(2) Bnouhachem et al. Journal of Inequalities and Applications (2016) 2016:297. Page 2 of 14. l explained in terms of ﬁnding z ∈ Z := Rn+ × Rm + × R such that. . z – z. . Q(z) ≥ ,. ∀z ∈ Z ,. (.). where ⎛ ⎞ u ⎜ ⎟ z = ⎝v⎠, λ. ⎞ f (u) – A λ ⎟ ⎜ Q(z) = ⎝ f (v) – A λ ⎠, A u + A v – b ⎛. (.). and A denotes the transpose of the matrix A . The problem (.)-(.) is referred as a structured variational inequality problem (in short, SVI). Yuan and Li [] developed the following logarithmic-quadratic proximal (LQP)-based decomposition method by applying the LQP terms to regularize the ADM subprobl lems: For a given zk = (uk , vk , λk ) ∈ Rn++ × Rm ++ × R , and μ ∈ (, ), the new iterative (uk+ , vk+ , λk+ ) is obtained via solving the following system: k k k k – f (u) – A = , λ – H A u + A v – b + R u – u + μ u – Uk u k k k – = , f (v) – A λ – H(A u + A v – b) + S v – v + μ v – Vk v λk+ = λk – H A uk + A vk – b ,. (.) (.) (.). where H ∈ Rl×l , R ∈ Rn×n , and S ∈ Rm×m are symmetric positive deﬁnite. Later, some LQP alternating direction methods have been proposed to make the LQP alternating direction method more practical, see, for example, [–] and the references therein. Each iteration of these methods contains a prediction and a correction, the predictor is obtained via solving (.)-(.) and the new iterate is obtained by a convex combination of the previous point and the one generated by a projection-type method along a descent direction. The main disadvantage of the methods proposed in [–] is that solving equation (.) requires the solution of equation (.). To overcome with this diﬃculty, Bnouhachem and Hamdi [] proposed a parallel descent LQP alternating direction method for solving SVI. In this paper, we propose a parallel descent LQP alternating direction method for solving the following structured variational inequality with three separable operators: Find y ∈ := {(u, v, w) : u ∈ Rn+ , v ∈ Rn+ , w ∈ Rn+ , A u + A v + A w = b} such that . y – y F(y) ≥ ,. ∀y ∈ ,. (.). with ⎛ ⎞ u ⎜ ⎟ y = ⎝ v ⎠, w. ⎛. ⎞ f (u) ⎜ ⎟ F(y) = ⎝ f (v) ⎠ , f (w). (.). where A ∈ Rm×n , A ∈ Rm×n , A ∈ Rm×n are given matrices, b ∈ Rm is a given vector, n and f : Rn+ → Rn , f : Rn+ → Rn , f : R+ → Rn are given monotone operators. By attaching a Lagrange multiplier vector λ ∈ Rm to the linear constraints A u + A v + A w = b,.

