# American Institute of Mathematical Sciences

2011, 7(4): 1013-1026. doi: 10.3934/jimo.2011.7.1013

## 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

Received  October 2010 Revised  July 2011 Published  August 2011

Recently, Han (Han D, Inexact operator splitting methods with self-adaptive strategy for variational inequality problems, Journal of Optimization Theory and Applications 132, 227-243 (2007)) proposed an inexact operator splitting method for solving variational inequality problems. It has advantage over the classical operator splitting method of Douglas-Peaceman-Rachford-Varga operator splitting methods (DPRV methods) and some of their variants, since it adopts a very flexible approximate rule in solving the subproblem in each iteration. However, its convergence is established under somewhat stringent condition that the underlying mapping $F$ is strongly monotone. In this paper, we mainly establish the global convergence of the method under weaker condition that the underlying mapping $F$ is monotone, which extends the fields of applications of the method relatively. Some numerical results are also presented to illustrate the method.
Citation: Zhili Ge, Gang Qian, Deren Han. Global convergence of an inexact operator splitting method for monotone variational inequalities. Journal of Industrial & Management Optimization, 2011, 7 (4) : 1013-1026. doi: 10.3934/jimo.2011.7.1013
##### 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), 421. [2] B. C. Eaves, On the basic theorem of complementarity,, Mathematical Programming, 1 (1971), 68. doi: 10.1007/BF01584073. [3] F. Facchinei and J. S. Pang, "Finite-Dimensional Variational Inequalities and Complementarity Problems,", Volumes I and II, (2003). [4] M. C. Ferris and J. S. Pang, Engineering and economic applications of complimentarity problems,, SIAM Review, 39 (1997), 669. doi: 10.1137/S0036144595285963. [5] A. Fischer, Solution of monotone complementarity problems with locally Lipschitzian functions,, Mathematical Programming, 76 (1997), 513. 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), 343. doi: 10.1006/jmaa.2001.7535. [7] D. R. Han and W. Y. Sun, A new modified Goldstein-Levitin-Polyak projection method for variational inequality problems,, Computers and Mathematics with Applications, 47 (2004), 1817. doi: 10.1016/j.camwa.2003.12.002. [8] D. R. Han, Inexact operator splitting methods with self-adaptive strategy for variational inequality problems,, Journal of Optimization Theory and Applications, 132 (2007), 227. doi: 10.1007/s10957-006-9060-5. [9] D. R. Han, W. Xu and H. Yang, An operator splitting method for variational inequalities with partially unknown mappings,, Numerische Mathematik, 111 (2008), 207. doi: 10.1007/s00211-008-0181-7. [10] B. S. He, Inexact implicit methods for monotone general variational inequalities,, Mathematical Programming, 86 (1999), 199. doi: 10.1007/s101070050086. [11] B. S. He, H. Yang, Q. Meng and D. R. Han, Modified Goldstein-Levitin-Polyak projection method for asymmetric strongly monotone variational inequalities,, Journal of Optimization Theory and Applications, 112 (2002), 129. doi: 10.1023/A:1013048729944. [12] B. S. He, L. Z. Liao and S. L. Wang, Self-adaptive operator splitting methods for monotone variational inequalities,, Numerische Mathematik, 94 (2003), 715. [13] M. Li and A. Bnouhachem, A modified inexact operator splitting method for monotone variational inequalities,, Journal of Global Optimization, 41 (2008), 417. doi: 10.1007/s10898-007-9229-y. [14] M. Aslam Noor, Y. J. Wang, and N. H. Xiu, Some new projection methods for variational inequalities,, Applied Mathematics and Computation, 137 (2003), 423. doi: 10.1016/S0096-3003(02)00148-0. [15] J. S. Pang and P. T. Harker, A damped-Newton method for the linear complementarity problem,, in, 26 (1990), 265. [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), 28. doi: 10.1137/0103003. [17] R. T. Rockafellar, Monotone operators and proximal point algorithm,, SIAM Journal on Control and Optimization, 14 (1976), 877. doi: 10.1137/0314056. [18] R. S. Varga, "Matrix Iterative Analysis,", Prentice-Hall, (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), 969. doi: 10.1016/S0898-1221(01)00213-9. [20] T. Zhu and Z. G. Yu, A simple proof for some important properties of the projection mapping,, Mathematical Inequalities and Applications, 7 (2004), 453.

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##### 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), 421. [2] B. C. Eaves, On the basic theorem of complementarity,, Mathematical Programming, 1 (1971), 68. doi: 10.1007/BF01584073. [3] F. Facchinei and J. S. Pang, "Finite-Dimensional Variational Inequalities and Complementarity Problems,", Volumes I and II, (2003). [4] M. C. Ferris and J. S. Pang, Engineering and economic applications of complimentarity problems,, SIAM Review, 39 (1997), 669. doi: 10.1137/S0036144595285963. [5] A. Fischer, Solution of monotone complementarity problems with locally Lipschitzian functions,, Mathematical Programming, 76 (1997), 513. 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), 343. doi: 10.1006/jmaa.2001.7535. [7] D. R. Han and W. Y. Sun, A new modified Goldstein-Levitin-Polyak projection method for variational inequality problems,, Computers and Mathematics with Applications, 47 (2004), 1817. doi: 10.1016/j.camwa.2003.12.002. [8] D. R. Han, Inexact operator splitting methods with self-adaptive strategy for variational inequality problems,, Journal of Optimization Theory and Applications, 132 (2007), 227. doi: 10.1007/s10957-006-9060-5. [9] D. R. Han, W. Xu and H. Yang, An operator splitting method for variational inequalities with partially unknown mappings,, Numerische Mathematik, 111 (2008), 207. doi: 10.1007/s00211-008-0181-7. [10] B. S. He, Inexact implicit methods for monotone general variational inequalities,, Mathematical Programming, 86 (1999), 199. doi: 10.1007/s101070050086. [11] B. S. He, H. Yang, Q. Meng and D. R. Han, Modified Goldstein-Levitin-Polyak projection method for asymmetric strongly monotone variational inequalities,, Journal of Optimization Theory and Applications, 112 (2002), 129. doi: 10.1023/A:1013048729944. [12] B. S. He, L. Z. Liao and S. L. Wang, Self-adaptive operator splitting methods for monotone variational inequalities,, Numerische Mathematik, 94 (2003), 715. [13] M. Li and A. Bnouhachem, A modified inexact operator splitting method for monotone variational inequalities,, Journal of Global Optimization, 41 (2008), 417. doi: 10.1007/s10898-007-9229-y. [14] M. Aslam Noor, Y. J. Wang, and N. H. Xiu, Some new projection methods for variational inequalities,, Applied Mathematics and Computation, 137 (2003), 423. doi: 10.1016/S0096-3003(02)00148-0. [15] J. S. Pang and P. T. Harker, A damped-Newton method for the linear complementarity problem,, in, 26 (1990), 265. [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), 28. doi: 10.1137/0103003. [17] R. T. Rockafellar, Monotone operators and proximal point algorithm,, SIAM Journal on Control and Optimization, 14 (1976), 877. doi: 10.1137/0314056. [18] R. S. Varga, "Matrix Iterative Analysis,", Prentice-Hall, (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), 969. doi: 10.1016/S0898-1221(01)00213-9. [20] T. Zhu and Z. G. Yu, A simple proof for some important properties of the projection mapping,, Mathematical Inequalities and Applications, 7 (2004), 453.
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