A new extragradient iteration algorithm for bilevel variational inequalities 2012

ACTA MATHEMATICA VIETNAMICA 95 Volume 37, Number 1, 2012, pp. 95–107 A NEW EXTRAGRADIENT ITERATION ALGORITHM FOR BILEVEL VARIATIONAL INEQUALITIES PHAM NGOC ANH Abstract. In this paper, we introduce an app roximation extragradient itera- tion method for solving bilevel variational inequalities involving two variational inequalities and we show that these problems can be solved by projection se- quences and fixed point techniques. We obtain a strong convergence of three iteration sequences generated by this method in a real Hilbert space. 1. Introduction Let H be a real Hilbert space with an inner product ·, · and the indu ced norm  · , and let C be a nonempty closed convex subset of H. We consider the bilevel variational inequalities (shortly BV I): Find x ∗ ∈ Sol(G, C) such that F (x ∗ ), x − x ∗  ≥ 0 ∀x ∈ Sol(G, C), where G : H → H, Sol(G, C) denotes the set of solutions of the following varia- tional inequalities: Find y ∗ ∈ C such that G(y ∗ ), y − y ∗  ≥ 0 ∀y ∈ C, and F : C → H. We denote by Sol(BV I) the set of solutions of (BV I). The problems (BV I) are also called to be quasivariational inequalities (see [8, 9, 10]). There problems are very interesting because they cover a class of mathematical programs with equilibrum constraints (see [12]), bilevel minimiza- tion problems (see [16]), variational inequalities and complementarity problems (see [1, 2, 5, 7, 13]). If F ≡ 0, then the b ilevel variational inequalities (BV I) become the f ollowing variational inequalities  shortly V I(G, C)  : Find x ∗ ∈ C such that G(x ∗ ), x − x ∗  ≥ 0 ∀x ∈ C. Suppose that f : H → R. It is well-known in convex programming that if f is convex and differentiable on Sol(G, C) then x ∗ is a solution to min{f(x) | x ∈ Sol(G, C)} Received November 19, 2010; in revised form July 7, 2011. 2010 Mathematics Subject Classification. 65K10, 90C25. Key words and phrases. Bilevel variational inequalities, monotonicity, Lipschitz continuous, extragradient algorithm. This work is supported by the Vietnam National Foundation for Science Technology Devel- opment (NAFOSTED). 96 PHAM NGOC ANH if and only if x ∗ is the solution to the variational inequalities V I  ∇f, Sol(G, C)  , where ∇f is the differentiation of f . Then the bilevel variational inequalities (BV I) are written by a form of mathematical programs with equilibrum con- straints:  min f (x) x ∈ {y ∗ | G(y ∗ ), z − y ∗  ≥ 0 ∀z ∈ C}. If f, g are two convex and differentiable functions, then the problems (BV I) (where F := ∇f and G := ∇g) become the following b ilevel minimization prob- lem (see [16]):  min f (x) x ∈ argmin{g(x) | x ∈ C}. In recent years, variational inequalities become an attractive field for many researchers and have many important applications in electricity markets, tr ans- portations, economics, nonlinear analysis (see [6, 9, 19]). Metho ds for solving vari- ational inequalities have been studied extensively. The extragradient algorithm for solving the variational inequalities V I(G, C) was introduced by Korpelevich