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 & 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.   Google Scholar

[2]

B. C. Eaves, On the basic theorem of complementarity,, Mathematical Programming, 1 (1971), 68.  doi: 10.1007/BF01584073.  Google Scholar

[3]

F. Facchinei and J. S. Pang, "Finite-Dimensional Variational Inequalities and Complementarity Problems,", Volumes I and II, (2003).   Google Scholar

[4]

M. C. Ferris and J. S. Pang, Engineering and economic applications of complimentarity problems,, SIAM Review, 39 (1997), 669.  doi: 10.1137/S0036144595285963.  Google Scholar

[5]

A. Fischer, Solution of monotone complementarity problems with locally Lipschitzian functions,, Mathematical Programming, 76 (1997), 513.  doi: 10.1007/BF02614396.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[10]

B. S. He, Inexact implicit methods for monotone general variational inequalities,, Mathematical Programming, 86 (1999), 199.  doi: 10.1007/s101070050086.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[15]

J. S. Pang and P. T. Harker, A damped-Newton method for the linear complementarity problem,, in, 26 (1990), 265.   Google Scholar

[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.  Google Scholar

[17]

R. T. Rockafellar, Monotone operators and proximal point algorithm,, SIAM Journal on Control and Optimization, 14 (1976), 877.  doi: 10.1137/0314056.  Google Scholar

[18]

R. S. Varga, "Matrix Iterative Analysis,", Prentice-Hall, (1962).   Google Scholar

[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.  Google Scholar

[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.   Google Scholar

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), 421.   Google Scholar

[2]

B. C. Eaves, On the basic theorem of complementarity,, Mathematical Programming, 1 (1971), 68.  doi: 10.1007/BF01584073.  Google Scholar

[3]

F. Facchinei and J. S. Pang, "Finite-Dimensional Variational Inequalities and Complementarity Problems,", Volumes I and II, (2003).   Google Scholar

[4]

M. C. Ferris and J. S. Pang, Engineering and economic applications of complimentarity problems,, SIAM Review, 39 (1997), 669.  doi: 10.1137/S0036144595285963.  Google Scholar

[5]

A. Fischer, Solution of monotone complementarity problems with locally Lipschitzian functions,, Mathematical Programming, 76 (1997), 513.  doi: 10.1007/BF02614396.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[10]

B. S. He, Inexact implicit methods for monotone general variational inequalities,, Mathematical Programming, 86 (1999), 199.  doi: 10.1007/s101070050086.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[15]

J. S. Pang and P. T. Harker, A damped-Newton method for the linear complementarity problem,, in, 26 (1990), 265.   Google Scholar

[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.  Google Scholar

[17]

R. T. Rockafellar, Monotone operators and proximal point algorithm,, SIAM Journal on Control and Optimization, 14 (1976), 877.  doi: 10.1137/0314056.  Google Scholar

[18]

R. S. Varga, "Matrix Iterative Analysis,", Prentice-Hall, (1962).   Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[1]

Lateef Olakunle Jolaoso, Maggie Aphane. Bregman subgradient extragradient method with monotone self-adjustment stepsize for solving pseudo-monotone variational inequalities and fixed point problems. Journal of Industrial & Management Optimization, 2020  doi: 10.3934/jimo.2020178

[2]

Xing-Bin Pan. Variational and operator methods for Maxwell-Stokes system. Discrete & Continuous Dynamical Systems - A, 2020, 40 (6) : 3909-3955. doi: 10.3934/dcds.2020036

[3]

Shipra Singh, Aviv Gibali, Xiaolong Qin. Cooperation in traffic network problems via evolutionary split variational inequalities. Journal of Industrial & Management Optimization, 2020  doi: 10.3934/jimo.2020170

[4]

Shengxin Zhu, Tongxiang Gu, Xingping Liu. AIMS: Average information matrix splitting. Mathematical Foundations of Computing, 2020, 3 (4) : 301-308. doi: 10.3934/mfc.2020012

[5]

