# American Institute of Mathematical Sciences

January  2012, 8(1): 67-86. doi: 10.3934/jimo.2012.8.67

## Global and global linear convergence of smoothing algorithm for the Cartesian $P_*(\kappa)$-SCLCP

 1 Department of Mathematics, School of Science, Tianjin University, Tianjin 300072, P.R. 2 Department of Mathematics, Xidian University, XiAn 710071, China

Received  October 2010 Revised  July 2011 Published  November 2011

In this paper, we consider the linear complementarity problem over Euclidean Jordan algebras with a Cartesian $P_*(\kappa)$-transformation, which is denoted by the Cartesian $P_*(\kappa)$-SCLCP. A smoothing algorithm is extended to solve the Cartesian $P_*(\kappa)$-SCLCP. We show that the algorithm is globally convergent if the problem concerned has a solution. In particular, we show that the algorithm is globally linearly convergent under a weak assumption.
Citation: 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
##### References:
 [1] R. W. Cottle, J.-S. Pang and R. E. Stone, "The Linear Complementarity Problems," Academic Press, Boston, 1992. [2] J. Faraut and A. Korányi, "Analysis on Symmetric Cones, Oxford Mathematical Monographs," Oxford University Press, New York, 1994. [3] L. Faybusovich, Euclidean Jordan algebras and interior-point algorithms, Positivity, 1 (1997), 331-357. [4] L. Faybusovich, Linear systems in Jordan algebras and primal-dual interior-point algorithms, J. Comput. Appl. Math., 86 (1997), 149-175. [5] L. Faybusovich and Y. Lu, Jordan-algebraic aspects of nonconvex optimization over symmetric cones, Appl. Math. Optim., 53 (2006), 67-77. [6] 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. [7] Z. H. Huang, The global linear and local quadratic convergence of a non-interior continuation algorithm for the LCP, IMA J. Numer. Anal., 25 (2005), 670-684. [8] Z. H. Huang, Locating a maximally complementary solution of the monotone NCP by using non-interior-point smoothing algorithms, Math. Meth. Oper. Res., 61 (2005), 41-55. [9] Z. H. Huang, S. L. Hu and J. Han, Global convergence of a smoothing algorithm for symmetric cone complementarity problems with a nonmonotone line search, Sci. China, Ser. A, 52 (2009), 833-848. [10] Z. H. Huang and T. Ni, Smoothing algorithms for complementarity problems over symmetric cones, Comput. Optim. Appl., 45 (2010), 557-579. [11] 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. [12] Z. H. Huang and J. Sun, A non-interior continuation algorithm for the $P_0$ or $P_*$ LCP with strong global and local convergence properties, Appl. Math. Optim., 52 (2005), 237-262. [13] Z. H. Huang and S. W. Xu, Convergence properties of a non-interior-point smoothing algorithm for the $P_*$ NCP, J. Ind. Manag. Optim., 3 (2007), 569-584. [14] M. Kojima, N. Megiddo, T. Noma and A. Yoshise, "A Unified Approach to Interior Point Algorithms for Linear Complementarity Problems," Lecture Note in Computer Science, 538, Springer, New York, 1991. [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. [16] L. C. Kong, L. Tunçel and N. H. Xiu, Fischer-Burmeister conplementarity function on Euclidean Jordan algebras, Pacific J. Optim., 6 (2010), 423-440. [17] X. H. Liu and Z. H. Huang, A smoothing Newton algorithm based on a one-parametric class of smoothing functions for linear programming over symmetric cones, Math. Meth. Oper. Res., 70 (2009), 385-404. [18] Y. J. Liu, L. W. Zhang and Y. H. Wang, Analysis of smoothing method for symmetric conic linear programming, J. Appl. Math. Comput., 22 (2006), 133-148. [19] 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. [20] S. H. Pan and J. S. Chen, A one-parametric class of merit functions for the symmetric cone complementarity problem, J. Math. Anal. Appl., 355 (2009), 195-215. [21] S. H. Pan and J. S. Chen, A regularization method for the second-order cone complementarity problem with the Cartesian $P_0$-property, Nonlinear Anal. - TMA, 70 (2009), 1475-1491. [22] 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. [23] S. H. Schimieta and F. Alizadeh, Associative and Jordan algebras, and polynomial time interior-point algorithms for symmetric cones, Math. Oper. Res., 26 (2001), 543-564. [24] S. H. Schimieta and F. Alizadeh, Extension of primal-dual interior-point algorithms to symmetric cones, Math. Program., 96 (2003), 409-438. [25] H. Völiaho, $P_*(\kappa)$-matrices are just sufficient, Linear Algebra Appl., 239 (1996), 103-108. [26] A. Yoshise, Interior point trajectories and a homogeneous model for nonlinear complementarity problems over symmetric cones, SIAM J. Optim., 17 (2006), 1129-1153. [27] Y. B. Zhao and D. Li, A globally and locally superlinearly convergent non-interior-point algorithm for $P_0$ LCPs, SIAM J Optim., 13 (2003), 1196-1221. [28] Y. B. Zhao and D. Li, A new path-following algorithm for nonlinear $P_*$ complementarity problems, Comput. Optim. Appl., 34 (2005), 183-214.

