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A new parallel splitting descent method for structured variational inequalities
1. | School of Mathematical Sciences, Jiangsu Key Laboratory for NSLSCS, Nanjing Normal University, Nanjing 210023, China, China |
2. | School of Mathematical Sciences, Jiangsu Key Labratory for NSLSCS, Nanjing Normal University, Nanjing, 210023 |
References:
[1] |
D. P. Bertsekas and E. M. Gafni, Projection methods for variational inequalities with application to the traffic assignment problem,, in Nondifferential and Variational Techniques in Optimization, (1982), 139.
|
[2] |
D. P. Bertsekas and J. N. Tsitsiklis, Parallel and Distributed Compution,, Numerical Methods, (1989). Google Scholar |
[3] |
M. D'Apuzzo, M. Marino, A. Migdalas, P. M. Pardalos and G. Toraldo, Parallel computing in global optimization,, in Handbook of Parallel Computing and Statistics (eds. E. J. Kontoghiorghes), (2006), 225.
doi: 10.1201/9781420028683.ch7. |
[4] |
J. Eckstein, Some saddle-function splitting methods for convex programming,, Optimization Methods Software, 4 (1994), 75.
doi: 10.1080/10556789408805578. |
[5] |
J. Eckstein and M. Fukushima, Some reformulation and applications of the alternating direction method of multipliers,, in Large Scale Optimization (Gainesville, (1993), 115.
|
[6] |
F. Facchinei and J. S. Pang, Finite-dimensional Variational Inequalities and Complementarity Problems,, Springer, (2003). Google Scholar |
[7] |
M. C. Ferris and J. S. Pang, Engineering and economic applications of comlementarity problems,, SIAM Review, 39 (1997), 669.
doi: 10.1137/S0036144595285963. |
[8] |
M. Fukushima, Application of the alternating direction method of multipliers to separable convex programming problems,, Comput. Optim. Appl., 1 (1992), 93.
doi: 10.1007/BF00247655. |
[9] |
D. Gabay, Applications of the method of multipliers to variational inequalities,, in Augmented Lagrangian Methods: Applications to the Numerical Solution of Boundary-Valued Problems (eds. M. Fortin and R. Glowinski), (1983), 299.
doi: 10.1016/S0168-2024(08)70034-1. |
[10] |
D. Gabay and B. Mercier, A dual algorithm for the solution of nonlinear variational problems via finite-element approximations,, Computers & Mathematics with Applications, 2 (1976), 17.
doi: 10.1016/0898-1221(76)90003-1. |
[11] |
R. Glowinski, Numerical Methods for Nonlinear Variational Problems,, Springer-Verlag, (1984).
|
[12] |
R. Glowinski and P. Le Tallec, Augmented Lagrangian and Operator-Splitting Methods in Nonlinear Mechanics,, SIAM Studies in Applied Mathematics, (1989).
doi: 10.1137/1.9781611970838. |
[13] |
D. R. Han, X. M. Yuan and W. X. Zhang, An augmented-Lagrangian-based parallel splitting method for separable convex programming with applications to image processing,, manuscript., (). Google Scholar |
[14] |
D. R. Han, X. M. Yuan, W. X. Zhang and X. J. Cai, An ADM-based splitting method for separable convex programming,, Computational Optimization and Applications, 54 (2013), 343.
doi: 10.1007/s10589-012-9510-y. |
[15] |
B.-S. He, Parallel splitting augmented Lagrangian methods for monotone structured variational inequalities,, Comput. Optim. Appl., 42 (2009), 195.
doi: 10.1007/s10589-007-9109-x. |
[16] |
B.-S. He, L. Z. Liao, H. Yang and D. R. Han, A new inexact alternating directions method for monotone variational inequalities,, Math. Program., 92 (2002), 103.
doi: 10.1007/s101070100280. |
[17] |
B.-S. He and L. Z. Liao, Improvements of some projection methods for monotone nonlinear variational inequalities,, J. Optim. Theory Appl., 112 (2002), 111.
doi: 10.1023/A:1013096613105. |
[18] |
B.-S. He, Y. Xu and X.-M. Yuan, A logarithmic-quadratic proximal prediction-correction method for structured monotone variational inequalities,, Comput. Optim. Appl., 35 (2006), 19.