Sören Bartels, Jakob Keck. Adaptive time stepping in elastoplasticity. Discrete & Continuous Dynamical Systems - S, 2021, 14 (1) : 71-88. doi: 10.3934/dcdss.2020323

[6]

Franck Davhys Reval Langa, Morgan Pierre. A doubly splitting scheme for the Caginalp system with singular potentials and dynamic boundary conditions. Discrete & Continuous Dynamical Systems - S, 2021, 14 (2) : 653-676. doi: 10.3934/dcdss.2020353

[7]

Andrew D. Lewis. Erratum for "nonholonomic and constrained variational mechanics". Journal of Geometric Mechanics, 2020, 12 (4) : 671-675. doi: 10.3934/jgm.2020033

[8]

José Madrid, João P. G. Ramos. On optimal autocorrelation inequalities on the real line. Communications on Pure & Applied Analysis, 2021, 20 (1) : 369-388. doi: 10.3934/cpaa.2020271

[9]

Hong Niu, Zhijiang Feng, Qijin Xiao, Yajun Zhang. A PID control method based on optimal control strategy. Numerical Algebra, Control & Optimization, 2021, 11 (1) : 117-126. doi: 10.3934/naco.2020019

[10]

Sumit Arora, Manil T. Mohan, Jaydev Dabas. Approximate controllability of a Sobolev type impulsive functional evolution system in Banach spaces. Mathematical Control & Related Fields, 2020  doi: 10.3934/mcrf.2020049

[11]

Jian Zhang, Tony T. Lee, Tong Ye, Liang Huang. An approximate mean queue length formula for queueing systems with varying service rate. Journal of Industrial & Management Optimization, 2021, 17 (1) : 185-204. doi: 10.3934/jimo.2019106

[12]

Constantine M. Dafermos. A variational approach to the Riemann problem for hyperbolic conservation laws. Discrete & Continuous Dynamical Systems - A, 2009, 23 (1&2) : 185-195. doi: 10.3934/dcds.2009.23.185

[13]

Indranil Chowdhury, Gyula Csató, Prosenjit Roy, Firoj Sk. Study of fractional Poincaré inequalities on unbounded domains. Discrete & Continuous Dynamical Systems - A, 2020  doi: 10.3934/dcds.2020394

[14]

Yantao Wang, Linlin Su. Monotone and nonmonotone clines with partial panmixia across a geographical barrier. Discrete & Continuous Dynamical Systems - A, 2020, 40 (6) : 4019-4037. doi: 10.3934/dcds.2020056

[15]

Mengyu Cheng, Zhenxin Liu. Periodic, almost periodic and almost automorphic solutions for SPDEs with monotone coefficients. Discrete & Continuous Dynamical Systems - B, 2021  doi: 10.3934/dcdsb.2021026

[16]

Philippe G. Ciarlet, Liliana Gratie, Cristinel Mardare. Intrinsic methods in elasticity: a mathematical survey. Discrete & Continuous Dynamical Systems - A, 2009, 23 (1&2) : 133-164. doi: 10.3934/dcds.2009.23.133

[17]

Mostafa Mbekhta. Representation and approximation of the polar factor of an operator on a Hilbert space. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020463

[18]

Ole Løseth Elvetun, Bjørn Fredrik Nielsen. A regularization operator for source identification for elliptic PDEs. Inverse Problems & Imaging, , () : -. doi: 10.3934/ipi.2021006

[19]

Li-Bin Liu, Ying Liang, Jian Zhang, Xiaobing Bao. A robust adaptive grid method for singularly perturbed Burger-Huxley equations. Electronic Research Archive, 2020, 28 (4) : 1439-1457. doi: 10.3934/era.2020076

[20]

Ke Su, Yumeng Lin, Chun Xu. A new adaptive method to nonlinear semi-infinite programming. Journal of Industrial & Management Optimization, 2020  doi: 10.3934/jimo.2021012

2019 Impact Factor: 1.366

Metrics

  • PDF downloads (34)
  • HTML views (0)
  • Cited by (3)

Other articles
by authors

[Back to Top]