(3) Bnouhachem et al. Journal of Inequalities and Applications (2016) 2016:297. Page 3 of 14. n. the problem (.)-(.) can be explained in terms of ﬁnding z ∈ Z := Rn+ × Rn+ × R+ × Rm such that . z – z. . Q(z) ≥ ,. ∀z ∈ Z ,. (.). where ⎛ ⎞ u ⎜v⎟ ⎜ ⎟ z=⎜ ⎟ ⎝w⎠ λ. ⎞ f (u) – A λ ⎟ ⎜ f (v) – A ⎟ ⎜ λ Q(z) = ⎜ ⎟. ⎠ ⎝ λ f (w) – A A u + A v + A w – b ⎛. and. (.). The problem (.)-(.) is referred as SVI . The main aim of this paper is to present the parallel descent LQP alternating direction method for solving SVI and to investigate the convergence rate of this method. We show that the proposed method has the O(/t) convergence rate. The iterative algorithm and results presented in this paper generalize, unify, and improve the previously known results in this area.. 2 The proposed method For any vector u ∈ Rn , u∞ = max{|u |, . . . , |un |}. Let D ∈ Rn×n be a symmetry positive deﬁnite matrix, we denote the D-norm of u by uD = uT Du. The following lemma provides a basic property of projection operator onto a closed convex subset of Rl . We denote by P,D (·) the projection operator under the D-norm, that is, P,D (v) = argmin v – uD : u ∈ . Lemma . Let D be a symmetry positive deﬁnite matrix and be a nonempty closed convex subset of Rl . Then . z – P,D [z]. D P,D [z] – v ≥ ,. ∀z ∈ Rl , v ∈ .. (.). We make the following standard assumptions. Assumption . f is monotone with respect to Rn+ , that is, (f (x) – f (y))T (x – y) ≥ , n ∀x, y ∈ Rn+ , f is monotone with respect to Rn+ , and f is monotone with respect to R+ . Assumption . The solution set of SVI , denoted by Z ∗ , is nonempty. We propose the following parallel LQP alternating direction method for solving SVI : Algorithm . √ Step . Given ε > , μ ∈ (, ), β ∈ ( , ), γ ∈ (, ) and n z = (u , v , w , λ ) ∈ Rn++ × Rn++ × R++ × Rm . Set k = ..

(4) Bnouhachem et al. Journal of Inequalities and Applications (2016) 2016:297. Page 4 of 14. n. Step . Compute z̃k = (ũk , ṽk , w̃k , λ̃k ) ∈ Rn++ × Rn++ × R++ × Rm by solving the following system: k k k f (u) – A λ – H A u + A v + A w – b + R u – uk + μ uk – Uk u– = , k k k f (v) – A λ – H A u + A v + A w – b + R v – vk + μ vk – Vk v– = , k k k f (w) – A λ – H A u + A v + A w – b + R w – wk + μ wk – Wk w– = , λ̃k = λk – βH A ũk + A ṽk + A w̃k – b ,. (.). (.). (.) (.). where H ∈ Rm×m , R ∈ Rn ×n , R ∈ Rn ×n and R ∈ Rn ×n are symmetric positive deﬁnite matrices. Uk , Vk , and Wk are positive deﬁnite diagonal matrices deﬁned by Uk = diag(uk , . . . , ukn ), Vk = diag(vk , . . . , vkn ), Wk = diag(wk , . . . , wkn ). Step . If max{uk – ũk ∞ , vk – ṽk ∞ , wk – w̃k ∞ , λk – λ̃k ∞ } < , then stop. Step . The new iterate zk+ (τk ) = (uk+ , vk+ , wk+ , λk+ ) is given by zk+ (τk ) = ( – σ )zk + σ PZ ,G zk – γ τk G– g zk , z̃k ,. σ ∈ (, ),. (.). where ϕ(zk , z̃k ) , zk – z̃k G ϕ zk , z̃k = zk – z̃k M. τk =. +. (.). T k λ – λ̃k A uk – ũk + A vk – ṽk + A wk – w̃k , (.) β. g(zk , z̃k ) ⎞ ⎛ –β – k k k k k k k k k f (ũk ) – A λ̃ + A H[A (u – ũ ) + A (v – ṽ ) + A (w – w̃ ) + β H (λ – λ̃ )] ⎜ f (ṽk ) – A λ̃k + A H[A (uk – ũk ) + A (vk – ṽk ) + A (wk – w̃k ) + –β H – (λk – λ̃k )] ⎟ ⎟ ⎜ β , =⎜ –β – k k k k k k k k k ⎟ ⎠ ⎝ f (w̃k ) – A + A H[A (u – ũ ) + A (v – ṽ ) + A (w – w̃ ) + H (λ – λ̃ )] λ̃ β k k k A ũ + A ṽ + A w̃ – b. (.) ⎛ ⎜ G=⎜ ⎝. ( + μ)R + A HA . ( + μ)R + A HA . ( + μ)R + A HA . and ⎛. R + A HA ⎜ ⎜ M=⎜ ⎝ Set k := k + and go to Step .. R + A HA . R + A HA . ⎞ ⎟ ⎟ ⎟. ⎠ – H β. ⎞ ⎟ ⎟, ⎠ – H β.

