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Smoothing Newton methods for symmetric cone linear complementarity problem with the Cartesian $P$/$P_0$-property
1. | Department of Applied Mathematics, Xidian University, Xi'an 710071, China |
2. | Department of Applied Mathematics, Xidian University, Xi'an, 710071, China |
3. | College of Mathematics and Information, Henan Normal University, Xinxiang 453007, China |
References:
[1] |
X. D. Chen, D. Sun and J. Sun, Complementarity functions and numerical experiments on smoothing Newton methods for second-order-cone complementarity problems, Comput. Optim. Appl., 25 (2003), 39-56.
doi: 10.1023/A:1022996819381. |
[2] |
X. Chen and H. D. Qi, Cartesian P-property and its applications to the semidefinite linear complementarity problem, Math. Program., Ser. A 106 (2006), 177-201. |
[3] |
X. J. Chen and Y. Y. Ye, On homotopy-smoothing methods for box-constrained variational inequalities, SIAM J. Control. Optim., 37 (1999), 589-616.
doi: 10.1137/S0363012997315907. |
[4] |
X. J. Chen, Smoothing methods for complementarity problems and their applications: A survey, J. Oper. Res. Soc. Japan, 43 (2000), 32-47.
doi: 10.1016/S0453-4514(00)88750-5. |
[5] |
X. J. Chen and S. H. Xiang, Computation of error bounds for P-matrix linear complementarity problems, Math. Program., Ser. A 106 (2006), 513-525. |
[6] |
X. H. Chen and C. F. Ma, A regularization smoothing Newtone method for solving nonlinear complementarity problem, Nonlinear Anal. Real World Appl., 10 (2009), 1702-1711.
doi: 10.1016/j.nonrwa.2008.02.010. |
[7] |
F. H. Clarke, "Optimization and Nonsmooth Analysis," Wiley Press, New York, 1983. |
[8] |
J. Faraut and A. Koranyi, "Analysis on Symmetric Cones," Clarendon Press, Oxford, 1994. |
[9] |
F. Facchinei and F. Pang, "Finite-dimensional Variational Ineqaulities and Complementarity Problems, vol. I and II," Springer-Verlag, New York, 2003. |
[10] |
F. Facchinei and C. Kanzow, Beyond monotonicity in regularization methods for nonlinear complementarity problems, SIAM J. Control Optim., 37 (1999), 1150-1161.
doi: 10.1137/S0363012997322935. |
[11] |
M. S. Gowda, R. Sznajder and J. Tao, Some P-properties for linear transformations on Euclidean Jordan algebras, Linear Algebra Appl., 393 (2004), 203-232.
doi: 10.1016/j.laa.2004.03.028. |
[12] |
D. R. Han, On the coerciveness of some merit functions for complementarity problems over symmetric cones, J. Math. Anal. Appl., 336 (2007), 727-737.
doi: 10.1016/j.jmaa.2007.03.003. |
[13] |
Z. H. Huang and T. Ni, Smoothing algorithms for complementarity problems over symmetric cones, Comput. Optim. Appl., 45 (2010), 557-579.
doi: 10.1007/s10589-008-9180-y. |
[14] |
Z. H. Huang, L. Q. Qi and D. F. Sun, Sub-quadratic convergence of a smoothing Newton algorithm for the $P_0$- and monotone LCP, Math. Program., Ser. A, 99 (2004), 423-441. |
[15] |
L. C. Kong, J. Sun, and N. H. Xiu, A regularized smoothing Newton method for symmetric cone complementarity problems, SIAM J. Optim., 19 (2008), 1028-1047.
doi: 10.1137/060676775. |
[16] |
L. C. Kong, L. Tuncel and N. H. Xiu, Vector-valued implicit Lagrangian for Symmetric cone complementarity problems, Asia-Pac. J. Oper. Res., 26 (2009), 199-233. |
[17] |
M. S. Lobo, L. Vandenberghe, S. Boyd and H. Lebret, Application of second-order cone programming, Linear Algebra Appl., 284 (1998), 193-228.
doi: 10.1016/S0024-3795(98)10032-0. |
[18] |
Y. J. Liu, L. W. Zhang and Y. H. Wang, Some properties of a class of merit functions for symmetric cone complementarity problems, Asia-Pac. J. Oper. Res., 23 (2006), 473-495. |
[19] |
Y. J. Liu, L. W. Zhang and M. J. Liu, Extension of smoothing functions to symmetric cone complementarity problems, Appl. Math. J. Chinese Univ. Ser. B, 22 (2007), 245-252.
