An algorithm for variational inequalities with equilibrium and fixed point constraints

Abstract: In this paper, we propose a new hybrid extragradient-viscosity algorithm for solving variational inequality problems, where the constraint set is the common elements of the set of solutions of a pseudomonotone equilibrium problem and the set of fixed points of a demicontractive mapping. Using the hybrid extragradientviscosity method and combining with hybrid plane cutting techniques, we obtain the algorithm for this problem. Under certain conditions on parameters, the convergence of the iteration sequences generated by the algorithms is obtained.


Introduction
Let ℝ n be a n-dimentional Euclidean space with the inner product ⟨⋅, ⋅⟩ and associated norm ‖ ⋅ ‖. Let C be a nonempty closed convex subset in ℝ n and G: C → ℝ n , T: C → C be operators, and f : C × C → ℝ be a bifunction satisfying f (x, x) = 0 for every x ∈ C. We consider the following variational inequality problem over the set is the common elements of the set of a pseudomonotone equilibrium problem and the set of fixed points of a demicontractive mapping (shortly VIEFP(C, f , T, G): (1) Find x * ∈ S such that ⟨G(x * ), y − x * ⟩ ≥ 0, ∀y ∈ S,

PUBLIC INTEREST STATEMENT
Variational inequality and equilibrium problems as well as fixed point problems are very useful and efficient tools in mathematics. They provided a unified framework for studying many problems arising in engineering, economics, and other fields. In this paper, we propose a new algorithm for solving variational inequality problems where the constraint set is the common elements of the set of solutions of a pseudomonotone equilibrium problem and the set of fixed points of a demicontractive mapping. One difficulty of this problem is that the constraint set is not given explicitly. The proposed method allows us to solve this problem by solving a sequence of convex programs in which they are much easier to solve.
where S = S f ∩ Fix(T), S f = {u ∈ C: f (u, y) ≥ 0, ∀y ∈ C}, i.e. S f is the solution set of the following equilibrium problem (EP(C, f ) for short) and Fix(T) is the fixed points of the mapping T, i.e. Fix(T) = {v ∈ C such that T(v) = v}.
We call problem (1) the upper problem and (2) the lower one. Problem (1) is a special case of mathematical programs with equilibrium constraints. Sources for such problems can be found in Luo, Pang, and Ralph (1996), Migdalas, Pardalos, and Varbrand (1988), Muu and Oettli (1992). Bilevel variational inequalities were considered in Anh, Kim, and Muu (2012), Moudafi (2010) and Yao, Liou, and Kang (2010) suggested the use of the proximal point method for monotone bilevel equilibrium problems, which contain monotone variational inequalites as a special case. Recently, Ding (2010) used the auxiliary problem principle to monotone bilevel equilibrium problems. In those papers, the lower problem is required to be monotone. In this case, the subproblems to be solved are monotone.
It should be noticed that the solution set S f of the lower problem (2) is convex whenever f is pseudomonotone on C. However, the main difficulty is that, even the constrained set S f is convex, it is not given explicitly as in a standard mathematical programming problem, and therefore the available methods of convex optimization and variational inequality cannot be applied directly to problem (1).
In this paper, we extend the hybrid extragradient-viscosity methods introduced by Maingé (2008b) for solving bilevel problem (1) when the lower problem is pseudomonotone with respect to its solution set equilibrium problems rather than monotone variational inequalities as in Maingé (2008b), the later pseudomonotonicity is somewhat more general than pseudomonotone. We show that the sequence of iterates generated by the proposed algorithm converges to the unique solution of the bilevel problem (1).
The paper is organized as follows. Section 2 contains some preliminaries on the Euclidean projection and equilibrium problems. Section 3 is devoted to presentation of the algorithm and its convergence. In Section 4, we describe a special case of variational inequalities with variational inequalities and fixed points constraints, where the lower variational inequality is pseudomonotone with respect to its solution set.