doi: 10.1007/s10589-006-6442-4. |
[19] |
B.-S. He, H. Yang and S. L. Wang, Altermating direction method with self-adaptive penalty parameters for monotone variational inequalities,, J. Optim. Theory Appl., 106 (2000), 337.
doi: 10.1023/A:1004603514434. |
[20] |
B.-S. He, M. Tao and X.-M. Yuan, Alternating direction method with Gaussian back substitution for separable convex programming,, SIAM J. Optim., 22 (2012), 313.
doi: 10.1137/110822347. |
[21] |
B.-S. He, M. Tao, M. H. Xu and X.-M. Yuan, Alternating directions based contraction method for generally separable linearly constrained convex programming problems,, manuscript, (2009). Google Scholar |
[22] |
Z. K. Jiang and X. M. Yuan, New parallel descent-like method for sloving a class of variational inequalities,, J. Optim. Theory Appl., 145 (2010), 311.
doi: 10.1007/s10957-009-9619-z. |
[23] |
D. Kinderlehrer and G. Stampacchia, An Introduction to Variational Inequalities and Their Application,, Pure and Applied Mathematics, (1980).
|
[24] |
A. Migdalas, P. M. Pardalos and S. Storøy, eds., Parallel Computing in Optimization,, Applied Optimization, (1997).
|
[25] |
A. Migdalas, G. Toraldo and V. Kumar, Nonlinear optimization and parallel computing,, Parallel Computing, 29 (2003), 375.
doi: 10.1016/S0167-8191(03)00013-9. |
[26] |
A. Nagurney and D. Zhang, Projected Dynamical Systems and Variational Inequalities with Applications,, International Series in Operations Research & Management Science, (1996).
doi: 10.1007/978-1-4615-2301-7. |
[27] |
P. M. Pardalos and S. Rajasekaran, eds., Advances in Randomized Parallel Computing,, Kluwer Academic Publishers, (1999).
doi: 10.1007/978-1-4613-3282-4. |
[28] |
J. Sun and S. Zhang, A modified alternating direction method for convex quadratically constrained quadratic semidefinite programs,, European J. Oper. Res., 207 (2010), 1210.
doi: 10.1016/j.ejor.2010.07.020. |
[29] |
K. Wang, D. R. Han and L. L. Xu, A parallel splitting method for separable convex programming,, J. Optim. Theory Appl., 159 (2013), 138.
doi: 10.1007/s10957-013-0277-9. |
show all references
References:
[1] |
D. P. Bertsekas and E. M. Gafni, Projection methods for variational inequalities with application to the traffic assignment problem,, in Nondifferential and Variational Techniques in Optimization, (1982), 139.
|
[2] |
D. P. Bertsekas and J. N. Tsitsiklis, Parallel and Distributed Compution,, Numerical Methods, (1989). Google Scholar |
[3] |
M. D'Apuzzo, M. Marino, A. Migdalas, P. M. Pardalos and G. Toraldo, Parallel computing in global optimization,, in Handbook of Parallel Computing and Statistics (eds. E. J. Kontoghiorghes), (2006), 225.
doi: 10.1201/9781420028683.ch7. |
[4] |
J. Eckstein, Some saddle-function splitting methods for convex programming,, Optimization Methods Software, 4 (1994), 75.
doi: 10.1080/10556789408805578. |
[5] |
J. Eckstein and M. Fukushima, Some reformulation and applications of the alternating direction method of multipliers,, in Large Scale Optimization (Gainesville, (1993), 115.
|
[6] |
F. Facchinei and J. S. Pang, Finite-dimensional Variational Inequalities and Complementarity Problems,, Springer, (2003). Google Scholar |
[7] |
M. C. Ferris and J. S. Pang, Engineering and economic applications of comlementarity problems,, SIAM Review, 39 (1997), 669.
doi: 10.1137/S0036144595285963. |
[8] |
M. Fukushima, Application of the alternating direction method of multipliers to separable convex programming problems,, Comput. Optim. Appl., 1 (1992), 93.
doi: 10.1007/BF00247655. |
[9] |
D. Gabay, Applications of the method of multipliers to variational inequalities,, in Augmented Lagrangian Methods: Applications to the Numerical Solution of Boundary-Valued Problems (eds. M. Fortin and R. Glowinski), (1983), 299.
doi: 10.1016/S0168-2024(08)70034-1. |
[10] |
D. Gabay and B. Mercier, A dual algorithm for the solution of nonlinear variational problems via finite-element approximations,, Computers & Mathematics with Applications, 2 (1976), 17.
