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A nonmonotone smoothing Newton algorithm for solving box constrained variational inequalities with a $P_0$ function
1. | College of Science, Civil Aviation University of China, Tianjin 300300, China |
2. | Department of Mathematics, School of Science, Tianjin University, Tianjin 300072 |
References:
[1] |
S. C. Billups, S. P. Dirkse and M. C. Soares, A comparison of algorithms for large scale mixed complementarity problems, Comput. Optim. Appl., 7 (1997), 3-25.
doi: 10.1023/A:1008632215341. |
[2] |
B. Chen and X. Chen, A global linear and local quadratic continuation smoothing method for variational inequalities with box constraints, Comput. Optim. Appl., 13 (2000), 131-158.
doi: 10.1023/A:1026546230851. |
[3] |
C. Chen and O. L. Mangasarian, A class of smoothing functions for nonlinear and mixed complementarity problems, Comput. Optim. Appl., 5 (1996), 97-138.
doi: 10.1007/BF00249052. |
[4] |
J.-S. Chen, The semismooth-related properties of a merit function and a descent method for the nonlinear complementarity problem, J. Global Optim., 36 (2006), 565-580.
doi: 10.1007/s10898-006-9027-y. |
[5] |
J.-S. Chen and P. H. Pan, A family of NCP functions and a descent method for the nonlinear complementarity problem, Comput. Optim. Appl., 40 (2008), 389-404.
doi: 10.1007/s10589-007-9086-0. |
[6] |
J.-S. Chen, S. H. Pan and T. C. Lin, A smoothing Newton method based on the generalized Fischer-Burmeister function for MCPs, Nonlinear Anal.: TMA, 72 (2010), 3739-3758.
doi: 10.1016/j.na.2010.01.012. |
[7] |
J.-S. Chen, S. H. Pan and C. Y. Yang, Numerical comparisons of two effective method for mixed complementarity problems, J. Comput. Appl. Math., 234 (2010), 667-683.
doi: 10.1016/j.cam.2010.01.004. |
[8] |
X. Chen, L. Qi and D. Sun, Global and superlinear convergence of the smoothing Newton method and its application to general box constrained variational inequalities, Math. Comput., 67 (1998), 519-540.
doi: 10.1090/S0025-5718-98-00932-6. |
[9] |
X. Chen and Y. Ye, On homotopy-smoothing methods for box-constrained variational inequalities, SIAM J. Control Optim., 37 (1999), 589-616.
doi: 10.1137/S0363012997315907. |
[10] |
F. Facchinei, A. Fischer and C. Kanzow, A semismooth Newton method for variational inequalities: The case of box constraints, in "Complementarity and Variational Problems: State of the Art" (eds. M. C. Ferris and J. S. Pang), SIAM: Philadelphia, (1997), 76-90. |
[11] |
M. Ferris, C. Kanzow and T. S. Munson, Feasible descent algorithms for mixed complentarity problems, Math. Program., 86 (1999), 475-497.
doi: 10.1007/s101070050101. |
[12] |
A. Fischer, Solution of monotone complementarity problems with Lipschitzian functions, Math. Program., 76 (1997), 513-532.
doi: 10.1007/BF02614396. |
[13] |
M. S. Gowda and J. J. Sznajder, Weak univalence and connectedness of inverse images of continuous functions, Math. Oper. Res., 24 (1999), 255-261.
doi: 10.1287/moor.24.1.255. |
[14] |
M. S. Gowda and M. A. Tawhid, Existence and limiting behavior of trajectories associated with $P_0$-equations, Comput. Optim. Appl., 12 (1999), 229-251.
doi: 10.1023/A:1008688302346. |
[15] |
S. L. Hu and Z. H. Huang, Smoothness of a class of generalized merit functions for the second-order cone complementarity problem, Pacific J. Optim., 6 (2010), 551-571. |
[16] |
S. L. Hu, Z. H. Huang and J.-S. Chen, Properties of a family of generalized NCP-functions and a derivative free algorithm for complementarity problems, J. Comput. Appl. Math., 230 (2009), 69-82.
doi: 10.1016/j.cam.2008.10.056. |
[17] |
S. L. Hu, Z. H. Huang and N. Lu, A non-monotone line search algorithm for unconstrained optimization, J. Sci. Comput., 42 (2010), 38-53.
doi: 10.1007/s10915-009-9314-0. |
[18] |
S. L. Hu, Z.H. Huang and P. Wang, A nonmonotone smoothing Newton algorithm for solving nonlinear complementarity problems, Optim. Methods Softw., 24 (2009), 447-460.