Preliminaries
In the rest of the paper, by P C we denote as the projection operator on C, that is The following well-known results on the projection operator onto a closed convex set will be used in the sequel.
In the sequel, we need the following blanket assumptions (A1) f (., y) is continuous on Ω for every y ∈ C; (A2) f (x, .) is convex on Ω for every x ∈ C; (A3) f is pseudomonotone on C with respect to the solution set S f of EP(C, f ); (A4) T is -demicontractive and closed mapping; (A5) G is L-Lipschitz and -strongly monotone on C; (B1) h(.) is -strongly convex, continuously differentiable on Ω; The following lemmas are well known from the auxiliary problem principle for equilibrium problems.
Lemma 2.3 (Mastroeni, 2003) Suppose that h is a continuously differentiable and strongly convex function on C with modulus > 0. Then, under assumptions (A1) and (A2), a point x * ∈ C is a solution of EP (C, f )

if and only if it is a solution to the equilibrium problem:
The function is called Bregman function. Such a function was used to define a generalized projection, called d-projection, which was used to develop algorithms for particular problems (see e.g. Censor & Lent, 1981).
An important case is h(x): = 1 2 ‖x‖ 2 . In this case, d-projection becomes the Euclidean one.

Lemma 2.4 (Mastroeni, 2003) Under assumptions (A1) and (A2), a point x * ∈ C is a solution of problem (AEP) if and only if
Note that since f (x, .) is convex and h is strongly convex, Problem (CP) is a strongly convex program.
For each z ∈ C, by 2 f (z, z), we denote the subgradient of the convex function f (z, .) at z, i.e. and each w ∈ 2 f (z, z) we define the halfspace H z as Note that when f (x, y) = ⟨F(x), y − x⟩, where F: C → ℝ n , this halfspace becomes the one introduced in Solodov and Svaiter (1999). The following lemma says that the hyperplane does not cut off any solution of problem EP(C, f ).

