A new extragradient iteration algorithm for bilevel variational inequalities (tt)

ACTA MATHEMATICA VIETNAMICA Volume 37, Number 1, 2012, pp 95–107 95 A NEW EXTRAGRADIENT ITERATION ALGORITHM FOR BILEVEL VARIATIONAL INEQUALITIES PHAM NGOC ANH Abstract In this paper, we introduce an approximation extragradient iteration method for solving bilevel variational inequalities involving two variational inequalities and we show that these problems can be solved by projection sequences and fixed point techniques We obtain a strong convergence of three iteration sequences generated by this method in a real Hilbert space Introduction Let H be a real Hilbert space with an inner product ·, · and the induced 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∗ ≥ ∀x ∈ Sol(G, C), where G : H → H, Sol(G, C) denotes the set of solutions of the following variational inequalities: Find y ∗ ∈ C such that G(y ∗ ), y − y ∗ ≥ ∀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 minimization problems (see [16]), variational inequalities and complementarity problems (see [1, 2, 5, 7, 13]) If F ≡ 0, then the bilevel variational inequalities (BV I) become the following variational inequalities shortly V I(G, C) : Find x∗ ∈ C such that G(x∗ ), x − x∗ ≥ ∀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 Development (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 constraints: f (x) x ∈ {y ∗ | G(y ∗ ), z − y ∗ ≥ ∀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 bilevel minimization problem (see [16]): 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, transportations, economics, nonlinear analysis (see [6, 9, 19]) Methods for solving variational 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 {xk } is defined by   x ∈ C, y k = P rC xk − ck G(xk ) ,   k+1 x = P rC xk − ck G(y k ) , and extended by many other authors (see [5, 9, 14, 18]) One of the main conditions 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 xk from the solution set Then the new iterate xk+1 is the projection of xk 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 multivalued variational inequalities V I(F, C), where F is a generalized monotone multifunction Next the authors constructed an appropriate hyperplane which separates the current 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 problem VI(G, C) is not explicitly given, the existing algorithms for variational inequalities can not be directly applied because the subproblems can not be implemented 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 space We are interested in finding a solution to bilevel variational inequalities (BV I) where the functions F and G satisfy the following usual conditions: A NEW EXTRAGRADIENT ITERATION ALGORITHM FOR BVI 97 (A1 ) G is monotone on C and F is β-strongly monotone on C, (A2 ) F is L1 -Lipschitz continuous on C, (A3 ) G is L2 -Lipschitz continuous on C, (A4 ) 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) Preliminaries We list some known definitions and properties of the projection 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 rC (·) with images P rC (x) = {y ∈ C | y − x = v − x } ∀x ∈ H v∈C 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 , (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 > (shortly L-Lipschitz) if for any x, y ∈ C, we have ϕ(x) − ϕ(y) ≤ L x − y If ϕ : C → C and L = then ϕ is called nonexpansive on C The projection P rC (·) has the following basic properties: (P roj1 ) P rC (x) − P rC (y) ≤ x − y ∀x, y ∈ H (P roj2 ) P rC (x) − P rC (y) ≤ P rC (x) − P rC (y), x − y ∀x, y ∈ H (P roj3 ) x − P rC (x), y − P rC (y) ≤ ∀y ∈ C, x ∈ H (P roj4 ) P rC (x) − y ≤ x − y − P rC (x) − x ∀y ∈ C, x ∈ H (P roj5 ) P rC (x) − P rC (y) ≤ x − y − P rC (x) − x + y − P rC (y) ∀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, x0 ∈ H, < λ ≤ sequences { k }, {βk }, {γk }, {δk }, {λk }, {αk } and {¯k } such that  {αk } ⊂ [m, n] for some m, n ∈ (0, 1), λk ≤ L12 ∀k ≥ 0,    ∞   ¯k < ∞, < lim inf βk < lim sup βk < 1, lim δk = 0, k→∞ k→∞ k→∞ k=0  ∞     k + βk + γk = ∀k ≥ 0, lim k = 0, k = ∞ k→∞ k=0 2β , L21 positive 98 PHAM NGOC ANH Step If xk ∈ Sol(BV I), then stop Otherwise compute y k = P rC xk − λk G(xk ) and z k = P rC xk − λk G(y k ) Step Inner iterations j = 0, 1, · · · Compute  k,0 k k  x = z − λF (z ), y k,j = P rC xk,j − δj G(xk,j ) ,   k,j+1 x = j xk,0 + βj xk,j + γj P rC xk,j − δj G(y k,j ) Find hk such that hk − lim xk,j ≤ ¯k and set xk+1 = αk xk +(1−αk )hk j→∞ Step Increase k by and go to Step Remark 2.3 If xk+1 = αk xk +(1−αk )hk is substituted for xk+1 = α ¯ k u+ β¯k xk + k n γ¯k h , where α ¯ k , β¯k , γ¯k ∈ [0, 1] for all k ≥ 0, u ∈ R and α ¯ k + β¯k + γ¯k = 1, then n Algorithm 2.2 becomes Algorithm 2.1 in R proposed by Anh et al in [3] Using this fixed point technique allows us to extend the result from a finite-dimensional space Rn to a real Hilbert space H Remark 2.4 Suppose that αk = δk = λ = Then we can choose hk = z k and it is easy to see that the sequence {xk } in Algorithm 2.2 is the well-known extragradient iteration sequence which was first introduced by Korpelevich in [11] Convergence results Let C be a nonempty closed convex subset of H, G : H → H be monotone and L2 -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 {xk } and {y k } be generated by   x ∈ H, y k = P rC xk − δk G(xk ) ,   k+1 x = k x0 + βk xk + γk SP rC xk − δk G(y k ) ∀k ≥ 0, where { k }, {βk }, {γk } and {δk } satisfy the following conditions:  δk > ∀k ≥ 0, lim δk = 0,   k→∞     + β + γ = ∀k ≥ 0,  k k k ∞  k = ∞, lim k = 0,  k→∞  k=1    0 < lim inf βk < lim sup βk < k→∞ k→∞ Under these conditions, Yao et al showed that the sequences {xk } and {y k } converge strongly to the same point P rSol(G,C)∩F ix(S) (x0 ) 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 (A1 ) − (A4 ) hold Then the sequence {xk,j } generated by Algorithm 2.2 converges strongly to the point P rSol(G,C) z k − λF (z k ) as j → ∞ Consequently, we have hk − P rSol(G,C) z k − λF (z k ) ≤ ¯k ∀k ≥ Lemma 3.2 Let sequences {xk } and {z k } be generated by Algorithm 2.2, G be L2 -Lipschitz and monotone on C, and x∗ ∈ Sol(G, C) Then, we have (3.1) z k − x∗ ≤ xk − x∗ − (1 − λk L2 ) xk − y k − (1 − λk L2 ) y k − z k Proof Let x∗ be a