in [11], where the iteration sequence {x k } is defined by      x 0 ∈ C, y k = P r C  x k − c k G(x k )  , x k+1 = P r C  x k − c k G(y k )  , and extended by many other authors (see [5, 9, 14, 18]). One of the main con- ditions ensures the convergence result of this method is that the cost mapping enjoys the Lipschitzian continuity property. However, such a condition is rather restrictive. In order to avoid it, the following Armijo-backtracking linesearch has been used to construct a hyperplane separating x k from the solution set. Then the new iterate x k+1 is the projection of x k onto this hyperplane. Recently, Anh and Kuno in [4] extended these results to generalized monotone nonlipschitzian multivalued variational inequalities. Precisely, the authors first used the interior proximal function to develop a convergent algorithm for the mu ltivalued varia- tional inequalities V I(F, C), where F is a generalized monotone multifunction. Next the authors constructed an appropriate hyperplane which separates the cur- rent iterative point from the solution set. Then the next iterate is the projection of the current iterate onto the intersection of the feasible set with the halfspace containing the solution set. Note that since the constraint set Sol(G, C) being the solution set of the prob- lem VI(G, C) is not explicitly given, the existing algorithms for variational in- equalities can not be directly applied because the subproblems can not be im- plemented by the available algorithms of convex programming. In this paper we extend results in [3] to the bilevel variational inequalities (BV I), but in a real Hilbert s pace. We are interested in findin g a solution to bilevel variational inequalities (BV I) where the functions F and G satisfy the following usual con- ditions: A NEW EXTRAGRADIENT ITERATION ALGORITHM FOR BVI 97 (A 1 ) G is monotone on C and F is β-strongly monotone on C, (A 2 ) F is L 1 -Lipschitz continuous on C, (A 3 ) G is L 2 -Lipschitz continuous on C, (A 4 ) The solution set of (BV I) denoted by Sol(BV I) is nonempty. In the next section, we give a new approximation extragradient algorithm for solving problems (BV I). 2. Preliminaries We list s ome known definitions and properties of the proj ection under the Euclidean norm which will be required in our following analysis. Definition 2.1. Let C be a nonempty closed convex subset in a real Hilbert space H. We denote the projection on C by P r C (·) with images P r C (x) = {y ∈ C | y − x = min v∈C v − x} ∀x ∈ H. The function ϕ : C → H is said to be (i) γ-strongly monotone on C if for any x, y ∈ C, we have ϕ(x) − ϕ(y), x − y ≥ γx − y 2 , (ii) monotone on C if for any x , y ∈ C, we have ϕ(x) − ϕ(y), x − y ≥ 0, (iii) Lipschitz on C with constant L > 0 (shortly L-Lipschitz) if for any x, y ∈ C, we have ϕ(x) − ϕ(y) ≤ Lx − y. If ϕ : C → C and L = 1 then ϕ is called nonexpansive on C. The projection P r C (·) has the following basic properties: (P roj 1 ) P r C (x) − P r C (y) ≤ x − y ∀x, y ∈ H. (P roj 2 ) P r C (x) − P r C (y) 2 ≤ P r C (x) − P r C (y), x − y ∀x, y ∈ H. (P roj 3 ) x − P r C (x), y − P r C (y) ≤ 0 ∀y ∈ C, x ∈ H. (P roj 4 ) P r C (x) − y 2 ≤ x − y 2 − P r C (x) − x 2 ∀y ∈ C, x ∈ H. (P roj 5 ) P r C (x)− P r C (y) 2 ≤ x −y 2 −P r C (x)− x+ y − P r C (y) 2 ∀x, y ∈ H. Now we are in a position to propose a new extragradient-type algorithm for (BV I). Algorithm 2.2. Initialization. Choose k = 0, x 0 ∈ H, 0 < λ ≤ 2β L 2 1 , positive sequences { k }, {β k }, {γ k }, {δ k }, {λ k }, {α k } and {¯ k } such that            {α k } ⊂ [m, n] for some m, n ∈ (0, 1), λ k ≤ 1 L 2 ∀k ≥ 0, lim k→∞ δ k = 0, ∞  k=0 ¯ k < ∞, 0 < lim inf k→∞ β k < lim sup k→∞ β k < 1,  k + β k + γ k = 1 ∀k ≥ 0, lim k→∞  k = 0, ∞  k=0  k = ∞. 98 PHAM NGOC ANH Step 1. If x k ∈ Sol(BV I), then stop. Otherwise compute y k = P r C  x k − λ k G(x k )  and z k = P r C  x k − λ k G(y k )  . Step 2. Inner iterations j = 0, 1, · · · . Compute      x k,0 = z k − λF (z k ), y k,j = P r C  x k,j − δ j G(x k,j )  , x k,j+1 =  j x k,0 + β j x k,j + γ j P r C  x k,j − δ j G(y k,j )  . Find h k such that h k − lim j→∞ x k,j  ≤ ¯ k and set x k+1 = α k x k +(1−α k )h k . Step 3. Increase k by 1 and go to Step 1. Remark 2.3. If x k+1 = α k x k +(1−α k )h k is substituted for x k+1 = ¯α k u+ ¯ β k x k + ¯γ k h k , where ¯α k , ¯ β k , ¯γ k ∈ [0, 1] for all k ≥ 0, u ∈ R n and ¯α k + ¯ β k + ¯γ k = 1, then Algorithm 2.2 becomes Algorithm 2.1 in R n proposed by Anh et al. in [3]. Using this fixed point technique allows us to extend the result from a finite-dimensional space R n to a real Hilbert space H. Remark 2.4. Suppose that α k = δ k = λ = 0. Then we can cho ose h k = z k and it is easy to see that the sequence {x k } in Algorithm 2.2 is the well-known extragradient iteration sequence which was firs t introduced by Korpelevich in [11]. 3. Convergence results Let C be a nonempty closed convex subset of H, G : H → H be monotone and L 2 -Lipschitz on C, and S : C → C be a nonexpansive mapping such that Sol(G, C) ∩ F ix(S) = ∅, where F ix(S) := {x ∈ C | S(x) = x} is the set of fixed points of S. Let the sequences {x k } and {y k } be generated by      x 0 ∈ H, y k = P r C  x k − δ k G(x k )  , x k+1 =  k x 0 + β k x k + γ k SP r C  x k − δ k G(y k )  ∀k ≥ 0, where { k }, {β k }, {γ k } and {δ k } satisfy the following conditions:                δ k > 0 ∀k ≥ 0, lim k→∞ δ k = 0,  k + β k + γ k = 1 ∀k ≥ 0, ∞  k=1  k = ∞, lim k→∞  k = 0, 0 < lim inf k→∞ β k < lim sup k→∞ β k < 1. Under these conditions, Yao et al. showed that the sequences {x k } and {y k } converge strongly to the same point P r Sol(G,C)∩F ix(S) (x 0 ) in [18]. Apply these iteration sequences with S being the identity mapping, we have the following lemma. A NEW EXTRAGRADIENT ITERATION ALGORITHM FOR BVI 99 Lemma 3.1. Suppose that the assumptions (A 1 ) − (A 4 ) hold. Then the sequence {x k,j } generated by Algorithm 2.2 converges strongly to the point Pr Sol(G,C)  z k − λF (z k )  as j → ∞. Consequently, we have h k − P r Sol(G,C)  z k − λF (z k )   ≤ ¯ k ∀k ≥ 0. Lemma 3.2. Let sequences {x k } and {z k } be generated by Algorithm 2.2, G be L 2 -Lipschitz and monotone on C, and x ∗ ∈ Sol(G, C). Then, we have (3.1) z k − x ∗  2 ≤ x k − x ∗  2 − (1 − λ k L 2 )x k − y k  2 − (1 − λ k L 2 )y k − z k  2 . Proof. Let x ∗ be a s olution to probems V I(G, C), x ∗ ∈ C and G(x ∗ ), x − x ∗  ≥ 0 ∀x ∈ C. Then, for each λ k > 0, x ∗ is a fixed point of mapping T (x) = P r C  x − λ k G(x)  on C (see [9 ]), i.e., x ∗ = P r C  x ∗ − λ k G(x ∗ )  . Substituting x by x k − λ k G(y k ) and y by x ∗ into (P roj 4 ), we get z k − x ∗  2 ≤x k − λ k G(y k ) − x ∗  2 − x k − λ k G(y k ) − z k  2 =x k − x ∗  2 − 2λ k G(y k ), x k − x ∗  + λ 2 k G(y k ) 2 − x k − z k  2 − λ 2 k G(y k  2 + 2λ k G(y k ), x k − z k  =x k − x ∗  2 − x k − z k  2 + 2λ k G(y k ), x ∗ − z k  =x k − x ∗  2 − x k − z k  2 + 2λ k G(y k ) − G(x ∗ ), x ∗ − y k  + 2λ k G(x ∗ ), x ∗ − y k  + 2λ k G(y k ), y k − z k  ≤x k − x ∗  2 − x k − z k  2 + 2λ k G(y k ), y k − z k .(3.2) The last inequality holds because y k ∈ C, x ∗ ∈ Sol(G, C) and G is monotone on C. Substituting x by x k − λ k G(x k ) and y by z k into (P roj 3 ), we have x k − λ k G(x k ) − y k , z k − y k  ≤ 0. 100 PHAM NGOC ANH Combining this with (3.2) and the Lipchitzian continuity of G on C with constant L 2 , we obtain z k − x ∗  2 ≤x k − x ∗  2 − (x k − y k ) + (y k − z k ) 2 + 2λ k G(y k ), y k − z k  =x k − x ∗  2 − x k − y k  2 − y k − z k  2 − 2x k − y k , y k − z k  + 2λ k G(y k ), y k − z k  =x k − x ∗  2 − x k − y k  2 − y k − z k  2 − 2x k − λ k G(y k ) − y k , y k − z k  =x k − x ∗  2 − x k − y k  2 − y k − z k  2 − 2x k − λ k G(x k ) − y k , y k − z k  + 2λ k G(x k ) − G(y k ), z k − y k  ≤x k − x ∗  2 − x k − y k  2 − y k − z k  2 + 2λ k G(x k ) − G(y k ), z k − y k  ≤x k − x ∗  2 − x k − y k  2 − y k − z k  2 + 2λ k G(x k ) − G(y k )z k − y k  ≤x k − x ∗  2 − x k − y k  2 − y k − z k  2 + 2λ k L 2 x k − y k z k − y k  ≤x k − x ∗  2 − x k − y k  2 − y k − z k  2 + λ k L 2  x k − y k  2 + z k − y k  2  ≤x k − x ∗  2 − (1 − λ k L 2 )x k − y k  2 − (1 − λ k L 2 )y k − z k  2 . This implies (3.1).  Lemma 3.3. Suppose that Assumptions (A 1 ) − (A 4 ) hold. Then, the sequence {x k } generated by Algorithm 2.2 is bounded. Proof. Suppose that x ∗ is a solution to problems (BV I), F (x ∗ ), x − x ∗  ≥ 0 ∀x ∈ Sol(G, C), we have x ∗ = P r Sol(G,C)  x ∗ − λF (x ∗ )  . Then, it follows from (P roj 1 ), β-strongly monotonicity and L 1 -Lipschitz conti- nuity of F , and 0 < λ ≤ 2β L 2 1 that P r Sol(G,C)  z k − λF (z k )  − x ∗  2 = P r Sol(G,C)  z k − λF (z k )  − P r Sol(G,C)  x ∗ − λF (x ∗ )   2 ≤ z k − λF (z k ) − x ∗ + λF (x ∗ ) 2 ≤ z k − x ∗  2 − 2λF (z k ) − F (x ∗ ), z k − x ∗  + λ 2 F (z k ) − F (x ∗ ) 2 ≤ (1 − 2βλ + λ 2 L 2 1 )z k − x ∗  2 ≤ z k − x ∗  2 .(3.3) A NEW EXTRAGRADIENT ITERATION ALGORITHM FOR BVI 101 It follows from λ k ≤ 1 L 2 and (3.1) that z k − x ∗  ≤ x k − x ∗ . Combining this with (3.3) and Assumptions 0 < λ ≤ 2β L 1 , ∞  k=0 ¯ k < +∞, we have x k+1 − x ∗  = α k x k + (1 − α k )h k − x ∗  ≤α k x k − x ∗  + (1 − α k )h k − x ∗  ≤α k x k − x ∗  + (1 − α k )h k − P r Sol(G,C)  z k − λF (z k )   + (1 − α k )P r Sol(G,C)  z k − λF (z k )  − x ∗  ≤α k x k − x ∗  + (1 − α k )¯ k + (1 − α k )z k − x ∗  ≤α k x k − x ∗  + (1 − α k )¯ k + (1 − α k )x k − x ∗  =x k − x ∗  + (1 − α k )¯ k (3.4) <x k − x ∗  + ¯ k ≤x 0 − x ∗  + +∞  k=0 ¯ k < + ∞. Therefore, the sequence {x k } is bounded. ✷ Lemma 3.4. (see [9]) Let {a k } and {b k } be two positive real sequences such that a k+1 ≤ a k + b k ∀k ≥ 0 and ∞  k=0 b k < +∞. Then there exists lim k→∞ a k = c. Lemma 3.5. Suppose that Assumptions (A 1 ) − (A 4 ) hold and the sequences {x k } and {z k } are generated by Algorithm 2.2. Then, we have x k+1 − x ∗  2 ≤x k − x ∗  2 + 2(1 − α k )¯ k z k − x ∗  + (1 − α k )¯ 2 k − (1 − α k )(1 − λ k L 2 )x k − y k  2 − (1 − α k )(1 − λ k L 2 )y k − z k  2 .(3.5) Consequently, we have lim k→∞ x k − y k  = lim k→∞ y k − z k  = lim k→∞ x k − z k  = 0. Proof. For each k ≥ 0, Lemma 3.1 shows th at there exists lim j→∞ x k,j = P r Sol(G,C)  z k − λF (z k )  . 102 PHAM NGOC ANH Combining this with 0 < λ ≤ 2β L 2 1 , (3.1), Lemma 3.1 and x ∗ ∈ Sol(BV I), for k ≥ 0 we have x k+1 − x ∗  2 =α k x k + (1 − α k )h k − x ∗  2 ≤α k x k − x ∗  2 + (1 − α k )h k − x ∗  2 ≤α k x k − x ∗  2 + (1 − α k )  P r Sol(G,C)  z k − λF (z k )  − x ∗  + ¯ k  2 =α k x k − x ∗  2 + (1 − α k ) × {P r Sol(G,C)  z k − λF (z k )  − P r Sol(G,C)  x ∗ − λF (x ∗ )   + ¯ k } 2 ≤α k x k − x ∗  2 + (1 − α k )   1 − 2ηλ + λ 2 L 2 1 z k − x ∗  + ¯ k  2 ≤α k x k − x ∗  2 + (1 − α k )  z k − x ∗  + ¯ k  2 =α k x k − x ∗  2 + (1 − α k )z k − x ∗  2 + 2(1 − α k )¯ k z k − x ∗  + (1 − α k )¯ 2 k ≤α k x k − x ∗  2 + 2(1 − α k )¯ k z k − x ∗  + (1 − α k )¯ 2 k + (1 − α k ) ×  x k − x ∗  2 − (1 − λ k L 2 )x k − y k  2 − (1 − λ k L 2 )y k − z k  2  =x k − x ∗  2 + 2(1 − α k )¯ k z k − x ∗  + (1 − α k )¯ 2 k − (1 − α k )(1 − λ k L 2 )x k − y k  2 − (1 − α k )(1 − λ k L 2 )y k − z k  2 . This implies (3.5). It follows from (3.4) that x k+1 − x ∗  ≤ x k − x ∗  + ¯ k . Combining this, ∞  k=0 ¯ k < +∞ and Lemma 3.4, there exists (3.6) lim k→∞ x k − x ∗  = c. Hence by (3.5), we have x k − y k  → 0 as k → ∞. Since λ k ≤ 1 L 2 , (3.5), (3.6) and {α k } ⊂ [m, n] for some m, n ∈ (0, 1), we obtain (1 − α k )(1 − λ k L 2 )z k − y k  2 ≤x k − x ∗  2 + 2(1 − α k )¯ k z k − x ∗  + (1 − α k )¯ 2 k − x k+1 − x ∗  2 , and hence z k − y k  → 0 as k → ∞. Con s equently, x k − z k  ≤ x k − y k  + y k − z k  ⇒ lim k→∞ x k − z k  = 0. ✷ Lemma 3.6. (see [15]) Let H be a real Hilbert space, {α k } be a sequence of real numbers such that 0 < a ≤ α k ≤ b < 1 for all k ≥ 0, and two sequences A NEW EXTRAGRADIENT ITERATION ALGORITHM FOR BVI 103 {x k }, {y k } in H such that          lim sup k→∞ x k  ≤ c, lim sup k→∞ y k  ≤ c, lim k→∞ α k x k + (1 − α k )y k  = c. Then, lim k→∞ x k − y k  = 0. Lemma 3.7. (see [17]) Let H be a real Hilbert space and C be a nonempty closed convex subset of H. Let {x k } be a sequence in H. Suppose that, for all x ∗ ∈ C, x k+1 − x ∗  ≤ x k − x ∗  ∀k ≥ 0. Then, the sequence {P r C (x k )} converges strongly to some ¯x ∈ C. Theorem 3.8. Suppose that Assumptions (A 1 ) − (A 4 ) hold. Then three se- quences {x k }, {y k } and {z k } generated by Algorithm 2.2 converge strongly to a solution x ∗ of problems (BV I). Moreover, we have x ∗ = lim k→∞ P r Sol(G,C) (x k ). Proof. It follows from (3.1), (3.3) and (3.6) that lim sup k→∞ h k − x ∗  ≤ lim sup k→∞ {h k − P r Sol(G,C)  z k − λF (z k )   + P r Sol(G,C)  z k − λF (z k )  − x ∗ } ≤ lim sup k→∞ {¯ k + z k − x ∗ } ≤ lim sup k→∞ {¯ k + x k − x ∗ } = c.