(5) Bnouhachem et al. Journal of Inequalities and Applications (2016) 2016:297. Page 5 of 14. Remark . As special cases, we can obtain some new LQP alternating methods as follows: (a) If uk+ = ũk , vk+ = ṽk , wk+ = w̃k and λk+ = λ̃k in (.), (.), (.) and (.), respectively, we obtain a new method which can be viewed as an extension of that proposed in [] for solving structured variational inequality with three separable operators in a parallel way. (b) If uk+ = ũk , vk+ = ṽk , wk+ = w̃k , λk+ = λ̃k , and β = in (.), (.), (.) and (.), respectively, we obtain a new method which can be viewed as an extension of that proposed in [] for solving structured variational inequality with three separable operators in a parallel wise. (c) If β = , the proposed method can be viewed as an extension of that proposed in [] for solving structured variational inequality with three separable operators. We need the following result in the convergence analysis of the proposed method. Lemma . ([]) Let q(u) ∈ Rn be a monotone mapping of u with respect to Rn+ and R ∈ Rn×n be a positive deﬁnite diagonal matrix. For a given uk > , if Uk := diag(uk , uk , . . . , ukn ) (the diagonal matrix with elements uk , uk , . . . , ukn ) and u– be an n-vector whose jth element is /uj , then the equation q(u) + R u – uk + μ uk – Uk u– = . (.). has a unique positive solution u. Moreover, for any v ≥ , we have (v – u) q(u) ≥. – μ k + μ u – u . u – vR – uk – vR + R . (.). The next theorem is useful for the convergence analysis. n. Theorem . For given zk ∈ Rn++ × Rn++ × R++ × Rm , let z̃k be generated by (.)-(.). Then . k. ϕ z , z̃. k. . √ β – zk – z̃k ≥ G β. (.). and √ β – . τk ≥ β. (.). Proof It follows from (.) that ϕ zk , z̃k = zk – z̃k M + λk – λ̃k A uk – ũk + A vk – ṽk + A wk – w̃k β k = u – ũk R + A uk – A ũk H + vk – ṽk R + A vk – A ṽk H . . + wk – w̃k R + A wk – A w̃k H + λk – λ̃k H – β + λk – λ̃k A uk – ũk + A vk – ṽk + A wk – w̃k . β. (.).

(6) Bnouhachem et al. Journal of Inequalities and Applications (2016) 2016:297. Page 6 of 14. By using the Cauchy-Schwarz inequality, we have k √ k k k λ – λ̃ A u – ũ ≥ – A u – ũ H + √ λ – λ̃ H – , √ k A vk – ṽk H + √ λk – λ̃k H – , λ – λ̃k A vk – ṽk ≥ – . . k . k. . k. k. . (.) (.). and . k. λ – λ̃. k . . k. k. A w – w̃. . k √ k k k ≥– A w – w̃ H + √ λ – λ̃ H – . . (.). Substituting (.), (.), and (.) into (.), we get . k. ϕ z , z̃. k. . ≥. ≥. =. ≥. √ β – A uk – A ũk + A vk – A ṽk + A wk – A w̃k H H H β √ – λk – λ̃k – + uk – ũk + vk – ṽk + wk – w̃k + H R R R β √ β – A uk – A ũk + A vk – A ṽk H H β + A wk – A w̃k H + λk – λ̃k H – β √ β – uk – ũk + vk – ṽk + wk – w̃k + R R R β √ β – zk – z̃k + ( – μ)uk – ũk G R β + ( – μ)vk – ṽk R + ( – μ)wk – w̃k R √ β – zk – z̃k . G β. Therefore, it follows from (.) and (.) that √ β – , τk ≥ β. (.). and this completes the proof.. . 3 Convergence rate Recall that Z ∗ can be characterized as (see (..) in p. of []) Z∗ =. ẑ ∈ Z : (z – ẑ) Q(z) ≥ . z∈Z. This implies that ẑ is an approximate solution of SVI with the accuracy > if it satisﬁes ẑ ∈ Z. and. sup (z – ẑ) Q(z) ≤ .. z∈Z. (.).