solution to probems V I(G, C), x∗ ∈ C and G(x∗ ), x − x∗ ≥ ∀x ∈ C Then, for each λk > 0, x∗ is a fixed point of mapping T (x) = P rC x − λk G(x) on C (see [9]), i.e., x∗ = P rC x∗ − λk G(x∗ ) Substituting x by xk − λk G(y k ) and y by x∗ into (P roj4 ), we get z k − x∗ ≤ xk − λk G(y k ) − x∗ = xk − x∗ − λ2k G(y k − xk − λk G(y k ) − z k − 2λk G(y k ), xk − x∗ + λ2k G(y k ) 2 − xk − z k + 2λk G(y k ), xk − z k = xk − x∗ − xk − z k + 2λk G(y k ), x∗ − z k = xk − x∗ − xk − z k + 2λk G(y k ) − G(x∗ ), x∗ − y k + 2λk G(x∗ ), x∗ − y k + 2λk G(y k ), y k − z k (3.2) ≤ xk − x∗ − xk − z k + 2λk G(y k ), y k − z k The last inequality holds because y k ∈ C, x∗ ∈ Sol(G, C) and G is monotone on C Substituting x by xk − λk G(xk ) and y by z k into (P roj3 ), we have xk − λk G(xk ) − y k , z k − y k ≤ 100 PHAM NGOC ANH Combining this with (3.2) and the Lipchitzian continuity of G on C with constant L2 , we obtain z k − x∗ ≤ xk − x∗ − (xk − y k ) + (y k − z k ) + 2λk G(y k ), y k − z k = xk − x∗ − xk − y k − yk − zk − xk − y k , y k − z k + 2λk G(y k ), y k − z k = xk − x∗ − xk − y k − yk − zk − xk − λk G(y k ) − y k , y k − z k = xk − x∗ − xk − y k − yk − zk − xk − λk G(xk ) − y k , y k − z k + 2λk G(xk ) − G(y k ), z k − y k ≤ xk − x∗ − xk − y k − yk − zk + 2λk G(xk ) − G(y k ), z k − y k ≤ xk − x∗ − xk − y k − yk − zk + 2λk G(xk ) − G(y k ) z k − y k ≤ xk − x∗ − xk − y k − yk − zk + 2λk L2 xk − y k ≤ xk − x∗ − xk − y k − yk − zk + λk L2 xk − y k ≤ xk − x∗ − (1 − λk L2 ) xk − y k zk − yk − (1 − λk L2 ) y k − z k + zk − yk This implies (3.1) Lemma 3.3 Suppose that Assumptions (A1 ) − (A4 ) hold Then, the sequence {xk } generated by Algorithm 2.2 is bounded Proof Suppose that x∗ is a solution to problems (BV I), F (x∗ ), x − x∗ ≥ ∀x ∈ Sol(G, C), we have x∗ = P rSol(G,C) x∗ − λF (x∗ ) Then, it follows from (P roj1 ), β-strongly monotonicity and L1 -Lipschitz conti2β nuity of F , and < λ ≤ L that P rSol(G,C) z k − λF (z k ) − x∗ = P rSol(G,C) z k − λF (z k ) − P rSol(G,C) x∗ − λF (x∗ ) ≤ z k − λF (z k ) − x∗ + λF (x∗ ) ≤ z k − x∗ 2 − 2λ F (z k ) − F (x∗ ), z k − x∗ + λ2 F (z k ) − F (x∗ ) ≤ (1 − 2βλ + λ2 L21 ) z k − x∗ (3.3) ≤ z k − x∗ 2 A NEW EXTRAGRADIENT ITERATION ALGORITHM FOR BVI It follows from λk ≤ L2 101 and (3.1) that z k − x∗ ≤ xk − x∗ Combining this with (3.3) and Assumptions < λ ≤ 2β L1 , ∞ ¯k < +∞, we have k=0 xk+1 − x∗ = αk xk + (1 − αk )hk − x∗ ≤αk xk − x∗ + (1 − αk ) hk − x∗ ≤αk xk − x∗ + (1 − αk ) hk − P rSol(G,C) z k − λF (z k ) + (1 − αk ) P rSol(G,C) z k − λF (z k ) − x∗ ≤αk xk − x∗ + (1 − αk )¯k + (1 − αk ) z k − x∗ ≤αk xk − x∗ + (1 − αk )¯k + (1 − αk ) xk − x∗ = xk − x∗ + (1 − αk )¯k (3.4) < xk − x∗ + ¯k +∞ ∗ ≤ x −x + ¯k k=0 < + ∞ Therefore, the sequence {xk } is bounded ✷ Lemma 3.4 (see [9]) Let {ak } and {bk } be two positive real sequences such that ∞ ak+1 ≤ ak + bk ∀k ≥ and bk < +∞ k=0 Then there exists lim ak = c k→∞ Lemma 3.5 Suppose that Assumptions (A1 ) − (A4 ) hold and the sequences {xk } and {z k } are generated by Algorithm 2.2 Then, we have xk+1 − x∗ (3.5) ≤ xk − x∗ + 2(1 − αk )¯k z k − x∗ + (1 − αk )¯2k − (1 − αk )(1 − λk L2 ) xk − y