doi: 10.1007/s11766-007-0214-5. |
[20] |
Y. J. Liu, L. W. Zhang and Y. H. Wang, Analysis of a smoothing method for symmetric conic linear programming, J. Appl. Math. Comput., 22 (2006), 133-148.
doi: 10.1007/BF02896466. |
[21] |
X. H. Liu and W. Z. Gu, Smoothing Newton algorithms based on a regularized one-parametric class of smoothing functions for generalized complementarity problems over symmetric cones, J. Ind. Manag Optim., 6 (2010), 363-380. |
[22] |
Z. Y. Luo and N. H. Xiu, Path-following interior point algorithms for the Cartesian $P$*$(\kappa)$-LCP over symmetric cones, Sci. China Ser. A, 52 (2009), 1769-1784.
doi: 10.1007/s11425-008-0174-0. |
[23] |
R. Mifflin, Semismooth and semiconvex functions in constrained optimization, SIAM J. Control Optim., 15 (1977), 959-972.
doi: 10.1137/0315061. |
[24] |
L. Q. Qi, D. F. Sun and L. G. Zhou, A new look at smoothing Newton methods for nonlinear complementarity problems and box constrained variational inequalities, Math. Program. Ser. A, 87 (2000), 1-35. |
[25] |
L. Q. Qi and J. Sun, A nonsmooth version of Newton's method, Math. Program. Ser. A, 58 (1993), 353-367.
doi: 10.1007/BF01581275. |
[26] |
D. F. Sun and J. Sun, Löwner's operator and spectral functions in euclidean Jordan algebras, Math. Oper. Res., 33 (2008), 421-445.
doi: 10.1287/moor.1070.0300. |
[27] |
A. Yoshise, Interior point trajectories and a homogeneous model for nonlinear complementarity problems over symmetric cones, SIAM J. Optim., 17 (2006), 1129-1153.
doi: 10.1137/04061427X. |
[28] |
L. P. Zhang and X. S. Zhang, Global linear and quadratic one-step smoothing Newton method for $P_0$-LCP, J. Global Optim., 25 (2003), 363-376.
doi: 10.1023/A:1022528320719. |
show all references
References:
[1] |
X. D. Chen, D. Sun and J. Sun, Complementarity functions and numerical experiments on smoothing Newton methods for second-order-cone complementarity problems, Comput. Optim. Appl., 25 (2003), 39-56.
doi: 10.1023/A:1022996819381. |
[2] |
X. Chen and H. D. Qi, Cartesian P-property and its applications to the semidefinite linear complementarity problem, Math. Program., Ser. A 106 (2006), 177-201. |
[3] |
X. J. Chen and Y. Y. Ye, On homotopy-smoothing methods for box-constrained variational inequalities, SIAM J. Control. Optim., 37 (1999), 589-616.
doi: 10.1137/S0363012997315907. |
[4] |
X. J. Chen, Smoothing methods for complementarity problems and their applications: A survey, J. Oper. Res. Soc. Japan, 43 (2000), 32-47.
doi: 10.1016/S0453-4514(00)88750-5. |
[5] |
X. J. Chen and S. H. Xiang, Computation of error bounds for P-matrix linear complementarity problems, Math. Program., Ser. A 106 (2006), 513-525. |
[6] |
X. H. Chen and C. F. Ma, A regularization smoothing Newtone method for solving nonlinear complementarity problem, Nonlinear Anal. Real World Appl., 10 (2009), 1702-1711.
doi: 10.1016/j.nonrwa.2008.02.010. |
[7] |
F. H. Clarke, "Optimization and Nonsmooth Analysis," Wiley Press, New York, 1983. |
[8] |
J. Faraut and A. Koranyi, "Analysis on Symmetric Cones," Clarendon Press, Oxford, 1994. |
[9] |
F. Facchinei and F. Pang, "Finite-dimensional Variational Ineqaulities and Complementarity Problems, vol. I and II," Springer-Verlag, New York, 2003. |
[10] |
F. Facchinei and C. Kanzow, Beyond monotonicity in regularization methods for nonlinear complementarity problems, SIAM J. Control Optim., 37 (1999), 1150-1161.