(7) Bnouhachem et al. Journal of Inequalities and Applications (2016) 2016:297. Page 7 of 14. Now we show that after t iterations of the proposed method, we can ﬁnd a ẑ ∈ Z such that (.) is satisﬁed with = O(/t). We introduce the following matrices, ⎛. I ⎜ ⎜ N =⎜ ⎝ –βHA. I –βHA. I –βHA. ⎞ ⎟ ⎟ ⎟ ⎠ βI. (.). and ⎛. ( + μ)R + A HA ⎜ ⎜ J =⎜ ⎝ –A. ( + μ)R + A HA –A. ( + μ)R + A HA –A. ⎞ ⎟ ⎟ ⎟. ⎠ H –. (.). By simple manipulations, we can ﬁnd that J = GN . Our analysis needs a new sequence deﬁned by ⎞ ⎛ ûk ⎜ v̂k ⎟ ⎜ ⎜ ⎟ ⎜ ẑk = ⎜ k ⎟ = ⎜ ⎝ ŵ ⎠ ⎝ ⎛. λ̂k. ⎞ ũk ⎟ ṽk ⎟ ⎟. ⎠ w̃k λk – H(A uk + A vk + A wk – b). (.). Based on (.) and (.), we can easily have zk – z̃k = N zk – ẑk .. (.). Using (.), (.), and (.), we obtain g zk , z̃k = Q ẑk .. (.). Lemma . Let ẑk be deﬁned by (.), z ∈ Z , and the matrix J be given by (.). Then . z – ẑk. k k Q ẑ – J z – ẑk ≥ –μuk – ûk R – μvk – v̂k R – μwk – ŵk R . (.) . . . Proof Applying Lemma . to (.), we get . u – ũk ≥. k k k k k f ũ – A λ – H A ũ + A v + A w – b. + μ ũk – u – uk – u + – μ uk – ũk . R R R . (.). Since k u – u = uk – ũk + ũk – u + ũk – u T R uk – ũk , R R R . . . we have . u – ũk. . R uk – ũk = ũk – uR – uk – uR + uk – ũk R . . (.).

(8) Bnouhachem et al. Journal of Inequalities and Applications (2016) 2016:297. Page 8 of 14. Adding (.) and (.), we obtain . k k k k ( + μ)R uk – ũk – f ũk + A λ – H A u + A v + A w – b k k + A ≤ μuk – ũk R . HA u – ũ. u – ũk. . (.). Similarly, applying Lemma . to (.), we get . v – ṽk ≥. k k k k k f ṽ – A λ – H A u + A ṽ + A w – b. + μ ṽk – v – vk – v + – μ vk – ṽk . R R R . (.). Similar to (.), we have . v – ṽk. . R vk – ṽk = ṽk – vR – vk – vR + vk – ṽk R . . (.). Adding (.) and (.), we have . k k k k ( + μ)R vk – ṽk – f ṽk + A λ – H A u + A v + A w – b k k ≤ μvk – ṽk R . + A HA v – ṽ. v – ṽk. . (.). Similarly, we have . k k k k ( + μ)R wk – w̃k – f w̃k + A λ – H A u + A v + A w – b k k ≤ μwk – w̃k R . + A (.) HA w – w̃. w – w̃k. . Then, by using the notation of ẑk in (.), (.), (.), and (.) can be written as . k k k u – ûk ( + μ)R uk – ûk – f (ûk ) + A λ̂ + A HA u – û ≤ μuk – ûk R , k k k k ( + μ)R vk – v̂k – f v̂k + A v – v̂ λ̂ + A HA v – v̂ ≤ μvk – v̂k R , . (.). (.). and . k k k ( + μ)R wk – ŵk – f ŵk + A w – ŵk λ̂ + A HA w – ŵ ≤ μwk – ŵk R . . (.). In addition, it follows from (.) that A ûk + A v̂k + A ŵk – b + H – λ̂k – λk – A ûk – uk – A v̂k – vk – A ŵk – wk = .. (.).