k − (1 − αk )(1 − λk L2 ) y k − z k Consequently, we have lim xk − y k = lim y k − z k = lim xk − z k = k→∞ k→∞ k→∞ Proof For each k ≥ 0, Lemma 3.1 shows that there exists lim xk,j = P rSol(G,C) z k − λF (z k ) j→∞ 102 PHAM NGOC ANH Combining this with < λ ≤ k ≥ we have 2β , L21 xk+1 − x∗ (3.1), Lemma 3.1 and x∗ ∈ Sol(BV I), for = αk xk + (1 − αk )hk − x∗ ≤αk xk − x∗ + (1 − αk ) hk − x∗ ≤αk xk − x∗ + (1 − αk ) =αk xk − x∗ + (1 − αk ) 2 P rSol(G,C) z k − λF (z k ) − x∗ + ¯k × { P rSol(G,C) z k − λF (z k ) − P rSol(G,C) x∗ − λF (x∗ ) ≤αk xk − x∗ + (1 − αk ) ≤αk xk − x∗ + (1 − αk ) =αk xk − x∗ + (1 − αk ) z k − x∗ − 2ηλ + λ2 L21 z k − x∗ + ¯k z k − x∗ + ¯k + ¯k }2 2 + 2(1 − αk )¯k z k − x∗ + (1 − αk )¯2k ≤αk xk − x∗ × xk − x∗ = xk − x∗ + 2(1 − αk )¯k z k − x∗ + (1 − αk )¯2k + (1 − αk ) − (1 − λk L2 ) xk − y k − (1 − λk L2 ) y k − z k + 2(1 − αk )¯k z k − x∗ + (1 − αk )¯2k − (1 − αk )(1 − λk L2 ) xk − y k − (1 − αk )(1 − λk L2 ) y k − z k This implies (3.5) It follows from (3.4) that xk+1 − x∗ ≤ xk − x∗ + ¯k ∞ ¯k < +∞ and Lemma 3.4, there exists Combining this, k=0 lim xk − x∗ = c (3.6) k→∞ Hence by (3.5), we have xk − y k → as k → ∞ Since λk ≤ and {αk } ⊂ [m, n] for some m, n ∈ (0, 1), we obtain (1 − αk )(1 − λk L2 ) z k − y k ≤ xk − x∗ L2 , (3.5), (3.6) + 2(1 − αk )¯k z k − x∗ + (1 − αk )¯2k − xk+1 − x∗ , and hence z k − y k → as k → ∞ Consequently, xk − z k ≤ xk − y k + y k − z k ⇒ lim xk − z k = k→∞ ✷ Lemma 3.6 (see [15]) Let H be a real Hilbert space, {αk } be a sequence of real numbers such that < a ≤ αk ≤ b < for all k ≥ 0, and two sequences A NEW EXTRAGRADIENT ITERATION ALGORITHM FOR BVI 103 {xk }, {y k } in H such that  lim sup xk ≤ c,     k→∞ lim sup y k ≤ c,  k→∞    lim αk xk + (1 − αk )y k = c k→∞ Then, lim xk − y k = k→∞ Lemma 3.7 (see [17]) Let H be a real Hilbert space and C be a nonempty closed convex subset of H Let {xk } be a sequence in H Suppose that, for all x∗ ∈ C, xk+1 − x∗ ≤ xk − x∗ ∀k ≥ Then, the sequence {P rC (xk )} converges strongly to some x ¯ ∈ C Theorem 3.8 Suppose that Assumptions (A1 ) − (A4 ) hold Then three sequences {xk }, {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 P rSol(G,C) (xk ) k→∞ Proof It follows from (3.1), (3.3) and (3.6) that lim sup hk − x∗ ≤ lim sup{ hk − P rSol(G,C) z k − λF (z k ) k→∞ k→∞ + P rSol(G,C) z k − λF (z k ) − x∗ } ≤ lim sup{¯k + z k − x∗ } k→∞ ≤ lim sup{¯k + xk − x∗ } k→∞ (3.7) = c Using xk+1 = αk xk + (1 − αk )hk and {αk } ⊂ [m, n] ⊂ (0, 1), we have (3.8) lim αk (xk − x∗ ) + (1 − αk )(hk − x∗ ) = lim xk+1 − x∗ = c k→∞ k→∞ Combining Lemma 3.6, (3.7) and (3.8), we have lim hk − xk = k→∞ Consequently, we get (3.9) lim xk+1 − xk = lim (1 − αk ) hk − xk = k→∞ k→∞ 104 PHAM NGOC ANH From (P roj1 ), it follows that P rSol(G,C) y k − λF (y k ) −xk+1 ≤ P rSol(G,C) y k − λF (y k ) − P rSol(G,C) z k − λF (z k ) + P rSol(G,C) z k − λF (z k ) − hk + hk − xk+1 ≤(1 + λL1 ) y k − z k + ¯k + hk − xk+1 αk xk − xk+1 =(1 + λL1 ) y k − z k + ¯k + − αk Then, we have P r Sol(G,C) xk − λF (xk ) − xk ≤ P rSol(G,C) xk − λF (xk ) − P rSol(G,C) z k − λF (z k ) + xk+1 − xk + P rSol(G,C) y k − λF (y k ) − xk+1 + P rSol(G,C) y k − λF (y k ) − P rSol(G,C) z k − λF (z k ) ≤(1 + λL1 ) xk − z k + (1 + λL1 ) y k − z k + xk+1 − xk + P rSol(G,C) y k − λF (y k ) − xk+1 ≤(1 + λL1 ) xk − z k + (1 + λL1 ) y k − z k + xk+1 − xk αk (1 + λL1 ) y k − z k + ¯k + xk − xk+1 − αk (3.10) ≤(1 + λL1 ) xk − z k + 2(1 + λL1 ) y k − z k + ¯k + xk − xk+1 − αk It follows from (3.9), (3.10) and Lemma 3.5 that (3.11) lim P rSol(G,C) xk − λF (xk ) − xk = k→∞ Lemma 3.3 shows that the sequence {xk } is bounded Then, there exists M > such that (3.12) P rSol(G,C) (xk − λF (xk )) − x∗ ≤ M ∀k ≥ Since (P roj1 ), F is β-strongly monotone and L1 -Lipschitz continuous, we have P rSol(G,C) (xk − λF (xk )) − x∗ = P rSol(G,C) (xk − λF (xk )) − P rSol(G,C) (x∗ − λF (x∗ )) ≤ xk − λF (xk ) − (x∗ − λF (x∗ )) = xk − x∗ − 2λ F (xk ) − F (x∗ ), xk − x∗ + λ2 F (xk ) − F (x∗ ) ≤ xk − x∗ 2 − 2λβ xk − x∗ + λ2 L21 xk − x∗ A NEW EXTRAGRADIENT ITERATION ALGORITHM FOR BVI 105 Combining this and (3.12), we have xk − x∗ = xk − P rSol(G,C) (xk − λF (xk )) + x∗ − P rSol(G,C) (xk − λF (xk )) + xk − P rSol(G,C) (xk − λF (xk )), P rSol(G,C) (xk − λF (xk )) − x∗ (3.13) ≤ xk − P rSol(G,C) (xk − λF (xk )) + 2M xk − P rSol(G,C) (xk − λF (xk )) + xk − x∗ Using this, (3.11) and λ < − 2λβ xk − x∗ 2β , L21 λ(2β − λL21 ) xk − x∗ + λ2 L21 xk − x∗ we get ≤ xk − P rSol(G,C) (xk − λF (xk )) + 2M xk − P rSol(G,C) (xk − λF (xk )) →0 as k → ∞ Thus, the sequence {xk } converges strongly to x∗ ∈ Sol(BV I) Then Lemma 3.5 implies that the sequences {xk }, {y k } and {z k } must converge strongly to the unique solution x∗ of problems (BV I) Now, we set tk = P rSol(G,C) (xk ) and xk → x∗ as k → ∞ Then, it follows from (P roj3 ) and x∗ ∈ C that x∗ − tk , tk − xk ≥ By Lemma 3.7 and (3.1), {tn } converges strongly to some x ¯ ∈ Sol(G, C) Therefore, we have lim x ¯∗ − tk , tk − xk ≥ ⇒ 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 } be the sequences generated by   x ∈ H, k y = P rC xk − λk G(xk ) ,   k+1 x = αk xk + (1 − αk )SP rC xk − λk G(y k ) ∀k ≥ 0, 106 PHAM NGOC ANH where {αk } and {δk } satisfy the following conditions:  0 < λk ≤ L1 ∀k ≥ 0, ∞  αk = ∞, lim αk = k=1 k→∞ Then {xk } and {y k } converge strongly to the same x ¯ ∈ Sol(G, C) Acknowledgments The author would like to thank the referees for their useful comments, remarks and suggestions References [1] P N Anh, An interior-quadratic proximal method for solving monotone generalized variational inequalities, East-West J Math 10 (2008), 81-100 [2] P N Anh, An interior proximal method for solving pseudomonotone nonlipschitzian multivalued variational inequalities, Nonlinear Anal Forum 14 (2009), 27-42 [3] P N Anh, J K Kim and L D Muu, An extragradient algorithm for solving bilevel variational inequalities, J Global Optim 52 (2012), 627-639 [4] P N Anh and T Kuno, A cutting hyperplane method for generalized monotone nonlipschitzian multivalued variational inequalities, in: Modeling, Simulation and Optimization of Complex Processes, Eds: H G Bock, H X Phu, R Rannacher, and J P Schloder, Springer, 2012 [5] P N Anh, L D Muu and J J Strodiot, Generalized projection method for non-Lipschitz