doi: 10.1016/0898-1221(76)90003-1. |
[11] |
R. Glowinski, Numerical Methods for Nonlinear Variational Problems,, Springer-Verlag, (1984).
|
[12] |
R. Glowinski and P. Le Tallec, Augmented Lagrangian and Operator-Splitting Methods in Nonlinear Mechanics,, SIAM Studies in Applied Mathematics, (1989).
doi: 10.1137/1.9781611970838. |
[13] |
D. R. Han, X. M. Yuan and W. X. Zhang, An augmented-Lagrangian-based parallel splitting method for separable convex programming with applications to image processing,, manuscript., (). Google Scholar |
[14] |
D. R. Han, X. M. Yuan, W. X. Zhang and X. J. Cai, An ADM-based splitting method for separable convex programming,, Computational Optimization and Applications, 54 (2013), 343.
doi: 10.1007/s10589-012-9510-y. |
[15] |
B.-S. He, Parallel splitting augmented Lagrangian methods for monotone structured variational inequalities,, Comput. Optim. Appl., 42 (2009), 195.
doi: 10.1007/s10589-007-9109-x. |
[16] |
B.-S. He, L. Z. Liao, H. Yang and D. R. Han, A new inexact alternating directions method for monotone variational inequalities,, Math. Program., 92 (2002), 103.
doi: 10.1007/s101070100280. |
[17] |
B.-S. He and L. Z. Liao, Improvements of some projection methods for monotone nonlinear variational inequalities,, J. Optim. Theory Appl., 112 (2002), 111.
doi: 10.1023/A:1013096613105. |
[18] |
B.-S. He, Y. Xu and X.-M. Yuan, A logarithmic-quadratic proximal prediction-correction method for structured monotone variational inequalities,, Comput. Optim. Appl., 35 (2006), 19.
doi: 10.1007/s10589-006-6442-4. |
[19] |
B.-S. He, H. Yang and S. L. Wang, Altermating direction method with self-adaptive penalty parameters for monotone variational inequalities,, J. Optim. Theory Appl., 106 (2000), 337.
doi: 10.1023/A:1004603514434. |
[20] |
B.-S. He, M. Tao and X.-M. Yuan, Alternating direction method with Gaussian back substitution for separable convex programming,, SIAM J. Optim., 22 (2012), 313.
doi: 10.1137/110822347. |
[21] |
B.-S. He, M. Tao, M. H. Xu and X.-M. Yuan, Alternating directions based contraction method for generally separable linearly constrained convex programming problems,, manuscript, (2009). Google Scholar |
[22] |
Z. K. Jiang and X. M. Yuan, New parallel descent-like method for sloving a class of variational inequalities,, J. Optim. Theory Appl., 145 (2010), 311.
doi: 10.1007/s10957-009-9619-z. |
[23] |
D. Kinderlehrer and G. Stampacchia, An Introduction to Variational Inequalities and Their Application,, Pure and Applied Mathematics, (1980).
|
[24] |
A. Migdalas, P. M. Pardalos and S. Storøy, eds., Parallel Computing in Optimization,, Applied Optimization, (1997).
|
[25] |
A. Migdalas, G. Toraldo and V. Kumar, Nonlinear optimization and parallel computing,, Parallel Computing, 29 (2003), 375.
doi: 10.1016/S0167-8191(03)00013-9. |
[26] |
A. Nagurney and D. Zhang, Projected Dynamical Systems and Variational Inequalities with Applications,, International Series in Operations Research & Management Science, (1996).
doi: 10.1007/978-1-4615-2301-7. |
[27] |
P. M. Pardalos and S. Rajasekaran, eds., Advances in Randomized Parallel Computing,, Kluwer Academic Publishers, (1999).
doi: 10.1007/978-1-4613-3282-4. |
[28] |
J. Sun and S. Zhang, A modified alternating direction method for convex quadratically constrained quadratic semidefinite programs,, European J. Oper. Res., 207 (2010), 1210.
doi: 10.1016/j.ejor.2010.07.020. |
[29] |
K. Wang, D. R. Han and L. L. Xu, A parallel splitting method for separable convex programming,, J. Optim. Theory Appl., 159 (2013), 138.
doi: 10.1007/s10957-013-0277-9. |
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