Lemma 2.7 Suppose the bifunction f satisfies the assumptions (A1) and (A2), the function h satisfies the assumption (B1). If {x k } ⊂ C is bounded and {y
and therefore In addition, Rockafellar (1970), the sequence {w k } is bounded; combining with the boundedness of {x k }, we have {y k } also bounded. Now, we prove the Lemma 2.7. Suppose contradict that {y k } is unbounded, i.e. there exists an subsequence {y k i } ⊆ {y k }, such that lim i→∞ ‖y k i ‖ = +∞. By the boundedness of {x k }, it implies {x k i } is also bounded; without loss of gerarality, we may assume that lim i→∞ x k i = x * . By the same argument as above, we have {y k i } is bounded, which we contradict. Therefore, {y k } is bounded. □ The following lemma is in Solodov and Svaiter (1999) (see also Dinh & Muu, 2015).
(2) w k ≠ 0 ∀k, indeed, at the beginning of Step 2, x k ≠ y k . By the Armijo linesearch rule and -strong convexity of h, we have Now, we are going to analyze the validity and convergence of the algorithm. Some parts in our proofs are based on the proof scheme in Maingé (2008b).
Proof First, we prove that there exists a positive integer m 0 such that Indeed, suppose by contradiction that for every positive integer m and z k,m = (1 − m )x k + m y k , there exists w k,m ∈ 2 f (z k,m , z k,m ), such that Since z k,m → x k as m → ∞, by Theorem 24.5 in Rockafellar (1970), the sequence {w k,m } ∞ m=1 is bounded. Thus, we may assume that w k,m →w for some w. Taking the limit as m → ∞, from z k,m → x k and w k,m →w, by Lemma 2.6, it follows that w ∈ 2 f (x k , x k ) and Since w ∈ 2 f (x k , x k ), we have Combining with (9) yields which contradicts to the fact that Thus, the linesearch is well defined. Now, we prove (8). For simplicity of notation, let d k : = x k − y k , H k : = H z k.
Since u k = P C∩H k (x k ) and x * ∈ S, by Lemma 2.5, x * ∈ C ∩ H k , we have which together with implies Replacing into (Equation 10), we obtain Substituting x k = z k + k d k into the last inequality, we get In addition, by the Armijo linesearch rule, using the -strong convexity of h, we have Note that x * ∈ H k can be written as: From Lemma 2.9, we have (Equation 12), we get where the last inequality follows from 0 < ≤ k ≤ 1− 2 . We have which together with (11) implies as desired. Proof From (13), we get In addition, Since G is L-Lipschitz and -strongly monotone, we have ∈ (0;1), and the last inequality deduced from (11). Combining with (15), we obtain Proof By definition of {x k } in Algorithm 1, we have Taking the limit as i → ∞ and by the closedness of mapping T, we get Now, we prove v ∈ S f . Indeed, from Lemma 3.2, {x k i } is bounded. Without loss of generality, we may assume that lim i→∞ x k i =x. We will consider two distinct cases: (18), one has lim i→∞ ‖y k i − x k i ‖ = 0, thus y k i →x and z k i →x.
According to the definition of y k i , we have by the continuity of h, ∇h, we get in the limit as i → ∞ that this fact shows that x ∈ S f .
By the linesearch rule and -strong convexity of h, we have From the boundedness of {w k i } and (18), it follows k i → 0, so that z k i = (1 − k i )x k i + k i y k i →x as i → ∞. Without loss of generality, we suppose that w k i →w ∈ 2 f (x,x) and y k i →ȳ as i → ∞.
We have letting i → ∞, we obtain in the limit that On the other hand, by the linesearch rule (6), for m k i − 1, there exists w Letting i → ∞ and combining with z →w ∈ 2 f (x,x) we obtain in the limit from (19) that (19) ⟨w Note that w ∈ f (x,x); it follows from the last inequality that Hence, which shows that x ∈ S f . (11), it implies taking the limit both sides of (20), we get lim i→∞ v k i =x. Hence, v =x. Therefore, v ∈ S. □ Now, we are in a position to prove the convergence of the proposed algorithm. By setting a k = ‖x k − x * ‖ 2 , and combining with the last inequalities, (21) becomes We will consider two distinct cases: Case 1. There exists k 0 , such that {a k } is decreasing when k ≥ k 0 .
Then, there exists lim k→∞ a k = a, taking the limit on both sides of (22), we get This implies lim k→∞ ‖v k − x * ‖ = a. In addition, lim k→∞ ‖x k+1 − v k ‖ 2 = 0, and lim Combining this fact with (23) and (24), we obtain By Lemma 3.3, we get ū ∈ S. Thus, Since G is -strongly monotone, one has Taking the limit as k → ∞ and remembering that a = lim ‖u k − x * ‖ 2 , we get If a > 0 , then by choosing = 1 2 a, from (25), it implies that there exists k 1 > 0, such that From (21), we get and thus summing up from k 1 to k, we have combining this fact with ∑ ∞ k=1 k = ∞ and ∑ ∞ k=1 2 k < ∞, we obtain lim inf a k = −∞, which is a contradiction.
Case 2. There exists a subsequence {a k i } i≥0 ⊂ {a k } k≥0 , such that a k i < a k i +1 for all i ≥ 0. In this situation, we consider the sequence of indices { (k)} defined as in Lemma 2.10. It follows that a (k)+1 − a (k) ≥ 0, which by (22) amounts to Therefore, From the boundedness of {v (k) }, without loss of generality, we may assume that v (k) →v. By Lemma 3.3, we get v ∈ S.

By (21), we get which implies
Since G is -strongly monotone, we have which combining with (26), we get so that which amounts to In addition, which together with (27), one has lim k→∞ x (k)+1 = x * , which means that lim k→∞ a (k)+1 = 0.

Application to variational inequalities with variational inequality and fixed point constraints
In this section, we consider the following variational inequality problem over the set that is the common elements of the solution set of a pseudomonotone variational inequality problem and the set of fixed points of a demicontractive mapping (shortly VIFP(C, F, T, G): where S = S F ∩ Fix(T), S F = {u ∈ C:⟨F(u), y − u⟩ ≥ 0, ∀y ∈ C}, i.e. S F is the solution set of the following variational inequality problems VIP(C, F) for short) and as before, Fix(T) is the fixed point of the mapping T. This problem was considered by Maingé (2008b).
In the sequel, we always suppose that Assumptions (A1), (A2), (A3), (A4), and (A5) are satisfied. The algorithm for this case takes the form: Similar to Theorem 3.1, we have the following theorem

Conclusion
We have proposed a hybrid extragradient-viscosity algorithm for solving strongly monotone variational inequality problems over the set that is common points of the set of solutions of a pseudomonotone equilibrium problem and the set of fixed points of a demicontractive mapping. The convergence of the proposed algorithm is obtained, and a special case of this problem is also considered.