An augmented Lagrangian-based parallel splitting method for a one-leader-two-follower game

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July 2016, 12(3): 879-890. doi: 10.3934/jimo.2016.12.879

An augmented Lagrangian-based parallel splitting method for a one-leader-two-follower game

1.	Department of Mathematics, Taiyuan Normal University, Taiyuan 030012, Shanxi Province, China

Received February 2014 Revised November 2014 Published September 2015

Abstract
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In this paper, we exploit a new parallel splitting method for the typical structured variational inequality problems which can be interpreted as a game with a leader and two followers. In the framework of this method, two followers decide their strategies simultaneously based on the instruction of the leader. Then, the leader improves his instruction by revising his own variable value according to the feedback information from the followers. The convergence of the method is established under some suitable conditions. Finally, we apply the proposed method to solve some application problems. Computational studies show that the method is reliable and efficient.

Keywords: separable structure, variational inequality problem, one-leader-two-follower game, Parallel splitting method, convergence..

Mathematics Subject Classification: Primary: 49M27, 65K15, 93B4.

Citation: Xihong Yan. An augmented Lagrangian-based parallel splitting method for a one-leader-two-follower game. Journal of Industrial and Management Optimization, 2016, 12 (3) : 879-890. doi: 10.3934/jimo.2016.12.879

References:

[1]	G. Chen and M. Teboulle, A proximal-based decomposition method for convex minimization problems, Mathematical Programming, 64 (1994), 81-101. doi: 10.1007/BF01582566.
[2]	J. Eckstein and D. P. Bertsekas, On the Douglas-Rachford splitting method and the proximal point algorithm for maximal monotone operators, Mathematical Programming, 55 (1992), 293-318. doi: 10.1007/BF01581204.
[3]	M. Fukushima, Application of the alternating direction method of multipliers to separable convex programming problems, Computational Optimization and Applications, 1 (1992), 93-111. doi: 10.1007/BF00247655.
[4]	D. Gabay and B. Mercier, A dual algorithm for the solution of nonlinear variational problems via finite element approximations, Computers and Mathematics with Applications, 2 (1976), 17-40. doi: 10.1016/0898-1221(76)90003-1.
[5]	R. Glowinski and P. Le Tallec, Augmented Lagrangian and Operator-Splitting Methods in Nonlinear Mechanics, SIAM Studies in Applied Mathematics, SIAM, Philadelphia, 1989. doi: 10.1137/1.9781611970838.
[6]	D. Han, H. He, H. Yang and X. Yuan, A customized Douglas-Rachford splitting algorithm for separable convex minimization with linear constraints, Numerische Mathematik, 127 (2014), 167-200. doi: 10.1007/s00211-013-0580-2.
[7]	B. S. He, Inexact implicit methods for monotone general variational inequalities, Mathematical Programming, 86 (1999), 199-217. doi: 10.1007/s101070050086.
[8]	B. S. He, Parallel splitting augmented Lagrangian methods for monotone structured variational inequalities, Computational Optimization and Applications, 42 (2009), 195-212. doi: 10.1007/s10589-007-9109-x.
[9]	B. S. He, L. Z. Liao, D. Han and H. Yang, A new inexact alternating directions method for monotone variational inequalities, Mathematical Programming 92 (2002), 103-118. doi: 10.1007/s101070100280.
[10]	B. S. He, Y. Xu and X. M. Yuan, A logarithmic-quadratic proximal prediction-correction method for structured monotone variational inequalities, Computational Optimization and Applications, 35 (2006), 19-46. doi: 10.1007/s10589-006-6442-4.
[11]	S. Kontogiorgis and R. Meyer, A variable-penalty alternating directions method for convex optimization, Mathematical Programming, 83 (1998), 29-53. doi: 10.1007/BF02680549.
[12]	A. Nagurney and D. Zhang, Projected Dynamical Systems and Variational Inequalities with Applications, Kluwer, Boston, 1996. doi: 10.1007/978-1-4615-2301-7.
[13]	M. Tao and X. Yuan, An inexact parallel splitting augmented Lagrangian method for monotone variational inequalities with separable structures, Computational Optimization and Applications, 52 (2012), 439-461. doi: 10.1007/s10589-011-9417-z.
[14]	P. Tseng, Alternating projection-proximal methods for convex programming and variational inequalities, SIAM Journal on Optimization, 7 (1997), 951-965. doi: 10.1137/S1052623495279797.
[15]	K. Wang, L. Xu and D. Han, A new parallel splitting descent method for structured variational inequalities, Journal of Industrial and Management Optimization, 10 (2014), 461-476. doi: 10.3934/jimo.2014.10.461.

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References:

[1]	G. Chen and M. Teboulle, A proximal-based decomposition method for convex minimization problems, Mathematical Programming, 64 (1994), 81-101. doi: 10.1007/BF01582566.
[2]	J. Eckstein and D. P. Bertsekas, On the Douglas-Rachford splitting method and the proximal point algorithm for maximal monotone operators, Mathematical Programming, 55 (1992), 293-318. doi: 10.1007/BF01581204.
[3]	M. Fukushima, Application of the alternating direction method of multipliers to separable convex programming problems, Computational Optimization and Applications, 1 (1992), 93-111. doi: 10.1007/BF00247655.
[4]	D. Gabay and B. Mercier, A dual algorithm for the solution of nonlinear variational problems via finite element approximations, Computers and Mathematics with Applications, 2 (1976), 17-40. doi: 10.1016/0898-1221(76)90003-1.
[5]	R. Glowinski and P. Le Tallec, Augmented Lagrangian and Operator-Splitting Methods in Nonlinear Mechanics, SIAM Studies in Applied Mathematics, SIAM, Philadelphia, 1989. doi: 10.1137/1.9781611970838.
[6]	D. Han, H. He, H. Yang and X. Yuan, A customized Douglas-Rachford splitting algorithm for separable convex minimization with linear constraints, Numerische Mathematik, 127 (2014), 167-200. doi: 10.1007/s00211-013-0580-2.
[7]	B. S. He, Inexact implicit methods for monotone general variational inequalities, Mathematical Programming, 86 (1999), 199-217. doi: 10.1007/s101070050086.
[8]	B. S. He, Parallel splitting augmented Lagrangian methods for monotone structured variational inequalities, Computational Optimization and Applications, 42 (2009), 195-212. doi: 10.1007/s10589-007-9109-x.
[9]	B. S. He, L. Z. Liao, D. Han and H. Yang, A new inexact alternating directions method for monotone variational inequalities, Mathematical Programming 92 (2002), 103-118. doi: 10.1007/s101070100280.
[10]	B. S. He, Y. Xu and X. M. Yuan, A logarithmic-quadratic proximal prediction-correction method for structured monotone variational inequalities, Computational Optimization and Applications, 35 (2006), 19-46. doi: 10.1007/s10589-006-6442-4.
[11]	S. Kontogiorgis and R. Meyer, A variable-penalty alternating directions method for convex optimization, Mathematical Programming, 83 (1998), 29-53. doi: 10.1007/BF02680549.
[12]	A. Nagurney and D. Zhang, Projected Dynamical Systems and Variational Inequalities with Applications, Kluwer, Boston, 1996. doi: 10.1007/978-1-4615-2301-7.
[13]	M. Tao and X. Yuan, An inexact parallel splitting augmented Lagrangian method for monotone variational inequalities with separable structures, Computational Optimization and Applications, 52 (2012), 439-461. doi: 10.1007/s10589-011-9417-z.
[14]	P. Tseng, Alternating projection-proximal methods for convex programming and variational inequalities, SIAM Journal on Optimization, 7 (1997), 951-965. doi: 10.1137/S1052623495279797.
[15]	K. Wang, L. Xu and D. Han, A new parallel splitting descent method for structured variational inequalities, Journal of Industrial and Management Optimization, 10 (2014), 461-476. doi: 10.3934/jimo.2014.10.461.

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