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Duality in linear programming: From trichotomy to quadrichotomy
Global convergence of an inexact operator splitting method for monotone variational inequalities
1.  School of Mathematical Sciences, Key Laboratory for NSLSCS of Jiangsu Province, Nanjing Normal University, Nanjing 210046, China, China 
2.  School of Computer Sciences, Nanjing Normal University, Nanjing 210097, China 
References:
[1] 
J. Douglas and H. H. Rachford, On the numerical solution of the heat conduction problem in 2 and 3 space variables, Transactions of American Mathematical Society, 82 (1956), 421439. 
[2] 
B. C. Eaves, On the basic theorem of complementarity, Mathematical Programming, 1 (1971), 6875. doi: 10.1007/BF01584073. 
[3] 
F. Facchinei and J. S. Pang, "FiniteDimensional Variational Inequalities and Complementarity Problems," Volumes I and II, Springer Verlag, Berlin, 2003. 
[4] 
M. C. Ferris and J. S. Pang, Engineering and economic applications of complimentarity problems, SIAM Review, 39 (1997), 669713. doi: 10.1137/S0036144595285963. 
[5] 
A. Fischer, Solution of monotone complementarity problems with locally Lipschitzian functions, Mathematical Programming, 76 (1997), 513532. doi: 10.1007/BF02614396. 
[6] 
D. R. Han and B. S. He, A new accuracy criterion for approximate proximal point algorithms, Journal of Mathematical Analysis and Applications, 263 (2001), 343354. doi: 10.1006/jmaa.2001.7535. 
[7] 
D. R. Han and W. Y. Sun, A new modified GoldsteinLevitinPolyak projection method for variational inequality problems, Computers and Mathematics with Applications, 47 (2004), 18171825. doi: 10.1016/j.camwa.2003.12.002. 
[8] 
D. R. Han, Inexact operator splitting methods with selfadaptive strategy for variational inequality problems, Journal of Optimization Theory and Applications, 132 (2007), 227243. doi: 10.1007/s1095700690605. 
[9] 
D. R. Han, W. Xu and H. Yang, An operator splitting method for variational inequalities with partially unknown mappings, Numerische Mathematik, 111 (2008), 207237. doi: 10.1007/s0021100801817. 
[10] 
B. S. He, Inexact implicit methods for monotone general variational inequalities, Mathematical Programming, 86 (1999), 199217. doi: 10.1007/s101070050086. 
[11] 
B. S. He, H. Yang, Q. Meng and D. R. Han, Modified GoldsteinLevitinPolyak projection method for asymmetric strongly monotone variational inequalities, Journal of Optimization Theory and Applications, 112 (2002), 129143. doi: 10.1023/A:1013048729944. 
[12] 
B. S. He, L. Z. Liao and S. L. Wang, Selfadaptive operator splitting methods for monotone variational inequalities, Numerische Mathematik, 94 (2003), 715737. 
[13] 
M. Li and A. Bnouhachem, A modified inexact operator splitting method for monotone variational inequalities, Journal of Global Optimization, 41 (2008), 417426. doi: 10.1007/s108980079229y. 
[14] 
M. Aslam Noor, Y. J. Wang, and N. H. Xiu, Some new projection methods for variational inequalities, Applied Mathematics and Computation, 137 (2003), 423435. doi: 10.1016/S00963003(02)001480. 
[15] 
J. S. Pang and P. T. Harker, A dampedNewton method for the linear complementarity problem, in "Computational Solution of Nonlinear Systems of Equations" (Fort Collins, CO, 1988), Lectures in Applied Mathematics, 26 Amer. Math. Soc., Providence, RI, (1990), 265284. 
[16] 
D. W. Peaceman and H. H. Rachford, The numerical solution of parabolic elliptic differential equations, Journal of the Society of Industry and Applied Mathematics, 3 (1955), 2841. doi: 10.1137/0103003. 
[17] 
R. T. Rockafellar, Monotone operators and proximal point algorithm, SIAM Journal on Control and Optimization, 14 (1976), 877898. doi: 10.1137/0314056. 
[18] 
R. S. Varga, "Matrix Iterative Analysis," PrenticeHall, Inc., Englewood Cliffs, New Jersey, 1962. 
[19] 
Y. Wang, N. Xiu and C. Wang, A new version of extragradient method for variational inequality problems, Computers and Mathematics with Applications, 42 (2001), 969979. doi: 10.1016/S08981221(01)002139. 
[20] 
T. Zhu and Z. G. Yu, A simple proof for some important properties of the projection mapping, Mathematical Inequalities and Applications, 7 (2004), 453456. 
show all references
References:
[1] 
J. Douglas and H. H. Rachford, On the numerical solution of the heat conduction problem in 2 and 3 space variables, Transactions of American Mathematical Society, 82 (1956), 421439. 
[2] 
B. C. Eaves, On the basic theorem of complementarity, Mathematical Programming, 1 (1971), 6875. doi: 10.1007/BF01584073. 
[3] 
F. Facchinei and J. S. Pang, "FiniteDimensional Variational Inequalities and Complementarity Problems," Volumes I and II, Springer Verlag, Berlin, 2003. 
[4] 
M. C. Ferris and J. S. Pang, Engineering and economic applications of complimentarity problems, SIAM Review, 39 (1997), 669713. doi: 10.1137/S0036144595285963. 
[5] 
A. Fischer, Solution of monotone complementarity problems with locally Lipschitzian functions, Mathematical Programming, 76 (1997), 513532. doi: 10.1007/BF02614396. 
[6] 
D. R. Han and B. S. He, A new accuracy criterion for approximate proximal point algorithms, Journal of Mathematical Analysis and Applications, 263 (2001), 343354. doi: 10.1006/jmaa.2001.7535. 
[7] 
D. R. Han and W. Y. Sun, A new modified GoldsteinLevitinPolyak projection method for variational inequality problems, Computers and Mathematics with Applications, 47 (2004), 18171825. doi: 10.1016/j.camwa.2003.12.002. 
[8] 
D. R. Han, Inexact operator splitting methods with selfadaptive strategy for variational inequality problems, Journal of Optimization Theory and Applications, 132 (2007), 227243. doi: 10.1007/s1095700690605. 
[9] 
D. R. Han, W. Xu and H. Yang, An operator splitting method for variational inequalities with partially unknown mappings, Numerische Mathematik, 111 (2008), 207237. doi: 10.1007/s0021100801817. 
[10] 
B. S. He, Inexact implicit methods for monotone general variational inequalities, Mathematical Programming, 86 (1999), 199217. doi: 10.1007/s101070050086. 
[11] 
B. S. He, H. Yang, Q. Meng and D. R. Han, Modified GoldsteinLevitinPolyak projection method for asymmetric strongly monotone variational inequalities, Journal of Optimization Theory and Applications, 112 (2002), 129143. doi: 10.1023/A:1013048729944. 
[12] 
B. S. He, L. Z. Liao and S. L. Wang, Selfadaptive operator splitting methods for monotone variational inequalities, Numerische Mathematik, 94 (2003), 715737. 
[13] 
M. Li and A. Bnouhachem, A modified inexact operator splitting method for monotone variational inequalities, Journal of Global Optimization, 41 (2008), 417426. doi: 10.1007/s108980079229y. 
[14] 
M. Aslam Noor, Y. J. Wang, and N. H. Xiu, Some new projection methods for variational inequalities, Applied Mathematics and Computation, 137 (2003), 423435. doi: 10.1016/S00963003(02)001480. 
[15] 
J. S. Pang and P. T. Harker, A dampedNewton method for the linear complementarity problem, in "Computational Solution of Nonlinear Systems of Equations" (Fort Collins, CO, 1988), Lectures in Applied Mathematics, 26 Amer. Math. Soc., Providence, RI, (1990), 265284. 
[16] 
D. W. Peaceman and H. H. Rachford, The numerical solution of parabolic elliptic differential equations, Journal of the Society of Industry and Applied Mathematics, 3 (1955), 2841. doi: 10.1137/0103003. 
[17] 
R. T. Rockafellar, Monotone operators and proximal point algorithm, SIAM Journal on Control and Optimization, 14 (1976), 877898. doi: 10.1137/0314056. 
[18] 
R. S. Varga, "Matrix Iterative Analysis," PrenticeHall, Inc., Englewood Cliffs, New Jersey, 1962. 
[19] 
Y. Wang, N. Xiu and C. Wang, A new version of extragradient method for variational inequality problems, Computers and Mathematics with Applications, 42 (2001), 969979. doi: 10.1016/S08981221(01)002139. 
[20] 
T. Zhu and Z. G. Yu, A simple proof for some important properties of the projection mapping, Mathematical Inequalities and Applications, 7 (2004), 453456. 
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