(9) Bnouhachem et al. Journal of Inequalities and Applications (2016) 2016:297. Page 9 of 14. Combining (.)-(.), we get ⎞ ⎞ ⎛ k T k k f (ûk ) – A u – ûk λ̂ – (( + μ)R + A HA )(u – û ) k k k ⎟ ⎜ v – v̂k ⎟ ⎜ f (v̂k ) – A λ̂ – (( + μ)R + A HA )(v – v̂ ) ⎟ ⎜ ⎟ ⎜ k k k ⎠ ⎝ w – ŵk ⎠ ⎝ f (ŵk ) – A λ̂ – (( + μ)R + A HA )(w – ŵ ) λ – λ̂k A ûk + A v̂k + A ŵk – b + A (uk – ûk ) + A (vk – v̂k ) + A (wk – ŵk ) – H – (λk – λ̂k ) ≥ –μuk – ûk R – μvk – v̂k R – μwk – ŵk R . (.) ⎛. . . . Recall the deﬁnition of J in (.), we obtain the assertion (.). The proof is completed. n. Lemma . For given zk ∈ Rn++ × Rn++ × R++ × Rm and z∗k := PZ ,G [zk – τk G– g(zk , z̃k )], we have γ τk z – ẑk Q(z) + z – zk G – z – z∗k G ≥ γ ( – γ )τk zk – z̃k G . . (.). Proof Since z∗k ∈ Z , substituting z = z∗k in (.), we get γ τk z∗k – ẑk Q ẑk ≥ γ τk z∗k – ẑk J zk – ẑk – μγ τk uk – ûk R – μγ τk vk – v̂k R k – μγ τk w – ŵk R k k k = γ τk z – ẑ J z – ẑk + γ τk z∗k – zk J zk – ẑk – μγ τk uk – ûk R – μγ τk vk – v̂k R – μγ τk wk – ŵk R k – – k k k k k – k N = γ τk z – z̃ JN z – z̃ + γ τk z∗ – z JN z – z̃k – γ τk μuk – ûk – γ τk μvk – v̂k – μγ τk wk – ŵk R. R. (.). R. = γ τk zk – z̃k N – G zk – z̃k – γ τk μuk – ûk R – γ τk μvk – v̂k R k – μγ τk w – ŵk R + γ τk z∗k – zk G zk – z̃k k – k k N M z – z̃k + γ τk z∗k – zk G zk – z̃k = γ τk z – z̃ ≥ γ τk ϕ zk , z̃k + γ τk z∗k – zk G zk – z̃k . ≥ γ τk ϕ zk , z̃k – zk – z∗k G – γ τk zk – z̃k G = γ ( – γ )τk zk – z̃k G – zk – z∗k G . Using (.), z∗k is the projection of zk – γ τk G– Q(ẑk ) on Z , it follows from (.) that . zk – γ τk G– Q ẑk – z∗k G z – z∗k ≤ ,. and consequently, we have γ τk z – z∗k Q ẑk ≥ zk – z∗k G z – z∗k .. ∀z ∈ Z ,. (.).