multivalued monotone variational inequalities, Acta Math Vietnam 34 (2009), 67-79 [6] P N Anh, L D Muu, V H Nguyen and J J Strodiot, Using the Banach contraction principle to implement the proximal point method for multivalued monotone variational inequalities, J Optim Theory Appl 124 (2005), 285-306 [7] T Q Bao and P Q Khanh, A projection-type algorithm for pseudomonotone nonlipschitzian multivalued variational inequalities, Generalized convexity, generalized monotonicity and applications.Nonconvex Optim Appl 77 Springer, New York, 2005, 113-129 [8] T Q Bao and P Q Khanh, Some algorithms for solving mixed variational inequalities, Acta Math Vietnam 31 (2006), 83-103 [9] F Facchinei and J S Pang, Finite-dimensional variational inequalities and complementarity problems, Springer-Verlag, NewYork, 2003 [10] F Giannessi, A Maugeri and P M Pardalos, Equilibrium problems: Nonsmooth optimization and variational inequality models, Kluwer, 2004 [11] G M Korpelevich, Extragradient method for finding saddle points and other problems, Ekonomika i Matematicheskie Metody 12 (1976), 747-756 [12] Z Q Luo, J S Pang and D Ralph, Mathematical programs with equilibrum constraints, Cambridge University Press, Cambridge, 1996 [13] A Moudafi, Proximal methods for a class of bilevel monotone equilibrum programs, J Global Optim 47 (2010), 287-292 [14] N Nadezhkina and W Takahashi, Weak convergence theorem by an extragradient method for nonexpansive mappings and monotone mappings, J Optim Theory Appl 128 (2006), 191-201 [15] J Schu, Weak and strong convergence to fixed points of asymptotically nonexpansive mappings, Bull Austral Math Soc 149 (1991), 153-159 [16] M Solodov, An explicit descent method for bilevel convex optimization, J Convex Anal 14 (2007), 227-237 A NEW EXTRAGRADIENT ITERATION ALGORITHM FOR BVI 107 [17] W Takahashi and M Toyoda, Weak convergence theorems for nonexpansive mappings and monotone mappings, J Theory Appl 118 (2003), 417-428 [18] Y Yao, Y C Liou, and J C Yao, An extragradient method for fixed point probelms and variational inequality programs, J Inequal Appl., (2007), Article ID 38752, 12 pages, doi:10.1155/2007/38752 [19] L C Zeng and J C Yao, Strong convergence theorem by an extragradient method for fixed point problems and variational inequality problems, Taiwanese J Math 10 (2006), 1293-1303 Department of Scientific Fundamentals Posts and Telecommunications Institute of Technology, Hanoi, Vietnam E-mail address: anhpn@ptit.edu.vn ... optimization, J Convex Anal 14 (2007), 227-237 A NEW EXTRAGRADIENT ITERATION ALGORITHM FOR BVI 107 [17] W Takahashi and M Toyoda, Weak convergence theorems for nonexpansive mappings and monotone mappings,... Finite-dimensional variational inequalities and complementarity problems, Springer-Verlag, NewYork, 2003 [10] F Giannessi, A Maugeri and P M Pardalos, Equilibrium problems: Nonsmooth optimization and variational. .. P N Anh, J K Kim and L D Muu, An extragradient algorithm for solving bilevel variational inequalities, J Global Optim 52 (2012), 627-639 [4] P N Anh and T Kuno, A cutting hyperplane method for