# American Institute of Mathematical Sciences

October  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 and 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), 421-439. [2] B. C. Eaves, On the basic theorem of complementarity, Mathematical Programming, 1 (1971), 68-75. doi: 10.1007/BF01584073. [3] F. Facchinei and J. S. Pang, "Finite-Dimensional 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), 669-713. doi: 10.1137/S0036144595285963. [5] A. Fischer, Solution of monotone complementarity problems with locally Lipschitzian functions, Mathematical Programming, 76 (1997), 513-532. 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), 343-354. 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), 1817-1825. 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), 227-243. 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), 207-237. doi: 10.1007/s00211-008-0181-7. [10] B. S. He, Inexact implicit methods for monotone general variational inequalities, Mathematical Programming, 86 (1999), 199-217. 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), 129-143. 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), 715-737. [13] M. Li and A. Bnouhachem, A modified inexact operator splitting method for monotone variational inequalities, Journal of Global Optimization, 41 (2008), 417-426. 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), 423-435. 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 "Computational Solution of Nonlinear Systems of Equations" (Fort Collins, CO, 1988), Lectures in Applied Mathematics, 26 Amer. Math. Soc., Providence, RI, (1990), 265-284. [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), 28-41. doi: 10.1137/0103003. [17] R. T. Rockafellar, Monotone operators and proximal point algorithm, SIAM Journal on Control and Optimization, 14 (1976), 877-898. doi: 10.1137/0314056. [18] R. S. Varga, "Matrix Iterative Analysis," Prentice-Hall, 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), 969-979. 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), 453-456.

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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), 421-439. [2] B. C. Eaves, On the basic theorem of complementarity, Mathematical Programming, 1 (1971), 68-75. doi: 10.1007/BF01584073. [3] F. Facchinei and J. S. Pang, "Finite-Dimensional 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), 669-713. doi: 10.1137/S0036144595285963. [5] A. Fischer, Solution of monotone complementarity problems with locally Lipschitzian functions, Mathematical Programming, 76 (1997), 513-532. 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), 343-354. 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), 1817-1825. 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), 227-243. 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), 207-237. doi: 10.1007/s00211-008-0181-7. [10] B. S. He, Inexact implicit methods for monotone general variational inequalities, Mathematical Programming, 86 (1999), 199-217. 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), 129-143. 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), 715-737. [13] M. Li and A. Bnouhachem, A modified inexact operator splitting method for monotone variational inequalities, Journal of Global Optimization, 41 (2008), 417-426. 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), 423-435. 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 "Computational Solution of Nonlinear Systems of Equations" (Fort Collins, CO, 1988), Lectures in Applied Mathematics, 26 Amer. Math. Soc., Providence, RI, (1990), 265-284. [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), 28-41. doi: 10.1137/0103003. [17] R. T. Rockafellar, Monotone operators and proximal point algorithm, SIAM Journal on Control and Optimization, 14 (1976), 877-898. doi: 10.1137/0314056. [18] R. S. Varga, "Matrix Iterative Analysis," Prentice-Hall, 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), 969-979. 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), 453-456.
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