(10) Bnouhachem et al. Journal of Inequalities and Applications (2016) 2016:297. Page 10 of 14. Using the identity x Gy = (xG – x – yG + yG ) to the right hand side of the last inequality, we obtain γ τk z – z∗k Q ẑk ≥ z – z∗k G – z – zk G + . zk – zk . ∗ G . (.). Adding (.) and (.), we get γ τk z – ẑk Q ẑk + z – zk G – z – z∗k G ≥ γ ( – γ )τk zk – z̃k G , . and by using the monotonicity of Q, we obtain (.) and the proof is completed. n. Lemma . Let zk ∈ Rn++ × Rn++ × R++ × Rm and zk+ (τk ) be generated by (.). Then γ σ τk z – ẑk Q(z) + z – zk G – z – zk+ (τk )G ≥ σ γ ( – γ )τk z – z̃k G . (.) Proof We have z – zk – z – zk+ (τk ) G G k k k = z – zG – z – σ z – z∗k – zG = σ zk – z G zk – z∗k – σ zk – z∗k G = σ zk – z∗k G – z – z∗k G zk – z∗k – σ zk – z∗k G .. (.). (.). Using the identity . z – z∗k. k G z – z∗k = z∗k – zG – zk – zG + zk – z∗k G , . we get k z – zk – z – zk G zk – zk = zk – z – zk – z . ∗ G ∗ ∗ ∗ G G. (.). Substituting (.) into (.), we obtain z – zk – z – zk+ (τk ) = σ z – zk – z – zk + σ ( – σ )zk – zk ∗ G ∗ G G G G ≥ σ z – zk G – z – z∗k G . Substituting (.) into (.), we obtain (.), the required result.. (.) . Theorem . Let z∗ be a solution of SVI and zk+ (τk ) be generated by (.). Then zk and z̃k are bounded, and k+ z (τk ) – z∗ ≤ zk – z∗ – czk – z̃k , G G G where σ γ ( – γ )(β – c := β . √. ). > .. (.).

(11) Bnouhachem et al. Journal of Inequalities and Applications (2016) 2016:297. Page 11 of 14. Proof Setting z = z∗ in (.), we obtain k+ z (τk ) – z∗ ≤ zk – z∗ – σ γ ( – γ )τ zk – z̃k + γ σ τk z∗ – ẑk Q z∗ k G G G ≤ zk – z∗ G – σ γ ( – γ )τk zk – z̃k G √ k σ γ ( – γ )(β – ) ∗ zk – z̃k . ≤ z –z G– G β Then we have k+ z (τk ) – z∗ ≤ zk – z∗ ≤ · · · ≤ z – z∗ , G G G and thus, {zk } is a bounded sequence. It follows from (.) that ∞ czk – z̃k G < +∞, k=. which means that lim zk – z̃k G = .. (.). k→∞. Since {zk } is a bounded sequence, we conclude that {z̃k } is also bounded.. . The global convergence of the proposed method can be proved by similar arguments as in []. Hence the proof is omitted. Theorem . The sequence {zk } generated by the proposed method converges to some z∞ which is a solution of SVI . Now, we are ready to present the O(/t) convergence rate of the proposed method. Theorem . For any integer t > , we have a ẑt ∈ Z which satisﬁes (ẑt – z) Q(z) ≤. z – z , G γ σ ϒt. ∀z ∈ Z ,. where. ẑt =. t k τk ẑ ϒt. and. ϒt =. k=. t . τk .. k=. Proof Summing the inequality (.) over k = , . . . , t, we obtain . t k=. γ σ τk z –. t k=. γ σ τk ẑ. k. Q(z) + z – z G ≥ . .

(12) Bnouhachem et al. Journal of Inequalities and Applications (2016) 2016:297. Page 12 of 14. Using the notations of ϒt and ẑt in the above inequality, we derive (ẑt – z) Q(z) ≤. z – z , G γ σ ϒt. ∀z ∈ Z .. Indeed, ẑt ∈ Z because it is a convex combination of ẑ , ẑ , . . . , ẑt . The proof is completed. Remark . It follows from (.) that ϒt ≥. √ β – (t + ). β. Suppose that, for any compact set D ⊂ Z , let d = sup{z – z G |z ∈ D}. For any given > , after at most . βd t= √ (β – )γ σ . . iterations, we have (ẑt – z)T Q(z) ≤ ,. ∀z ∈ D.. That is, the O(/t) convergence rate is established in an ergodic sense.. 4 Preliminary computational results In this section, we present some numerical examples to illustrate the proposed method. We consider the following optimization problem with matrix variables, which is studied in []: min U – CF U ∈ S+n , where · F is the matrix Fröbenius norm, i.e., CF = (. (.) n n i=. / j= |Cij | ). and. S+n = M ∈ Rn×n : M = M, M . It has been shown in [] that solving problem (.) is equivalent to the following variational inequality problem: Find X ∗ = (U ∗ , V ∗ , Z∗ ) ∈ = S+n × S+n × Rn×n such that ⎧ ∗ ∗ ∗ ⎪ ⎪ ⎨U – U , (U – C) – Z ≥ , V – V ∗ , (V ∗ – C) + Z∗ ≥ , ∀X = (U, V , Z) ∈ , ⎪ ⎪ ⎩ ∗ U – V ∗ = .. (.). The problem (.) is a special case of (.)-(.) with matrix variables where A = In×n , A = –In×n , b = , f (U) = U – C, f (V ) = V – C, and W = S+n × S+n × Rn×n . For simpliﬁcation, we take R = r In×n , R = r In×n , and H = In×n where r > and r > are scalars. In all tests.

(13) Bnouhachem et al. Journal of Inequalities and Applications (2016) 2016:297. Page 13 of 14. Table 1 Numerical results for problem (4.1) with r1 = 0.5, r2 = 5 Dimension of the problem. The proposed method. The method in [18]. The method in [17]. k. CPU (Sec.). k. CPU (Sec.). k. CPU (Sec.). 100 300 500 700. 43 48 50 52. 0.83 3.98 11.54 29.91. 49 53 56 57. 0.96 4.85 13.27 34.33. 71 79 82 85. 2.47 6.33 20.2 44.06. Table 2 Numerical results for problem (4.1) with r1 = 1, r2 = 10 Dimension of the problem. The proposed method. The method in [18]. The method in [17]. k. CPU (Sec.). k. CPU (Sec.). k. CPU (Sec.). 100 300 500 700. 106 119 125 129. 0.87 6.85 25.85 53.19. 109 123 128 132. 1.18 7.54 29.71 58.06. 124 140 147 152. 2.61 9.06 37.25 64.35. we take μ = ., β = ., C = rand(n), and (U , V , Z ) = (In×n , In×n , n×n ) as the initial point in the test. The iteration is stopped as soon as max U k – Ũ k , V k – Ṽ k , Zk – Z̃k ≤ – . All codes were written in Matlab, we compare the proposed method with those in [] and []. The iteration numbers, denoted by k, and the computational time for the problem (.) with diﬀerent dimensions are given in Tables -. Tables - show that the proposed method is more ﬂexible and eﬃcient for the problem tested.. 5 Conclusions In this paper, we proposed a new modiﬁed logarithmic-quadratic proximal alternating direction method for solving structured variational inequalities with three separable operators. The prediction point is obtained by solving series of related systems of nonlinear equations in a parallel way. Global convergence of the proposed method is proved under mild assumptions. We have proved the O(/t) convergence rate of the parallel LQP alternating direction method. Some preliminary numerical examples are reported to verify the eﬀectiveness of the proposed method in practice. Competing interests The authors declare that they have no competing interests. Authors’ contributions All authors contributed equally and signiﬁcantly in writing this article. All authors read and approved the ﬁnal manuscript. Author details 1 School of Management Science and Engineering, Nanjing University, Nanjing, 210093, P.R. China. 2 Laboratoire d’Ingénierie des Systémes et Technologies de l’Information, ENSA, Ibn Zohr University, Agadir, BP 1136, Morocco. 3 Department of Mathematics, King Abdulaziz University, P.O. Box 80203, Jeddah 21589, Saudi Arabia. 4 Department of Mathematics, Aligarh Muslim University, Aligarh 202002, India. 5 Department of Mathematics & Statistics, King Fahd University of Petroleum & Minerals, Dhahran, Saudi Arabia. Acknowledgements In this research, the second author was funded by the Deanship of Scientiﬁc Research (DSR), King Abdulaziz University, Jeddah. Therefore, he acknowledges with thanks DSR for the technical and ﬁnancial support. The research part of the.

