Iterative algorithms based on the viscosity approximation method for equilibrium and constrained convex minimization problem
© Tian and Liu; licensee Springer 2012
Received: 20 March 2012
Accepted: 24 October 2012
Published: 7 November 2012
The gradient-projection algorithm (GPA) plays an important role in solving constrained convex minimization problems. Based on the viscosity approximation method, we combine the GPA and averaged mapping approach to propose implicit and explicit composite iterative algorithms for finding a common solution of an equilibrium and a constrained convex minimization problem for the first time in this paper. Under suitable conditions, strong convergence theorems are obtained.
MSC:46N10, 47J20, 74G60.
Let H be a real Hilbert space with inner product and norm . Let C be a nonempty closed convex subset of H. Let be a nonexpansive mapping, namely , for all . The set of fixed points of T is denoted by .
We denoted the set of solutions of EP by . Given a mapping , let for all , then if and only if for all , that is, z is a solution of the variational inequality. Numerous problems in physics, optimizations, and economics reduce to find a solution of (1.1). Some methods have been proposed to solve the equilibrium problem; see, for instance, [1–3] and the references therein.
is a contraction; hence the sequence defined by the algorithm (1.3) converges in norm to the unique minimizer of (1.2). However, if the gradient ∇g fails to be strongly monotone, the operator W defined by (1.5) would fail to be contractive; consequently, the sequence generated by the algorithm (1.3) may fail to converge strongly (see ). If ∇g is Lipschitzian, then the algorithms (1.3) and (1.4) can still converge in the weak topology under certain conditions.
Recently, Xu  proposed an explicit operator-oriented approach to the algorithm (1.4); that is, an averaged mapping approach. He gave his averaged mapping approach to the GPA (1.4) and the relaxed gradient-projection algorithm. Moreover, he constructed a counterexample which shows that the algorithm (1.3) does not converge in norm in an infinite-dimensional space and also presented two modifications of GPA which are shown to have strong convergence [20, 21].
where and for each . He proved that the sequences converge strongly to a minimizer of the constrained convex minimization problem, which also solves a certain variational inequality.
The purpose of the paper is to study the iterative method for finding the common solution of an equilibrium problem and a constrained convex minimization problem. Based on the viscosity approximation method, we combine the GPA and averaged mapping approach to propose implicit and explicit composite iterative method for finding the common element of the set of solutions of an equilibrium problem and the solution set of a constrained convex minimization problem. We also prove some strong convergence theorems.
Throughout this paper, we always assume that C is a nonempty closed convex subset of a Hilbert space H. We use ‘⇀’ for weak convergence and ‘→’ for strong convergence.
holds for every with .
In order to solve the equilibrium problem for a bifunction , let us assume that ϕ satisfies the following conditions:
(A1) , for all ;
(A2) ϕ is monotone, that is, for all ;
(A3) for all , ;
(A4) for each fixed , the function is convex and lower semicontinuous.
Let us recall the following lemmas which will be useful for our paper.
Lemma 2.1 
- (2)is firmly nonexpansive; that is,
is closed and convex.
where is nonexpansive. Obviously, projections are firmly nonexpansive.
where and is nonexpansive. More precisely, when (2.1) holds, we say that T is α-averaged.
Clearly, a firmly nonexpansive mapping is a -averaged map.
Proposition 2.1 
If for some and if U is averaged and V is nonexpansive, then T is averaged.
T is firmly nonexpansive if and only if the complement I-T is firmly nonexpansive.
If for some , U is firmly nonexpansive and V is nonexpansive, then T is averaged.
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 α-averaged, where .
if and only if , .
if and only if , .
Consequently, is nonexpansive and monotone.
The so-called demiclosedness principle for nonexpansive mappings will be used.
Lemma 2.4 (Demiclosedness principle )
Let be a nonexpansive mapping with . If is a sequence in C that converges weakly to x and if converges strongly to y, then . In particular, if , then .
Next, we introduce monotonicity of a nonlinear operator.
- (a)monotone if
- (b)β-strongly monotone if there exists such that
- (c)ν-inverse strongly monotone (for short, ν-ism) if there exists such that
It can be easily seen that if G is nonexpansive, then is monotone; and the projection map is a 1-ism.
The inverse strongly monotone (also referred to as co-coercive) operators have been widely used to solve practical problems in various fields, for instance, in traffic assignment problems; see, for example, [29, 30] and reference therein.
The following proposition summarizes some results on the relationship between averaged mappings and inverse strongly monotone operators.
Proposition 2.2 
T is nonexpansive if and only if the complement is -ism.
If T is ν-ism, then for , γT is -ism.
T is averaged if and only if the complement is ν-ism for some . Indeed, for , T is α-averaged if and only if is -ism.
Lemma 2.5 
3 Main results
In this paper, we always assume that is a real-valued convex function and ∇g is an L-Lipschitzian mapping with . Since the Lipschitz continuity of ∇g implies that it is indeed inverse strongly monotone, its complement can be an averaged mapping. Consequently, the GPA can be rewritten as the composite of a projection and an averaged mapping, which is again an averaged mapping. This shows that an averaged mapping plays an important role in the gradient-projection algorithm.
where is nonexpansive and .
Then converges strongly, as (), to a point which solves the variational inequality (3.1).
and hence is bounded. From (3.2), we also derive that is bounded.
This shows that .
and hence . From (A3), we have for all and hence . Therefore, .
Since , it follows from (3.4) that as .
This completes the proof. □
, , ;
, , , ;
, (), .
Then converges strongly to a point which solves the variational inequality (3.1).
and hence is bounded. From (3.5), we also derive that is bounded.
This together with the boundedness of implies that is bounded.
Then, , it follows that .
Since is bounded, without loss of generality, we may assume that . By the same argument as in the proof of Theorem 3.1, we have .
where , and .
It is easy to see that , , and by (3.12). Hence, by Lemma 2.5, the sequence converges strongly to q. This completes the proof. □
Methods for solving the equilibrium problem and the constrained convex minimization problem have extensively been studied respectively in a Hilbert space. But to the best of our knowledge, it would probably be the first time in the literature that we introduce implicit and explicit algorithms for finding the common element of the set of solutions of an equilibrium problem and the set of solutions of a constrained convex minimization problem, which also solves a certain variational inequality.
The authors wish to thank the referees for their helpful comments, which notably improved the presentation of this manuscript. This work was supported in part by The Fundamental Research Funds for the Central Universities (the Special Fund of Science in Civil Aviation University of China: No. ZXH2012K001), and by the Science Research Foundation of Civil Aviation University of China (No. 2012KYM03).
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