Appl. Math. Inf. Sci. 9, No. 3, 1259-1264 (2015) 1259 Applied Mathematics & Information Sciences An International Journal http://dx.doi.org/10.12785/amis/090319 A Hybrid LQP Alternating Direction Method for Solving Variational Inequality Problems with Separable Structure Abdellah Bnouhachem 1,2,∗ and Abdelouahed Hamdi 3 1 School of Management Science and Engineering, Nanjing University, Nanjing, 210093, P.R. China. 2 Ibn Zohr University, ENSA, BP 1136, Agadir, Morocco. 3 Department of Mathematics, Statistics and Physics College of Arts and Sciences Qatar University,PB 2713, Doha, Qatar Received: 20 Jul. 2014, Revised: 21 Oct. 2014, Accepted: 22 Oct. 2014 Published online: 1 May 2015 Abstract: In this paper, we presented a logarithmic-quadratic proximal alternating direction method for structured variational inequalities. The method generates the new iterate by searching the optimal step size along the descent direction. Global convergence of the new method is proved under certain assumptions. Keywords: Variational inequalities, monotone operator, logarithmic-quadratic proximal method, projection method, alternating direction method. 2010 AMS Subject Classification: 49J40, 65N30 1 Introduction The problem concerned in this paper is the following variational inequalities, find u ∈ Ω such that: (u ′ − u) T F (u) ≥ 0, ∀ u ′ ∈ Ω , (1) with u =  x y  , F (u)=  f (x) g(y)  , (2) and Ω = {(x, y)|x ∈ R n ++ , y ∈ R m ++ , Ax + By = b} (3) where A ∈ R l ×n , B ∈ R l ×m are given matrices, b ∈ R l is a given vector, and f : R n ++ → R n , g : R m ++ → R m are given monotone operators. Studies and applications of such problems can be found in [7, 9, 10, 11, 12, 13, 14]. By attaching a Lagrange multiplier vector λ ∈ R l to the linear constraints Ax + By = b, the problem (1)-(3) can be explained as find w ∈ W such that: (w ′ − w) T Q(w) ≥ 0, ∀w ′ ∈ W , (4) where w =   x y λ   Q(w)=   f (x) − A T λ g(y) − B T λ Ax + By − b   , (5) W = R n ++ × R m ++ × R l . (6) Problem (4)-(6) is referred to as SVI (structured variational inequalities). The alternating direction method (ADM) is a powerful method for solving the structured problem (4)-(6), since it decomposes the original problems into a series subproblems with lower scale, which was originally proposed by Gabay and Mercier [11] and Gabay [10]. The classical proximal alternating direction method (PADM) [6, 8, 15] is an effective numerical approach for solving variational inequalities with separable structure. To make the PADM more efficient and practical, He et al. [15] proposed a modified PADM as following. For given (x k , y k , λ k ) ∈ R n ++ × R m ++ × R l , the new iterative (x k+1 , y k+1 , λ k+1 ) is obtained via the following steps: Step 1.Solve the following inequality to obtain x k+1 : (x ′ − x k+1 ) T { f (x k+1 ) − A T [λ k − H k (Ax k+1 + By k − b)] +R k (x k+1 − x k )}≥ 0, ∀x ′ ∈ R n ++ (7) Step 2.Solve the following inequality to obtain y k+1 : (y ′ − y k+1 ) T {g(y k+1 ) − B T [λ k − H k (Ax k+1 + By k+1 − b)] +S k (y k+1 − y k )}≥ 0, ∀y ′ ∈ R m ++ (8) ∗ Corresponding author e-mail: babedallah@yahoo.com c  2015 NSP Natural Sciences Publishing Cor.