Since , for any , we obtain
From (A2), we have
and hence
Since and , it follows from (A4) that for any .
Let , , , then we have and hence .
Thus, from (A1) and (A4), we have
and hence . From (A3), we have for any , hence . Therefore, .
On the other hand, we note that
Hence, we obtain from (3.3) and (3.5) that
It follows that
In particular,
Since and , it follows from (3.7) that as .
Next, we show that z solves the variational inequality (3.2). Observe that .
Hence, we conclude that
Since is nonexpansive, we have is monotone. Note that for any given ,
Now, replacing n with in the above inequality, and letting , since , , and , we have
It follows that is a solution of the variational inequality (3.2). Further, by the uniqueness of the solution of the variational inequality (3.2), we conclude that as .
The variational inequality (3.2) can be rewritten as
By Lemma 2.4, it is equivalent to the following fixed-point equation:
This completes the proof. □
Theorem 3.2 Let C be a nonempty, closed and convex subset of Hilbert space H. Let ϕ be a bifunction from satisfying (A1)-(A4), and let be a real-valued convex function, and assume that the gradient ∇f is -ism with a constant . Assume that . Let be Lipschitzian with a constant , and let be a strongly positive bounded linear operator with a coefficient , and . Let the sequences and be generated by and
where , , and . Let , and satisfy the following conditions:
(C1) , , ;
(C2) , , , ;
(C3) , , .
Then the sequence converges strongly to a point , which solves the variational inequality (3.2).
Proof It is well known that:
-
(a)
solves the minimization problem (1.5) if and only if for each fixed , solves the fixed-point equation
It is clear that , i.e., .
-
(b)
The gradient ∇f is -ism.
-
(c)
is averaged for , in particular, the following relation holds:
Since , we may assume that . Now, we first show that is bounded. Indeed, pick , since , by Lemma 2.7, we know that
Thus, we derive from (3.5) that
Since , there exists a real number such that . Thus,
By induction, we have
Hence is bounded. From (3.9), we also derive that is bounded.
Next, we show that .
Indeed, since
we have
So, we obtain that
for some appropriate constant such that
Thus, we get
Since both and are bounded, we can take a constant such that
Consequently,
From and , we note that
and
Putting in (3.11) and in (3.12), we have
and
So, from (A2), we have
and hence
Since , without loss of generality, let us assume that there exists a real number a such that for all . Thus, we have
thus,
where .
From (3.10) and (3.13), we obtain
where . Hence, by Lemma 2.1, we have
Then, from (3.13), (3.14) and , we have
For any , as the same proof of Theorem 3.1, we have
Then, from (3.5) and (3.16), by the same argument as in the proof of Theorem 3.1, we derive that
Since both and are bounded, , , and , we have
Next,
and then
It follows that .
Now, we show that
where is a unique solution of the variational inequality (3.2).
Indeed, since is bounded, without loss of generality, we may assume that such that
By (3.17) and , we derive that .
Note that
Hence, by , we get .
In terms of Lemma 2.2, we get .
Then, by the same argument as in the proof of Theorem 3.1, we have .
Since is the solution of the variational inequality (3.2), we derive from (3.19) that
Finally, we show that .
As a matter of fact,
So, from (3.5) and (3.9), we derive
It follows that
Since is bounded, we can take a constant such that
Then, we obtain that
where .
By (3.20) and , we get . Now applying Lemma 2.1 to (3.21) concludes that as . □
4 Application
In this section, we give an application of Theorem 3.2 to the split feasibility problem (SFP for short) which was introduced by Censor and Elfving [25]. Since its inception in 1994, the split feasibility problem (SFP) has received much attention (see [21, 26, 27]) due to its applications in signal processing and image reconstruction, with particular progress in intensity-modulated radiation therapy.
The SFP can mathematically be formulated as the problem of finding a point x with the property
where C and Q are nonempty closed convex subsets of Hilbert spaces and , respectively. is a bounded linear operator.
It is clear that is a solution to the split feasibility problem (4.1) if and only if and . We define the proximity function f by
and consider the constrained convex minimization problem
Then solves the split feasibility problem (4.1) if and only if solves the minimization problem (4.2) with the minimize equal to 0. Byrne [7] introduced the so-called CQ algorithm to solve the SFP.
where . He obtained that the sequence generated by (4.3) converges weakly to a solution of the SFP.
In order to obtain a strong convergence iterative sequence to solve the SFP, we propose the following algorithm:
where . Let satisfy the following conditions:
-
(i)
and ;
-
(ii)
for all n,
where is Lipschitzian with a constant and is a strongly positive bounded linear operator with a constant . Suppose that . We can show that the sequence generated by (4.4) converges strongly to a solution of SFP (4.1) if the sequence and the sequence of parameters satisfy appropriate conditions.
Applying Theorem 3.2, we obtain the following result.
Theorem 4.1 Assume that the split feasibility problem (4.1) is consistent. Let the sequence be generated by (4.4). Where the sequence and the sequence satisfy the conditions (C1)-(C3). Then the sequence converges strongly to a solution of the split feasibility problem (4.1).
Proof By the definition of the proximity function f, we have
and ∇f is Lipschitz continuous, i.e.,
where .
Set ; consequently,
Then the iterative scheme (4.4) is equivalent to
where . satisfy the following conditions:
-
(i)
and ;
-
(ii)
for all n.
Due to Theorem 3.2, we have the conclusion immediately. □
5 Conclusion
Methods for solving the equilibrium problem (EP) and the constrained convex minimization problem have been extensively studied, respectively, in a Hilbert space. In 2012, Tian and Liu proposed an iterative method for finding a common solution of an EP and a constrained convex minimization problem. But, in this paper, it is the first time that we combine the regularized gradient-projection algorithm and the averaged mappings approach to propose implicit and explicit algorithms for finding the common solution of an EP and a constrained convex minimization problem, which also solves a certain variational inequality.
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Acknowledgements
The authors wish to thank the referees for their helpful comments, which notably improved the presentation of this manuscript. This work was supported by the Fundamental Research Funds for the Central Universities (No. ZXH2012K001).
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Tian, M., Huang, LH. Regularized gradient-projection methods for equilibrium and constrained convex minimization problems. J Inequal Appl 2013, 243 (2013). https://doi.org/10.1186/1029-242X-2013-243
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DOI: https://doi.org/10.1186/1029-242X-2013-243