Iterative Schemes for Convex Minimization Problems with Constraints

We first introduce and analyze one implicit iterative algorithm for finding a solution of the minimization problem for a convex and continuously Fr´echet differentiable functional, with constraints of several problems: the generalized mixed equilibrium problem, the system of generalized equilibrium problems, and finitely many variational inclusions in a real Hilbert space. We prove strong convergence theorem for the iterative algorithm under suitable conditions. On the other hand, we also propose another implicit iterative algorithm for finding a fixed point of infinitely many nonexpansive mappings with the same constraints, and derive its strong convergence under mild assumptions.


Introduction
Let be a nonempty closed convex subset of a real Hilbert space and let be the metric projection of onto . Let : → be a nonlinear mapping on . We denote by Fix( ) the set of fixed points of and by R the set of all real numbers. A mapping is called strongly positive on if there exists a constant > 0 such that ⟨ , ⟩ ≥ ‖ ‖ 2 , ∀ ∈ . (1) A mapping : → is called -Lipschitz continuous if there exists a constant ≥ 0 such that In particular, if = 1, then is called a nonexpansive mapping; if ∈ [0, 1), then is called a contraction.
We denote by ( , ) the solution set of the variational inclusion (11). In 1998, Huang [4] studied problem (11) in the case where is maximal monotone and is strongly monotone and Lipschitz continuous with ( ) = = .
Let : → R be a convex and continuously Fréchet differentiable functional. Consider the convex minimization problem (CMP) of minimizing over the constraint set : We denote by Γ the set of minimizers of CMP (12). Very recently, Ceng and Al-Homidan [5] introduced an implicit iterative algorithm for finding a common solution of the CMP (12), finitely many GMEPs and finitely many variational inclusions, and derived its strong convergence under appropriate conditions.
Algorithm CA (see [5,Theorem 18] where ( − ∇ ) = +(1− ) (here is nonexpansive, = ((2 − )/4) ∈ (0, (1/2)) for each ∈ (0, (2/ ))), and the following conditions hold: Motivated and inspired by the above facts, we first introduce and analyze one implicit iterative algorithm for finding a solution of the CMP (12) with constraints of several problems: the GMEP (4), the SGEP (8), and finitely many variational inclusions in a real Hilbert space. We prove strong convergence theorem for the iterative algorithm under suitable conditions. The iterative algorithm is based on Korpelevich's extragradient method, hybrid steepest-descent method in [6], viscosity approximation method, averaged mapping approach to the GPA in [7], and strongly positive bounded linear operator technique. On the other hand, we also propose another implicit iterative algorithm for finding a fixed point of infinitely many nonexpansive mappings with the same constraints. We derive its strong convergence under mild assumptions.

Preliminaries
Throughout this paper, we assume that is a real Hilbert space whose inner product and norm are denoted by ⟨⋅, ⋅⟩ and ‖ ⋅ ‖, respectively. Let be a nonempty closed convex subset of . We write ⇀ to indicate that the sequence { } converges weakly to and → to indicate that the sequence { } converges strongly to . Moreover, we use ( ) to denote the weak -limit set of the sequence { }; that is, Recall that a mapping : → is called (ii) -strongly monotone if there exists a constant > 0 such that (iii) -inverse-strongly monotone if there exists a constant > 0 such that It is obvious that if is -inverse-strongly monotone, then is monotone and (1/ )-Lipschitz continuous.
The metric (or nearest point) projection from onto is the mapping : → which assigns to each point ∈ the unique point ∈ satisfying the property Some important properties of projections are gathered in the following proposition.
If is an -inverse-strongly monotone mapping of into , then it is obvious that is (1/ )-Lipschitz continuous. If ≤ 2 , then it is easy to see that − is a nonexpansive mapping from to .
alternatively, is firmly nonexpansive if and only if can be expressed as where : → is nonexpansive; projections are firmly nonexpansive.
It can be easily seen that if is nonexpansive, then − is monotone. It is also easy to see that a projection is 1-ism.

Definition 3. A mapping
: → is said to be an averaged mapping if it can be written as the average of the identity and a nonexpansive mapping; that is, where ∈ (0, 1) and : → is nonexpansive. More precisely, when the last equality holds, we say that isaveraged. Thus firmly nonexpansive mappings (in particular, projections) are (1/2)-averaged mappings.
for some ∈ (0, 1) and if is averaged and is nonexpansive, then is averaged.

(ii) is firmly nonexpansive if and only if the complement
− is firmly nonexpansive.
We need some facts and tools in a real Hilbert space which are listed as lemmas below.