doi: 10.1080/10556780902769862. |
[19] |
Z. H. Huang, L. Qi and D. Sun, Sub-quadratic convergence of a smoothing Newton algorithm for the $P_0$- and monotone LCP, Math. Program., 99 (2004), 423-441.
doi: 10.1007/s10107-003-0457-8. |
[20] |
C. Kanzow and M. Fukushima, Theoretical and numerical investigation of the D-gap function for box constrained variational inequalities, Math. Program., 83 (1998), 55-87.
doi: 10.1007/BF02680550. |
[21] |
C. Kanzow and S. Petra, Projected filter trust region methods for a semismooth least squares formulation of mixed complementarity problems, Optim. Methods Softw., 22 (2007), 713-735.
doi: 10.1080/10556780701296455. |
[22] |
D. Li and M. Fukushima, Smoothing Newton and quasi-Newton methods for mixed complementarity problems, Comput. Optim. Appl., 17 (2000), 203-230.
doi: 10.1023/A:1026502415830. |
[23] |
J. M. Peng, C. Kanzow and M. Fukushima, A hybrid Josephy-Newton method for solving box constrained variational inequality problems via the D-gap function, Optim. Methods Softw., 10 (1999), 687-710.
doi: 10.1080/10556789908805734. |
[24] |
H. D. Qi, A regularized smoothing Newton method for box constrained variational inequality problems with $P_0$-functions, SIAM J. Optim., 10 (2000), 315-330.
doi: 10.1137/S1052623497324047. |
[25] |
L. Qi, D. Sun and G. Zhou, A new look at smoothing Newton methods for nonlinear complementarity problems and box constrained variational inequality problems, Math. Program., 87 (2000), 1-35. |
[26] |
L. Qi and J. Sun, A nonsmooth version of Newton's method, Math. Program., 58 (1993), 353-367.
doi: 10.1007/BF01581275. |
[27] |
G. Ravindran and M. S. Gowda, Regularization of $P_0$-functions in box variational inequality problems, SIAM J. Optim., 11 (2001), 748-760.
doi: 10.1137/S1052623497329567. |
[28] |
D. Sun and R. S. Womersley, A new unconstrained differentiable merit function for box constrained variational inequality problems and a damped Guass-Newton method, SIAM J. Optim., 9 (1999), 388-413.
doi: 10.1137/S1052623496314173. |
show all references
References:
[1] |
S. C. Billups, S. P. Dirkse and M. C. Soares, A comparison of algorithms for large scale mixed complementarity problems, Comput. Optim. Appl., 7 (1997), 3-25.
doi: 10.1023/A:1008632215341. |
[2] |
B. Chen and X. Chen, A global linear and local quadratic continuation smoothing method for variational inequalities with box constraints, Comput. Optim. Appl., 13 (2000), 131-158.
doi: 10.1023/A:1026546230851. |
[3] |
C. Chen and O. L. Mangasarian, A class of smoothing functions for nonlinear and mixed complementarity problems, Comput. Optim. Appl., 5 (1996), 97-138.
doi: 10.1007/BF00249052. |
[4] |
J.-S. Chen, The semismooth-related properties of a merit function and a descent method for the nonlinear complementarity problem, J. Global Optim., 36 (2006), 565-580.
doi: 10.1007/s10898-006-9027-y. |
[5] |
J.-S. Chen and P. H. Pan, A family of NCP functions and a descent method for the nonlinear complementarity problem, Comput. Optim. Appl., 40 (2008), 389-404.
doi: 10.1007/s10589-007-9086-0. |
[6] |
J.-S. Chen, S. H. Pan and T. C. Lin, A smoothing Newton method based on the generalized Fischer-Burmeister function for MCPs, Nonlinear Anal.: TMA, 72 (2010), 3739-3758.
doi: 10.1016/j.na.2010.01.012. |
[7] |
J.-S. Chen, S. H. Pan and C. Y. Yang, Numerical comparisons of two effective method for mixed complementarity problems, J. Comput. Appl. Math., 234 (2010), 667-683.
doi: 10.1016/j.cam.2010.01.004. |
[8] |
X. Chen, L. Qi and D. Sun, Global and superlinear convergence of the smoothing Newton method and its application to general box constrained variational inequalities, Math. Comput., 67 (1998), 519-540.
doi: 10.1090/S0025-5718-98-00932-6. |
[9] |
X. Chen and Y. Ye, On homotopy-smoothing methods for box-constrained variational inequalities, SIAM J. Control Optim., 37 (1999), 589-616.
