# American Institute of Mathematical Sciences

• Previous Article
Global optimality conditions for some classes of polynomial integer programming problems
• JIMO Home
• This Issue
• Next Article
A hybrid particle swarm optimization and tabu search algorithm for order planning problems of steel factories based on the Make-To-Stock and Make-To-Order management architecture
January  2011, 7(1): 53-66. doi: 10.3934/jimo.2011.7.53

## 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

Received  April 2010 Revised  September 2010 Published  January 2011

A smoothing Newton method based on the CHKS smoothing function for a class of non-monotone symmetric cone linear complementarity problem (SCLCP) with the Cartesian $P$-property and a regularization smoothing Newton method for SCLCP with the Cartesian $P_0$-property are proposed. The existence of Newton directions and the boundedness of iterates, two important theoretical issues encountered in the smoothing method, are showed for these two classes of non-monotone SCLCP. Finally, we show that these two algorithms are globally convergent. Moreover, they are globally linear and locally quadratic convergent under a nonsingular assumption.
Citation: Li-Xia Liu, Sanyang Liu, Chun-Feng Wang. Smoothing Newton methods for symmetric cone linear complementarity problem with the Cartesian $P$/$P_0$-property. Journal of Industrial and Management Optimization, 2011, 7 (1) : 53-66. doi: 10.3934/jimo.2011.7.53
##### 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.
 [1] Yan Li, Liping Zhang. A smoothing Newton method preserving nonnegativity for solving tensor complementarity problems with $P_0$ mappings. Journal of Industrial and Management Optimization, 2022  doi: 10.3934/jimo.2022041 [2] Na Zhao, Zheng-Hai Huang. A nonmonotone smoothing Newton algorithm for solving box constrained variational inequalities with a $P_0$ function. Journal of Industrial and Management Optimization, 2011, 7 (2) : 467-482. doi: 10.3934/jimo.2011.7.467 [3] Behrouz Kheirfam. A weighted-path-following method for symmetric cone linear complementarity problems. Numerical Algebra, Control and Optimization, 2014, 4 (2) : 141-150. doi: 10.3934/naco.2014.4.141 [4] Zheng-Hai Huang, Nan Lu. Global and global linear convergence of smoothing algorithm for the Cartesian $P_*(\kappa)$-SCLCP. Journal of Industrial and Management Optimization, 2012, 8 (1) : 67-86. doi: 10.3934/jimo.2012.8.67 [5] Xiao-Hong Liu, Wei-Zhe Gu. Smoothing Newton algorithm based on a regularized one-parametric class of smoothing functions for generalized complementarity problems over symmetric cones. Journal of Industrial and Management Optimization, 2010, 6 (2) : 363-380. doi: 10.3934/jimo.2010.6.363 [6] Yi Zhang, Liwei Zhang, Jia Wu. On the convergence properties of a smoothing approach for mathematical programs with symmetric cone complementarity constraints. Journal of Industrial and Management Optimization, 2018, 14 (3) : 981-1005. doi: 10.3934/jimo.2017086 [7] Xin-He Miao, Jein-Shan Chen. Error bounds for symmetric cone complementarity problems. Numerical Algebra, Control and Optimization, 2013, 3 (4) : 627-641. doi: 10.3934/naco.2013.3.627 [8] Yanhong Yuan, Hongwei Zhang, Liwei Zhang. A smoothing Newton method for generalized Nash equilibrium problems with second-order cone constraints. Numerical Algebra, Control and Optimization, 2012, 2 (1) : 1-18. doi: 10.3934/naco.2012.2.1 [9] Shunsuke Hayashi. Inexact sequential injective algorithm for weakly univalent vector equation and its application to regularized smoothing Newton algorithm for mixed second-order cone complementarity problems. Numerical Algebra, Control and Optimization, 2022  doi: 10.3934/naco.2022024 [10] Peili Li, Xiliang Lu, Yunhai Xiao. Smoothing Newton method for $\ell^0$-$\ell^2$ regularized linear inverse problem. Inverse Problems and Imaging, 2022, 16 (1) : 153-177. doi: 10.3934/ipi.2021044 [11] Shuang Chen, Li-Ping Pang, Dan Li. An inexact semismooth Newton method for variational inequality with symmetric cone constraints. Journal of Industrial and Management Optimization, 2015, 11 (3) : 733-746. doi: 10.3934/jimo.2015.11.733 [12] Fengming Ma, Yiju Wang, Hongge Zhao. A potential reduction method for the generalized linear complementarity problem over a polyhedral cone. Journal of Industrial and Management Optimization, 2010, 6 (1) : 259-267. doi: 10.3934/jimo.2010.6.259 [13] Jianguo Huang, Sen Lin. A $C^0P_2$ time-stepping virtual element method for linear wave equations on polygonal meshes. Electronic Research Archive, 2020, 28 (2) : 911-933. doi: 10.3934/era.2020048 [14] Li Chu, Bo Wang, Jie Zhang, Hong-Wei Zhang. Convergence analysis of a smoothing SAA method for a stochastic mathematical program with second-order cone complementarity constraints. Journal of Industrial and Management Optimization, 2021, 17 (4) : 1863-1886. doi: 10.3934/jimo.2020050 [15] ShiChun Lv, Shou-Qiang Du. A new smoothing spectral conjugate gradient method for solving tensor complementarity problems. Journal of Industrial and Management Optimization, 2022, 18 (6) : 4111-4127. doi: 10.3934/jimo.2021150 [16] Zheng-Hai Huang, Jie Sun. A smoothing Newton algorithm for mathematical programs with complementarity constraints. Journal of Industrial and Management Optimization, 2005, 1 (2) : 153-170. doi: 10.3934/jimo.2005.1.153 [17] Zhengyong Zhou, Bo Yu. A smoothing homotopy method based on Robinson's normal equation for mixed complementarity problems. Journal of Industrial and Management Optimization, 2011, 7 (4) : 977-989. doi: 10.3934/jimo.2011.7.977 [18] Wei-Zhe Gu, Li-Yong Lu. The linear convergence of a derivative-free descent method for nonlinear complementarity problems. Journal of Industrial and Management Optimization, 2017, 13 (2) : 531-548. doi: 10.3934/jimo.2016030 [19] Jian Lu, Huaiyu Jian. Topological degree method for the rotationally symmetric $L_p$-Minkowski problem. Discrete and Continuous Dynamical Systems, 2016, 36 (2) : 971-980. doi: 10.3934/dcds.2016.36.971 [20] Lianjun Zhang, Lingchen Kong, Yan Li, Shenglong Zhou. A smoothing iterative method for quantile regression with nonconvex $\ell_p$ penalty. Journal of Industrial and Management Optimization, 2017, 13 (1) : 93-112. doi: 10.3934/jimo.2016006

2021 Impact Factor: 1.411