(3.7) Using x k+1 = α k x k + (1 − α k )h k and {α k } ⊂ [m, n] ⊂ (0, 1), we have (3.8) lim k→∞ α k (x k − x ∗ ) + (1 − α k )(h k − x ∗ ) = lim k→∞ x k+1 − x ∗  = c. Combining L emm a 3.6, (3.7) and (3.8), we have lim k→∞ h k − x k  = 0. Consequently, we get (3.9) lim k→∞ x k+1 − x k  = lim k→∞ (1 − α k )h k − x k  = 0. 104 PHAM NGOC ANH From (P roj 1 ), it follows that P r Sol(G,C)  y k − λF (y k )  −x k+1  ≤P r Sol(G,C)  y k − λF (y k )  − P r Sol(G,C)  z k − λF (z k )   + P r Sol(G,C)  z k − λF (z k )  − h k  + h k − x k+1  ≤(1 + λL 1 )y k − z k  + ¯ k + h k − x k+1  =(1 + λL 1 )y k − z k  + ¯ k + α k 1 − α k x k − x k+1 . Then, we have P r Sol(G,C)  x k − λF (x k )  − x k  ≤P r Sol(G,C)  x k − λF (x k )  − P r Sol(G,C)  z k − λF (z k )   + x k+1 − x k  + P r Sol(G,C)  y k − λF (y k )  − x k+1  + P r Sol(G,C)  y k − λF (y k )  − P r Sol(G,C)  z k − λF (z k )   ≤(1 + λL 1 )x k − z k  + (1 + λL 1 )y k − z k  + x k+1 − x k  + P r Sol(G,C)  y k − λF (y k )  − x k+1  ≤(1 + λL 1 )x k − z k  + (1 + λL 1 )y k − z k  + x k+1 − x k  (1 + λL 1 )y k − z k  + ¯ k + α k 1 − α k x k − x k+1  ≤(1 + λL 1 )x k − z k  + 2(1 + λL 1 )y k − z k  + ¯ k + 1 1 − α k x k − x k+1 .(3.10) It follows from (3.9), (3.10) and Lemma 3.5 that (3.11) lim k→∞ P r Sol(G,C)  x k − λF (x k )  − x k  = 0. Lemma 3.3 shows that the sequence {x k } is bounded . Then, there exists M > 0 such th at (3.12) P r Sol(G,C) (x k − λF (x k )) − x ∗  ≤ M ∀k ≥ 0. Since (P roj 1 ), F is β-strongly monotone and L 1 -Lipschitz continuous, we have P r Sol(G,C) (x k − λF (x k )) − x ∗  =P r Sol(G,C) (x k − λF (x k )) − P r Sol(G,C) (x ∗ − λF (x ∗ )) 2 ≤x k − λF (x k ) − (x ∗ − λF (x ∗ )) 2 =x k − x ∗  2 − 2λF (x k ) − F (x ∗ ), x k − x ∗  + λ 2 F (x k ) − F (x ∗ ) 2 ≤x k − x ∗  2 − 2λβx k − x ∗  2 + λ 2 L 2 1 x k − x ∗  2 . 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There¯ fore, we have lim x∗ − tk , tk − xk ≥ 0 ⇒ x∗ − x, x − x∗ ≥ 0, ¯ ¯ ¯ k→∞ and x∗ ≡ x Thus the sequences {xk }, {y k } and {z k } converge strongly to x∗ , ¯ where x∗ = lim P rSol(G,C) (xk ) k→∞ As a direct consequence of Theorem 3.8, we obtain the following corollary Corollary 3.9 Let C be a nonempty closed convex subset of H, G : H → H be monotone and L-Lipschitz continuous Let {xk } and {y k . ACTA MATHEMATICA VIETNAMICA 95 Volume 37, Number 1, 2012, pp. 95–107 A NEW EXTRAGRADIENT ITERATION ALGORITHM FOR BILEVEL VARIATIONAL INEQUALITIES PHAM NGOC ANH Abstract. In this paper,. recent years, variational inequalities become an attractive field for many researchers and have many important applications in electricity markets, tr ans- portations, economics, nonlinear analysis. we introduce an app roximation extragradient itera- tion method for solving bilevel variational inequalities involving two variational inequalities and we show that these problems can be solved