(14) Bnouhachem et al. Journal of Inequalities and Applications (2016) 2016:297. Page 14 of 14. third author was done during his visit to King Fahd University of Petroleum & Minerals (KFUPM), Dhahran, Saudi Arabia. He is grateful to KFUPM for providing excellent research facilities to carry out his part of this research. Received: 3 October 2016 Accepted: 8 November 2016 References 1. Fortin, M, Glowinski, R: Augmented Lagrangian Methods: Applications to the Solution of Boundary-Valued Problems. North-Holland, Amsterdam (1983) 2. Gabay, D: Applications of the method of multipliers to variational inequalities. In: Fortin, M, Glowinski, R (eds.) Augmented Lagrangian Methods: Applications to the Solution of Boundary-Valued Problems. Studies in Mathematics and Its Applications, vol. 15, pp. 299-331. North-Holland, Amsterdam, The Netherlands (1983) 3. Gabay, D, Mercier, B: A dual algorithm for the solution of nonlinear variational problems via ﬁnite-element approximations. Comput. Math. Appl. 2, 17-40 (1976) 4. Glowinski, R: Numerical Methods for Nonlinear Variational Problems. Springer, New York (1984) 5. Glowinski, R, Le Tallec, P: Augmented Lagrangian and Operator-Splitting Methods in Nonlinear Mechanics. SIAM Studies in Applied Mathematics. SIAM, Philadelphia, PA (1989) 6. He, BS, Yang, H: Some convergence properties of a method of multipliers for linearly constrained monotone variational inequalities. Oper. Res. Lett. 23, 151-161 (1998) 7. Hou, LS: On the O(1/t) convergence rate of the parallel descent-like method and parallel splitting augmented Lagrangian method for solving a class of variational inequalities. Appl. Math. Comput. 219, 5862-5869 (2013) 8. Jiang, ZK, Bnouhachem, A: A projection-based prediction-correction method for structured monotone variational inequalities. Appl. Math. Comput. 202, 747-759 (2008) 9. Kontogiorgis, S, Meyer, RR: A variable-penalty alternating directions method for convex optimization. Math. Program. 83, 29-53 (1998) 10. Tao, M, Yuan, XM: On the O(1/t) convergence rate of alternating direction method with logarithmic-quadratic proximal regularization. SIAM J. Optim. 22(4), 1431-1448 (2012) 11. Wang, K, Xu, LL, Han, DR: A new parallel splitting descent method for structured variational inequalities. J. Ind. Manag. Optim. 10(2), 461-476 (2014) 12. Yuan, XM, Li, M: An LQP-based decomposition method for solving a class of variational inequalities. SIAM J. Optim. 21(4), 1309-1318 (2011) 13. Bnouhachem, A: On LQP alternating direction method for solving variational. J. Inequal. Appl. 2014, 80 (2014) 14. Bnouhachem, A, Ansari, QH: A descent LQP alternating direction method for solving variational inequality problems with separable structure. Appl. Math. Comput. 246, 519-532 (2014) 15. Bnouhachem, A, Benazza, H, Khalfaoui, M: An inexact alternating direction method for solving a class of structured variational inequalities. Appl. Math. Comput. 219, 7837-7846 (2013) 16. Bnouhachem, A, Xu, MH: An inexact LQP alternating direction method for solving a class of structured variational inequalities. Comput. Math. Appl. 67, 671-680 (2014) 17. Li, M: A hybrid LQP-based method for structured variational inequalities. J. Comput. Math. 89(10), 1412-1425 (2012) 18. Bnouhachem, A, Hamdi, A: Parallel LQP alternating direction method for solving variational inequality problems with separable structure. J. Inequal. Appl. 2014, 392 (2014) 19. Facchinei, F, Pang, JS: Finite-Dimensional Variational Inequalities and Complementarity Problems, Vol. I and II. Springer Series in Operations Research. Springer, New York (2003).

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