doi: 10.1137/S0363012997315907. |
[10] |
F. Facchinei, A. Fischer and C. Kanzow, A semismooth Newton method for variational inequalities: The case of box constraints, in "Complementarity and Variational Problems: State of the Art" (eds. M. C. Ferris and J. S. Pang), SIAM: Philadelphia, (1997), 76-90. |
[11] |
M. Ferris, C. Kanzow and T. S. Munson, Feasible descent algorithms for mixed complentarity problems, Math. Program., 86 (1999), 475-497.
doi: 10.1007/s101070050101. |
[12] |
A. Fischer, Solution of monotone complementarity problems with Lipschitzian functions, Math. Program., 76 (1997), 513-532.
doi: 10.1007/BF02614396. |
[13] |
M. S. Gowda and J. J. Sznajder, Weak univalence and connectedness of inverse images of continuous functions, Math. Oper. Res., 24 (1999), 255-261.
doi: 10.1287/moor.24.1.255. |
[14] |
M. S. Gowda and M. A. Tawhid, Existence and limiting behavior of trajectories associated with $P_0$-equations, Comput. Optim. Appl., 12 (1999), 229-251.
doi: 10.1023/A:1008688302346. |
[15] |
S. L. Hu and Z. H. Huang, Smoothness of a class of generalized merit functions for the second-order cone complementarity problem, Pacific J. Optim., 6 (2010), 551-571. |
[16] |
S. L. Hu, Z. H. Huang and J.-S. Chen, Properties of a family of generalized NCP-functions and a derivative free algorithm for complementarity problems, J. Comput. Appl. Math., 230 (2009), 69-82.
doi: 10.1016/j.cam.2008.10.056. |
[17] |
S. L. Hu, Z. H. Huang and N. Lu, A non-monotone line search algorithm for unconstrained optimization, J. Sci. Comput., 42 (2010), 38-53.
doi: 10.1007/s10915-009-9314-0. |
[18] |
S. L. Hu, Z.H. Huang and P. Wang, A nonmonotone smoothing Newton algorithm for solving nonlinear complementarity problems, Optim. Methods Softw., 24 (2009), 447-460.
doi: 10.1080/10556780902769862. |
[19] |
Z. H. Huang, L. Qi and D. Sun, Sub-quadratic convergence of a smoothing Newton algorithm for the $P_0$- and monotone LCP, Math. Program., 99 (2004), 423-441.
doi: 10.1007/s10107-003-0457-8. |
[20] |
C. Kanzow and M. Fukushima, Theoretical and numerical investigation of the D-gap function for box constrained variational inequalities, Math. Program., 83 (1998), 55-87.
doi: 10.1007/BF02680550. |
[21] |
C. Kanzow and S. Petra, Projected filter trust region methods for a semismooth least squares formulation of mixed complementarity problems, Optim. Methods Softw., 22 (2007), 713-735.
doi: 10.1080/10556780701296455. |
[22] |
D. Li and M. Fukushima, Smoothing Newton and quasi-Newton methods for mixed complementarity problems, Comput. Optim. Appl., 17 (2000), 203-230.
doi: 10.1023/A:1026502415830. |
[23] |
J. M. Peng, C. Kanzow and M. Fukushima, A hybrid Josephy-Newton method for solving box constrained variational inequality problems via the D-gap function, Optim. Methods Softw., 10 (1999), 687-710.
doi: 10.1080/10556789908805734. |
[24] |
H. D. Qi, A regularized smoothing Newton method for box constrained variational inequality problems with $P_0$-functions, SIAM J. Optim., 10 (2000), 315-330.
doi: 10.1137/S1052623497324047. |
[25] |
L. Qi, D. Sun and G. Zhou, A new look at smoothing Newton methods for nonlinear complementarity problems and box constrained variational inequality problems, Math. Program., 87 (2000), 1-35. |
[26] |
L. Qi and J. Sun, A nonsmooth version of Newton's method, Math. Program., 58 (1993), 353-367.
doi: 10.1007/BF01581275. |
[27] |
G. Ravindran and M. S. Gowda, Regularization of $P_0$-functions in box variational inequality problems, SIAM J. Optim., 11 (2001), 748-760.
doi: 10.1137/S1052623497329567. |
[28] |
D. Sun and R. S. Womersley, A new unconstrained differentiable merit function for box constrained variational inequality problems and a damped Guass-Newton method, SIAM J. Optim., 9 (1999), 388-413.
doi: 10.1137/S1052623496314173. |
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