Lemma 8.
Let be a real inner product space. Then there holds the following inequality: Lemma 9. Let : → be a monotone mapping. In the context of the variational inequality problem the characterization of the projection (see Proposition 1(i)) implies The following lemma can be easily proved and, therefore, we omit the proof.
That is, − is strongly monotone with constant − 1.
Let be a nonempty closed convex subset of a real Hilbert space . We introduce some notations. Let be a number in (0, 1] and let > 0. Associating with a nonexpansive mapping : → , we define the mapping : → by where : → is an operator such that, for some positive constants , > 0, is -Lipschitzian and -strongly monotone on ; that is, satisfies the conditions for all , ∈ .
Let : → be a monotone, -Lipschitz-continuous mapping and let V be the normal cone to at V ∈ ; that is, Define It is well known that is maximal monotone and 0 ∈ V if and only if V ∈ ( , ). Assume that : ( ) ⊂ → 2 is a maximal monotone mapping. Then, for > 0, associated with , the resolvent operator , can be defined as From Huang [4] (see also [13]), there holds the following property for the resolvent operator , : → ( ).
Lemma 16 (see [14]). Let be a maximal monotone mapping with ( ) = . Then for any given > 0, ∈ is a solution of problem (11) if and only if ∈ satisfies Lemma 17 (see [13]). Let be a maximal monotone mapping with ( ) = and let : → be a strongly monotone, continuous, and single-valued mapping. Then for each ∈ , the equation ∈ ( + ) has a unique solution for > 0.

Convex Minimization Problems with Constraints
In this section, we will introduce and analyze one implicit iterative algorithm for finding a solution of the CMP (12) with constraints of several problems: the GMEP (4), the SGEP (8), and finitely many variational inclusions in a real Hilbert space. We po prove strong convergence theorem for the iterative algorithm under suitable conditions.
Finally, we prove that { } converges strongly as → (2/ ) (⇔ → 0) to ∈ Ω, which is the unique solution in Ω to the VIP (43). In fact, we note that, for ∈ Ω with ⇀ , By (48), (57), and Lemma 14, we obtain that In particular, we have Since ⇀ and lim → ∞ = 0, it follows from (112) that → as → ∞. Now we show that solves the VIP (43). As a matter of fact, from (55) and (59) we obtain that, for any ∈ Ω, which immediately implies that Since → 0 and → , we get By Minty's lemma, is a solution in Ω to the VIP (43). In terms of the uniqueness of solutions of VIP (43), we deduce that = and → as → ∞. So, every weak convergence subsequence of { } converges strongly to the unique solution of VIP (43). Therefore, { } converges strongly to the unique solution of VIP (43). This completes the proof.
→ R is strongly convex with constant > 0 and its derivative is Lipschitz continuous with constant ] > 0 such that the function → ⟨ − , ( )⟩ is weakly upper semicontinuous for each ∈ .

Fixed Point Problems with Constraints
In this section, we will introduce and analyze another implicit iterative algorithm for solving the fixed point problem of infinitely many nonexpansive mappings with constraints of several problems: the GMEP (4), the SGEP (8), and finitely many variational inclusions in a real Hilbert space. We prove strong convergence theorem for the iterative algorithm under mild assumptions.
where is the -mapping defined by (9). Suppose that the following conditions hold: → R is strongly convex with constant > 0 and its derivative is Lipschitz continuous with constant ] > 0 such that the function → ⟨ − , ( )⟩ is weakly upper semicontinuous for each ∈ ; (ii) for each ∈ , there exist a bounded subset ⊂ and ∈ such that for any ∉ , (v) 0 < lim inf → ∞ ≤ lim sup → ∞ < 2 .
Finally, we prove that { } converges strongly to ∈ Ω, which is the unique solution in Ω to the VIP (136). In fact, we note that, for ∈ Ω with ⇀ , Utilizing the arguments similar to those in the proof of Theorem 19, we obtain that which hence leads to Since ⇀ and lim → ∞ = 0, it follows from (152) that → as → ∞. Now we show that solves the VIP (136). As a matter of fact, from (142) and (145) we obtain that, for any ∈ Ω, Since → 0 and → , we get ⟨( − ) , − ⟩ ≤ 0, ∀ ∈ Ω.
By Minty's lemma, is a solution in Ω to the VIP (136). In terms of the uniqueness of solutions of VIP (136), we deduce that = and → as → ∞. So, every weak convergence subsequence of { } converges strongly to the unique solution of VIP (136). Therefore, { } converges strongly to the unique solution of VIP (136). This completes the proof.
where is the -mapping defined by (9). Suppose that the following conditions hold: → R is strongly convex with constant > 0 and its derivative is Lipschitz continuous with constant ] > 0 such that the function → ⟨ − , ( )⟩ is weakly upper semicontinuous for each ∈ ; (ii) for each ∈ , there exist a bounded subset ⊂ and ∈ such that, for any ∉ , (v) 0 < lim inf → ∞ ≤ lim sup → ∞ < 2 .
where is the -mapping defined by (9). Suppose that the following conditions hold: → R is strongly convex with constant > 0 and its derivative is Lipschitz continuous with constant ] > 0 such that the function → ⟨ − , ( )⟩ is weakly upper semicontinuous for each ∈ ; (ii) for each ∈ , there exist a bounded subset ⊂ and ∈ such that, for any ∉ , (v) 0 < lim inf → ∞ ≤ lim sup → ∞ < 2 .
Assume that (Θ, ) is firmly nonexpansive. Then { } converges strongly to a point ∈ Ω, which is a unique solution in Ω to the VIP: