American Institute of Mathematical Sciences

April  2014, 10(2): 461-476. doi: 10.3934/jimo.2014.10.461

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

Received  October 2012 Revised  August 2013 Published  October 2013

In this paper, we propose a new parallel splitting descent method for solving a class of variational inequalities with separable structure. The new method can be applied to solve convex optimization problems in which the objective function is separable with three operators and the constraint is linear. In the framework of the new algorithm, we adopt a new descent strategy by combining two descent directions and resolve the descent direction which is different from the methods in He (Comput. Optim. Appl., 2009, 42: 195-212.) and Wang et al. (submitted to J. Optimiz. Theory App.). Theoretically, we establish the global convergence of the new method under mild assumptions. In addition, we apply the new method to solve problems in management science and traffic equilibrium problem. Numerical results indicate that the new method is efficient and reliable.
Citation: Kai Wang, Lingling Xu, Deren Han. A new parallel splitting descent method for structured variational inequalities. Journal of Industrial & Management Optimization, 2014, 10 (2) : 461-476. doi: 10.3934/jimo.2014.10.461
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). [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). [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., (). [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). [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). [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). [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., (). [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). [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.
 [1] Xi-Hong Yan. A new convergence proof of augmented Lagrangian-based method with full Jacobian decomposition for structured variational inequalities. Numerical Algebra, Control & Optimization, 2016, 6 (1) : 45-54. doi: 10.3934/naco.2016.6.45 [2] Bingsheng He, Xiaoming Yuan. Linearized alternating direction method of multipliers with Gaussian back substitution for separable convex programming. Numerical Algebra, Control & Optimization, 2013, 3 (2) : 247-260. doi: 10.3934/naco.2013.3.247 [3] Xihong Yan. An augmented Lagrangian-based parallel splitting method for a one-leader-two-follower game. Journal of Industrial & Management Optimization, 2016, 12 (3) : 879-890. doi: 10.3934/jimo.2016.12.879 [4] Russell E. Warren, Stanley J. Osher. Hyperspectral unmixing by the alternating direction method of multipliers. Inverse Problems & Imaging, 2015, 9 (3) : 917-933. doi: 10.3934/ipi.2015.9.917 [5] Anis Theljani, Ke Chen. An augmented lagrangian method for solving a new variational model based on gradients similarity measures and high order regulariation for multimodality registration. Inverse Problems & Imaging, 2019, 13 (2) : 309-335. doi: 10.3934/ipi.2019016 [6] Foxiang Liu, Lingling Xu, Yuehong Sun, Deren Han. A proximal alternating direction method for multi-block coupled convex optimization. Journal of Industrial & Management Optimization, 2019, 15 (2) : 723-737. doi: 10.3934/jimo.2018067 [7] Sohana Jahan. Supervised distance preserving projection using alternating direction method of multipliers. Journal of Industrial & Management Optimization, 2017, 13 (5) : 1-17. doi: 10.3934/jimo.2019029 [8] Dan Xue, Wenyu Sun, Hongjin He. A structured trust region method for nonconvex programming with separable structure. Numerical Algebra, Control & Optimization, 2013, 3 (2) : 283-293. doi: 10.3934/naco.2013.3.283 [9] Hongxia Yin. An iterative method for general variational inequalities. Journal of Industrial & Management Optimization, 2005, 1 (2) : 201-209. doi: 10.3934/jimo.2005.1.201 [10] Chunlin Wu, Juyong Zhang, Xue-Cheng Tai. Augmented Lagrangian method for total variation restoration with non-quadratic fidelity. Inverse Problems & Imaging, 2011, 5 (1) : 237-261. doi: 10.3934/ipi.2011.5.237 [11] Wei Zhu, Xue-Cheng Tai, Tony Chan. Augmented Lagrangian method for a mean curvature based image denoising model. Inverse Problems & Imaging, 2013, 7 (4) : 1409-1432. doi: 10.3934/ipi.2013.7.1409 [12] Xueyong Wang, Yiju Wang, Gang Wang. An accelerated augmented Lagrangian method for multi-criteria optimization problem. Journal of Industrial & Management Optimization, 2017, 13 (5) : 1-9. doi: 10.3934/jimo.2018136 [13] Li Jin, Hongying Huang. Differential equation method based on approximate augmented Lagrangian for nonlinear programming. Journal of Industrial & Management Optimization, 2017, 13 (5) : 1-15. doi: 10.3934/jimo.2019053 [14] Yue Lu, Ying-En Ge, Li-Wei Zhang. An alternating direction method for solving a class of inverse semi-definite quadratic programming problems. Journal of Industrial & Management Optimization, 2016, 12 (1) : 317-336. doi: 10.3934/jimo.2016.12.317 [15] Zhongming Wu, Xingju Cai, Deren Han. Linearized block-wise alternating direction method of multipliers for multiple-block convex programming. Journal of Industrial & Management Optimization, 2018, 14 (3) : 833-855. doi: 10.3934/jimo.2017078 [16] Yuan Shen, Lei Ji. Partial convolution for total variation deblurring and denoising by new linearized alternating direction method of multipliers with extension step. Journal of Industrial & Management Optimization, 2019, 15 (1) : 159-175. doi: 10.3934/jimo.2018037 [17] Yoshifumi Aimoto, Takayasu Matsuo, Yuto Miyatake. A local discontinuous Galerkin method based on variational structure. Discrete & Continuous Dynamical Systems - S, 2015, 8 (5) : 817-832. doi: 10.3934/dcdss.2015.8.817 [18] Yunhai Xiao, Soon-Yi Wu, Bing-Sheng He. A proximal alternating direction method for $\ell_{2,1}$-norm least squares problem in multi-task feature learning. Journal of Industrial & Management Optimization, 2012, 8 (4) : 1057-1069. doi: 10.3934/jimo.2012.8.1057 [19] Wei Zhu. A numerical study of a mean curvature denoising model using a novel augmented Lagrangian method. Inverse Problems & Imaging, 2017, 11 (6) : 975-996. doi: 10.3934/ipi.2017045 [20] Xiantao Xiao, Liwei Zhang, Jianzhong Zhang. On convergence of augmented Lagrangian method for inverse semi-definite quadratic programming problems. Journal of Industrial & Management Optimization, 2009, 5 (2) : 319-339. doi: 10.3934/jimo.2009.5.319

2017 Impact Factor: 0.994

Metrics

• PDF downloads (9)
• HTML views (0)
• Cited by (6)

Other articlesby authors

• on AIMS
• on Google Scholar

[Back to Top]