Steepest-Descent Approach to Triple Hierarchical Constrained Optimization Problems

We introduce and analyze a hybrid steepest-descent algorithm by combining Korpelevich’s extragradient method, the steepest-descent method, and the averaged mapping approach to the gradient-projection algorithm. It is proven that under appropriate assumptions, the proposed algorithm converges strongly to the unique solution of a triple hierarchical constrained optimization problem (THCOP) over the common fixed point set of finitely many nonexpansive mappings, with constraints of finitely many generalized mixed equilibrium problems (GMEPs), finitely many variational inclusions, and a convex minimization problem (CMP) in a real Hilbert space.


Introduction
Let be a real Hilbert space with inner product ⟨⋅, ⋅⟩ and norm ‖ ⋅ ‖; let be a nonempty closed convex subset of 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 -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.
The solution set of VIP (2) is denoted by VI( , ).
On the other hand, let be a single-valued mapping of into and be a set-valued mapping with ( ) = . Considering the following variational inclusion, find a point ∈ such that 0 ∈ + .
We denote by ( , ) the solution set of the variational inclusion (9). Let a set-valued mapping : ( ) ⊂ → 2 be maximal monotone. We define the resolvent operator , : → ( ) associated with and as follows: where is a positive number. Let and be two nonexpansive mappings. In 2009, Yao et al. [9] considered the following hierarchical VIP: find hierarchically a fixed point of , which is a solution to the VIP for monotone mapping − ; namely, find̃∈ Fix( ) such that ⟨( − )̃, −̃⟩ ≥ 0, ∀ ∈ Fix ( ) . (11) The solution set of the hierarchical VIP (11) is denoted by Λ. It is not hard to check that solving the hierarchical VIP (11) is equivalent to the fixed point problem of the composite mapping Fix( ) ; that is, find̃∈ such that = Fix( )̃. The authors [9] introduced and analyzed the following iterative algorithm for solving the hierarchical VIP (11): In this paper, we introduce and study the following triple hierarchical constrained optimization problem (THCOP) with constraints of the CMP (8), finitely many GMEPs and finitely many variational inclusions.

Preliminaries
Throughout this paper, we assume that is a real Hilbert space of which 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, (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. Moreover, we also have that, for all , V ∈ and > 0, So, if ≤ 2 , then − is a nonexpansive mapping from to .
The metric 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.

Proposition 2.
For given ∈ and ∈ : (This implies that is nonexpansive and monotone.) Next we list some elementary conclusions for the mixed equilibrium problem where MEP(Θ, ) is the solution set.
Proposition 3 (see [10]  In the following, we recall some facts and tools in a real Hilbert space .

Lemma 4. Let be a real inner product space. Then there holds the following inequality
Lemma 5. Let be a real Hilbert space. Then the following hold: Abstract and Applied Analysis Definition 6. 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 (particularly, projections) are 1/2-averaged mappings.

(ii) is firmly nonexpansive if and only if the complement
− is firmly nonexpansive.
(v) If the mappings { } =1 are averaged and have a common fixed point, then The notation Fix( ) denotes the set of all fixed points of the mapping ; that is, Fix( ) = { ∈ : = }.

Let
: → R be a convex functional with -Lipschitz continuous gradient ∇ . It is well known that the gradient-projection algorithm (GPA) generates a sequence { } determined by the gradient ∇ and the metric projection : or more generally, where, in both (26) and (27), the initial guess 0 is taken from arbitrarily, and the parameters or are positive real numbers. The convergence of algorithms (26) and (27) depends on the behavior of the gradient ∇ .
Lemma 9 (see [12, Demiclosedness principle]). Let be a nonempty closed convex subset of a real Hilbert space . Let be a nonexpansive self-mapping on . Then − is demiclosed. That is, whenever { } is a sequence in weakly converging to some ∈ and the sequence {( − ) } strongly converges to some , it follows that ( − ) = . Here is the identity operator of .
→ be a monotone mapping. In the context of the variational inequality problem the characterization of the projection (see Proposition 2(i)) implies 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 , ∈ .
. is a contraction provided by 0 < < 2 / 2 ; that is, Lemma 12 (see [13]). Let { } be a sequence of nonnegative numbers satisfying the conditions where { } and { } are sequences of real numbers such that Then lim → ∞ = 0.
Recall that a Banach space is said to satisfy Opial's property [12] if, for any given sequence { } ⊂ which converges weakly to an element ∈ , there holds the inequality (34) Abstract and Applied Analysis 5 It is well known that every Hilbert space satisfies Opial's property in [12].
Finally, recall that a set-valued mapping : ( ) ⊂ → 2 is called monotone if for all , ∈ ( ), ∈ , and ∈ imply A set-valued mapping is called maximal monotone if is monotone and ( + ) ( ) = for each > 0, where is the identity mapping of . We denote by ( ) the graph of . It is known that a monotone mapping is maximal if and only if, for ( , ) ∈ × , ⟨ − , − ⟩ ≥ 0, for every ( , ) ∈ ( ), implies ∈ . Let : → be a monotone, -Lipschitzcontinuous mapping and let V be the normal cone to at V ∈ ; that is, Then,̃is maximal monotone such that Let : ( ) ⊂ → 2 be a maximal monotone mapping. Let , > 0 be two positive numbers.
Lemma 13 (see [14]). There holds the resolvent identity For , > 0, there holds the following relation that Based on Huang [15], there holds the following property for the resolvent operator , : → ( ).
Lemma 15 (see [16]). Let be a maximal monotone mapping with ( ) = . Then for any given > 0, ∈ is a solution of problem (10) if and only if ∈ satisfies Lemma 16 (see [17]). 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.

Main Results
In this section, we will introduce and analyze a hybrid steepest-descent algorithm for finding a solution of the THCOP (13) with constraints of several problems: the CMP (8), finitely many GMEPs, and finitely many variational inclusions in a real Hilbert space. This algorithm is based on Korpelevich's extragradient method, the steepest-descent method, and the averaged mapping approach to the gradientprojection algorithm. We prove the strong convergence of the proposed algorithm to a unique solution of THCOP (13) under suitable conditions. Throughout this paper, let { } =1 be nonexpansive mappings : → with ≥ 1 an integer. We write [ ] := mod , for integer ≥ 1, with the mod function taking values in the set {1, 2, . . . , } (i.e., if = + for some integers ≥ 0 and 0 ≤ < , then The following is to state and prove the main result in this paper.
Step 1. We prove that { } is bounded.
In Theorem 18, putting ( ) ≡ 0, ∀ ∈ , we obtain that Γ = and = which is the identity mapping of . Hence Theorem 18 reduces to the following.
Then the following hold: In Theorem 18, putting = 1 and = 1, we obtain the following.
and that the following conditions are satisfied: