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Neural network smoothing approximation method for stochastic variational inequality problems
1. | School of Economics, Southwest University for Nationalities, Chengdu, Sichuan 610041, China |
2. | Department of Mathematics, Sichuan University, Chengdu, Sichuan 610064 |
References:
[1] |
R. P. Agdeppa, N. Yamashita and M. Fukushima, Convex expected residual models for stochastic affine variational inequality problems and its application to the traffic equilibrium problem,, Pacific Journal of Optimization, 6 (2010), 3.
|
[2] |
R. J. Aumann, Integrals of set-value function,, Journal of Mathematical Analysis and Applications, 12 (1965), 1.
doi: 10.1016/0022-247X(65)90049-1. |
[3] |
B. T. Chen and P. T. Harker, Smooth approximations to nonlinear complementarity problems,, SIAM Journal on Optimization, 7 (1997), 403.
doi: 10.1137/S1052623495280615. |
[4] |
X. Chen and M. Fukushima, Expected residual minimization method for stochastic linear complementarity problems,, Mathematics of Operations Research, 30 (2005), 1022.
doi: 10.1287/moor.1050.0160. |
[5] |
X. Chen and G. H. Lin, CVaR-based formulation and approximation method for Stochastic variational inequalities,, Numerical Algebra, 1 (2011), 35.
doi: 10.3934/naco.2011.1.35. |
[6] |
X. Chen, C. Zhang and M. Fukushima, Robust solution of monotone stochastic linear complementarity problems,, Mathematical Programming, 117 (2009), 51.
doi: 10.1007/s10107-007-0163-z. |
[7] |
F. H. Clarke, Optimization and Nonsmooth Analysis,, Wiley, (1983).
|
[8] |
M. Fukushima, Equivalent differentiable optimization problems and descent methods for asymmetric variational inequality problems,, Mathematical Programming, 53 (1992), 99.
doi: 10.1007/BF01585696. |
[9] |
F. Facchinei and J. S. Pang, Finite-Dimensional Variational Inequalities and Complementarity Problems,, Springer, (2003).
doi: 10.1007/b97544. |
[10] |
H. Fang, X. Chen and M. Fukushima, Stochastic $R_0$ matrix linear complementarity problems,, SIAM Journal on Optimization, 18 (2007), 482.
doi: 10.1137/050630805. |
[11] |
P. T. Harker and J. S. Pang, Finite-dimensional variational inequality and nonlinear complementarity problems: A survey of theory, algorithms and applications,, Mathematical Programming, 48 (1990), 161.
doi: 10.1007/BF01582255. |
[12] |
W. W. Hogan, Energy policy models for project independence,, Computers and Operations Research, 2 (1975), 251.
doi: 10.1016/0305-0548(75)90008-8. |
[13] |
H. Jiang and H. F. Xu, Stochastic approximation approaches to the stochastic variational inequality problem,, IEEE Transactions on Automatic Control, 53 (2008), 1462.
doi: 10.1109/TAC.2008.925853. |
[14] |
D. Kinderlehre and G. Stampacchia, An Intruduction to Variational Inequalities and Their Aplications,, Academic Press, (1980). Google Scholar |
[15] |
G. H. Lin, X. Chen and M. Fukushima, New restricted NCP function and their applications to stochastic NCP and stochastic MPEC,, Optimization, 56 (2007), 641.
doi: 10.1080/02331930701617320. |
[16] |
G. H. Lin and M. Fukushima, Stochastic equilibrium problems and stochastic mathematical programs with equilibrium constraints: A survey,, Pacific Journal of Optimization, 6 (2010), 455.
|
[17] |
G. H. Lin and M. Fukushima, New reformulations for stochastic nonlinear complementarity peoblems,, Optimization Methods and Software, 21 (2006), 551.
doi: 10.1080/10556780600627610. |
[18] |
C. Ling, L. Qi, G. Zhou and L. Caccetta, The SC' property of an expected residual function arising from stochastic complementarity problems,, Operations Research Letters, 36 (2008), 456.
doi: 10.1016/j.orl.2008.01.010. |
[19] |
M. J. Luo and G. H. Lin, Expected residual minimization method for stochastic variational inequality problems,, Journal of Optimization Theory and Applications, 140 (2009), 103.
doi: 10.1007/s10957-008-9439-6. |
[20] |
M. J. Luo and G. H. Lin, Convergence results of the ERM method for nonlinear stochastic variational inequality problems,, Journal of Optimization Theory and Applications, 142 (2009), 569.
doi: 10.1007/s10957-009-9534-3. |
[21] |
F. W. Meng, J. Sun and M. Goh, Stochastic optimization problems with CVaR risk measure and their sample average approximation,, Journal of Optimization Theory and Applications, 146 (2010), 399.
doi: 10.1007/s10957-010-9676-3. |
[22] |
L. Q. Qi, D. F. Sun and G. L. Zhou, A new look at smoothing Newton methods for nonlinear complementarity problems and box constrained variational inequalities,, Mathematical Programming, 87 (2000), 1.
|
[23] |
R. T. Rockafellar and S. Uryasev, Optimization of conditional value-at-risk,, Journal of Risk, 2 (2000), 493. Google Scholar |
[24] |
A. Ruszczynski and A. Shapiro, Stochastic Programming,, Elsevier, (2003).
|
[25] |
A. Shapiro, Stochastic Programming by Monte Carlo Simulation Methods,, Stochastic Programming E-Print Series, (2000). Google Scholar |
[26] |
M. Z. Wang, M. M. Ali and G. H. Lin, Sample average approximation method for stochastic complementarity problems with applications to supply chain supernetworks,, Journal of Industrial and Management Optimization, 7 (2011), 317.
doi: 10.3934/jimo.2011.7.317. |
[27] |
D. De Wolf and Y. Smeers, A stochastic version of a Stackelberg-Nash-Cournot equilibrium model,, Management Science, 43 (1997), 190. Google Scholar |
[28] |
H. Xu, Sample average approximation methods for a class of stochastic variational inequality problems,, Asia-Pacific Journal of Operational Research, 27 (2010), 103.
doi: 10.1142/S0217595910002569. |
[29] |
H. Xu and D. Zhang, Smooth sample average approximation of stationary points in nonsmooth stochastic optimization and applications,, Mathematical Programming, 119 (2009), 371.
doi: 10.1007/s10107-008-0214-0. |
[30] |
C. Zhang and X. Chen, Stochastic nonlinear complementarity problem and applications to traffic equilibrium under uncertainty,, Journal of Optimization Theory and Applications, 137 (2008), 277.
doi: 10.1007/s10957-008-9358-6. |
show all references
References:
[1] |
R. P. Agdeppa, N. Yamashita and M. Fukushima, Convex expected residual models for stochastic affine variational inequality problems and its application to the traffic equilibrium problem,, Pacific Journal of Optimization, 6 (2010), 3.
|
[2] |
R. J. Aumann, Integrals of set-value function,, Journal of Mathematical Analysis and Applications, 12 (1965), 1.
doi: 10.1016/0022-247X(65)90049-1. |
[3] |
B. T. Chen and P. T. Harker, Smooth approximations to nonlinear complementarity problems,, SIAM Journal on Optimization, 7 (1997), 403.
doi: 10.1137/S1052623495280615. |
[4] |
X. Chen and M. Fukushima, Expected residual minimization method for stochastic linear complementarity problems,, Mathematics of Operations Research, 30 (2005), 1022.
doi: 10.1287/moor.1050.0160. |
[5] |
X. Chen and G. H. Lin, CVaR-based formulation and approximation method for Stochastic variational inequalities,, Numerical Algebra, 1 (2011), 35.
doi: 10.3934/naco.2011.1.35. |
[6] |
X. Chen, C. Zhang and M. Fukushima, Robust solution of monotone stochastic linear complementarity problems,, Mathematical Programming, 117 (2009), 51.
doi: 10.1007/s10107-007-0163-z. |
[7] |
F. H. Clarke, Optimization and Nonsmooth Analysis,, Wiley, (1983).
|
[8] |
M. Fukushima, Equivalent differentiable optimization problems and descent methods for asymmetric variational inequality problems,, Mathematical Programming, 53 (1992), 99.
doi: 10.1007/BF01585696. |
[9] |
F. Facchinei and J. S. Pang, Finite-Dimensional Variational Inequalities and Complementarity Problems,, Springer, (2003).
doi: 10.1007/b97544. |
[10] |
H. Fang, X. Chen and M. Fukushima, Stochastic $R_0$ matrix linear complementarity problems,, SIAM Journal on Optimization, 18 (2007), 482.
doi: 10.1137/050630805. |
[11] |
P. T. Harker and J. S. Pang, Finite-dimensional variational inequality and nonlinear complementarity problems: A survey of theory, algorithms and applications,, Mathematical Programming, 48 (1990), 161.
doi: 10.1007/BF01582255. |
[12] |
W. W. Hogan, Energy policy models for project independence,, Computers and Operations Research, 2 (1975), 251.
doi: 10.1016/0305-0548(75)90008-8. |
[13] |
H. Jiang and H. F. Xu, Stochastic approximation approaches to the stochastic variational inequality problem,, IEEE Transactions on Automatic Control, 53 (2008), 1462.
doi: 10.1109/TAC.2008.925853. |
[14] |
D. Kinderlehre and G. Stampacchia, An Intruduction to Variational Inequalities and Their Aplications,, Academic Press, (1980). Google Scholar |
[15] |
G. H. Lin, X. Chen and M. Fukushima, New restricted NCP function and their applications to stochastic NCP and stochastic MPEC,, Optimization, 56 (2007), 641.
doi: 10.1080/02331930701617320. |
[16] |
G. H. Lin and M. Fukushima, Stochastic equilibrium problems and stochastic mathematical programs with equilibrium constraints: A survey,, Pacific Journal of Optimization, 6 (2010), 455.
|
[17] |
G. H. Lin and M. Fukushima, New reformulations for stochastic nonlinear complementarity peoblems,, Optimization Methods and Software, 21 (2006), 551.
doi: 10.1080/10556780600627610. |
[18] |
C. Ling, L. Qi, G. Zhou and L. Caccetta, The SC' property of an expected residual function arising from stochastic complementarity problems,, Operations Research Letters, 36 (2008), 456.
doi: 10.1016/j.orl.2008.01.010. |
[19] |
M. J. Luo and G. H. Lin, Expected residual minimization method for stochastic variational inequality problems,, Journal of Optimization Theory and Applications, 140 (2009), 103.
doi: 10.1007/s10957-008-9439-6. |
[20] |
M. J. Luo and G. H. Lin, Convergence results of the ERM method for nonlinear stochastic variational inequality problems,, Journal of Optimization Theory and Applications, 142 (2009), 569.
doi: 10.1007/s10957-009-9534-3. |
[21] |
F. W. Meng, J. Sun and M. Goh, Stochastic optimization problems with CVaR risk measure and their sample average approximation,, Journal of Optimization Theory and Applications, 146 (2010), 399.
doi: 10.1007/s10957-010-9676-3. |
[22] |
L. Q. Qi, D. F. Sun and G. L. Zhou, A new look at smoothing Newton methods for nonlinear complementarity problems and box constrained variational inequalities,, Mathematical Programming, 87 (2000), 1.
|
[23] |
R. T. Rockafellar and S. Uryasev, Optimization of conditional value-at-risk,, Journal of Risk, 2 (2000), 493. Google Scholar |
[24] |
A. Ruszczynski and A. Shapiro, Stochastic Programming,, Elsevier, (2003).
|
[25] |
A. Shapiro, Stochastic Programming by Monte Carlo Simulation Methods,, Stochastic Programming E-Print Series, (2000). Google Scholar |
[26] |
M. Z. Wang, M. M. Ali and G. H. Lin, Sample average approximation method for stochastic complementarity problems with applications to supply chain supernetworks,, Journal of Industrial and Management Optimization, 7 (2011), 317.
doi: 10.3934/jimo.2011.7.317. |
[27] |
D. De Wolf and Y. Smeers, A stochastic version of a Stackelberg-Nash-Cournot equilibrium model,, Management Science, 43 (1997), 190. Google Scholar |
[28] |
H. Xu, Sample average approximation methods for a class of stochastic variational inequality problems,, Asia-Pacific Journal of Operational Research, 27 (2010), 103.
doi: 10.1142/S0217595910002569. |
[29] |
H. Xu and D. Zhang, Smooth sample average approximation of stationary points in nonsmooth stochastic optimization and applications,, Mathematical Programming, 119 (2009), 371.
doi: 10.1007/s10107-008-0214-0. |
[30] |
C. Zhang and X. Chen, Stochastic nonlinear complementarity problem and applications to traffic equilibrium under uncertainty,, Journal of Optimization Theory and Applications, 137 (2008), 277.
doi: 10.1007/s10957-008-9358-6. |
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