doi: 10.1137/S0363012997322935. |
[11] |
M. S. Gowda, R. Sznajder and J. Tao, Some P-properties for linear transformations on Euclidean Jordan algebras, Linear Algebra Appl., 393 (2004), 203-232.
doi: 10.1016/j.laa.2004.03.028. |
[12] |
D. R. Han, On the coerciveness of some merit functions for complementarity problems over symmetric cones, J. Math. Anal. Appl., 336 (2007), 727-737.
doi: 10.1016/j.jmaa.2007.03.003. |
[13] |
Z. H. Huang and T. Ni, Smoothing algorithms for complementarity problems over symmetric cones, Comput. Optim. Appl., 45 (2010), 557-579.
doi: 10.1007/s10589-008-9180-y. |
[14] |
Z. H. Huang, L. Q. Qi and D. F. Sun, Sub-quadratic convergence of a smoothing Newton algorithm for the $P_0$- and monotone LCP, Math. Program., Ser. A, 99 (2004), 423-441. |
[15] |
L. C. Kong, J. Sun, and N. H. Xiu, A regularized smoothing Newton method for symmetric cone complementarity problems, SIAM J. Optim., 19 (2008), 1028-1047.
doi: 10.1137/060676775. |
[16] |
L. C. Kong, L. Tuncel and N. H. Xiu, Vector-valued implicit Lagrangian for Symmetric cone complementarity problems, Asia-Pac. J. Oper. Res., 26 (2009), 199-233. |
[17] |
M. S. Lobo, L. Vandenberghe, S. Boyd and H. Lebret, Application of second-order cone programming, Linear Algebra Appl., 284 (1998), 193-228.
doi: 10.1016/S0024-3795(98)10032-0. |
[18] |
Y. J. Liu, L. W. Zhang and Y. H. Wang, Some properties of a class of merit functions for symmetric cone complementarity problems, Asia-Pac. J. Oper. Res., 23 (2006), 473-495. |
[19] |
Y. J. Liu, L. W. Zhang and M. J. Liu, Extension of smoothing functions to symmetric cone complementarity problems, Appl. Math. J. Chinese Univ. Ser. B, 22 (2007), 245-252.
doi: 10.1007/s11766-007-0214-5. |
[20] |
Y. J. Liu, L. W. Zhang and Y. H. Wang, Analysis of a smoothing method for symmetric conic linear programming, J. Appl. Math. Comput., 22 (2006), 133-148.
doi: 10.1007/BF02896466. |
[21] |
X. H. Liu and W. Z. Gu, Smoothing Newton algorithms based on a regularized one-parametric class of smoothing functions for generalized complementarity problems over symmetric cones, J. Ind. Manag Optim., 6 (2010), 363-380. |
[22] |
Z. Y. Luo and N. H. Xiu, Path-following interior point algorithms for the Cartesian $P$*$(\kappa)$-LCP over symmetric cones, Sci. China Ser. A, 52 (2009), 1769-1784.
doi: 10.1007/s11425-008-0174-0. |
[23] |
R. Mifflin, Semismooth and semiconvex functions in constrained optimization, SIAM J. Control Optim., 15 (1977), 959-972.
doi: 10.1137/0315061. |
[24] |
L. Q. Qi, D. F. Sun and L. G. Zhou, A new look at smoothing Newton methods for nonlinear complementarity problems and box constrained variational inequalities, Math. Program. Ser. A, 87 (2000), 1-35. |
[25] |
L. Q. Qi and J. Sun, A nonsmooth version of Newton's method, Math. Program. Ser. A, 58 (1993), 353-367.
doi: 10.1007/BF01581275. |
[26] |
D. F. Sun and J. Sun, Löwner's operator and spectral functions in euclidean Jordan algebras, Math. Oper. Res., 33 (2008), 421-445.
doi: 10.1287/moor.1070.0300. |
[27] |
A. Yoshise, Interior point trajectories and a homogeneous model for nonlinear complementarity problems over symmetric cones, SIAM J. Optim., 17 (2006), 1129-1153.
doi: 10.1137/04061427X. |
[28] |
L. P. Zhang and X. S. Zhang, Global linear and quadratic one-step smoothing Newton method for $P_0$-LCP, J. Global Optim., 25 (2003), 363-376.
doi: 10.1023/A:1022528320719. |
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