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##### References:
 [1] R. W. Cottle, J.-S. Pang and R. E. Stone, "The Linear Complementarity Problems," Academic Press, Boston, 1992. [2] J. Faraut and A. Korányi, "Analysis on Symmetric Cones, Oxford Mathematical Monographs," Oxford University Press, New York, 1994. [3] L. Faybusovich, Euclidean Jordan algebras and interior-point algorithms, Positivity, 1 (1997), 331-357. [4] L. Faybusovich, Linear systems in Jordan algebras and primal-dual interior-point algorithms, J. Comput. Appl. Math., 86 (1997), 149-175. [5] L. Faybusovich and Y. Lu, Jordan-algebraic aspects of nonconvex optimization over symmetric cones, Appl. Math. Optim., 53 (2006), 67-77. [6] 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. [7] Z. H. Huang, The global linear and local quadratic convergence of a non-interior continuation algorithm for the LCP, IMA J. Numer. Anal., 25 (2005), 670-684. [8] Z. H. Huang, Locating a maximally complementary solution of the monotone NCP by using non-interior-point smoothing algorithms, Math. Meth. Oper. Res., 61 (2005), 41-55. [9] Z. H. Huang, S. L. Hu and J. Han, Global convergence of a smoothing algorithm for symmetric cone complementarity problems with a nonmonotone line search, Sci. China, Ser. A, 52 (2009), 833-848. [10] Z. H. Huang and T. Ni, Smoothing algorithms for complementarity problems over symmetric cones, Comput. Optim. Appl., 45 (2010), 557-579. [11] 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. [12] Z. H. Huang and J. Sun, A non-interior continuation algorithm for the $P_0$ or $P_*$ LCP with strong global and local convergence properties, Appl. Math. Optim., 52 (2005), 237-262. [13] Z. H. Huang and S. W. Xu, Convergence properties of a non-interior-point smoothing algorithm for the $P_*$ NCP, J. Ind. Manag. Optim., 3 (2007), 569-584. [14] M. Kojima, N. Megiddo, T. Noma and A. Yoshise, "A Unified Approach to Interior Point Algorithms for Linear Complementarity Problems," Lecture Note in Computer Science, 538, Springer, New York, 1991. [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. [16] L. C. Kong, L. Tunçel and N. H. Xiu, Fischer-Burmeister conplementarity function on Euclidean Jordan algebras, Pacific J. Optim., 6 (2010), 423-440. [17] X. H. Liu and Z. H. Huang, A smoothing Newton algorithm based on a one-parametric class of smoothing functions for linear programming over symmetric cones, Math. Meth. Oper. Res., 70 (2009), 385-404. [18] Y. J. Liu, L. W. Zhang and Y. H. Wang, Analysis of smoothing method for symmetric conic linear programming, J. Appl. Math. Comput., 22 (2006), 133-148. [19] 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. [20] S. H. Pan and J. S. Chen, A one-parametric class of merit functions for the symmetric cone complementarity problem, J. Math. Anal. Appl., 355 (2009), 195-215. [21] S. H. Pan and J. S. Chen, A regularization method for the second-order cone complementarity problem with the Cartesian $P_0$-property, Nonlinear Anal. - TMA, 70 (2009), 1475-1491. [22] 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. [23] S. H. Schimieta and F. Alizadeh, Associative and Jordan algebras, and polynomial time interior-point algorithms for symmetric cones, Math. Oper. Res., 26 (2001), 543-564. [24] S. H. Schimieta and F. Alizadeh, Extension of primal-dual interior-point algorithms to symmetric cones, Math. Program., 96 (2003), 409-438. [25] H. Völiaho, $P_*(\kappa)$-matrices are just sufficient, Linear Algebra Appl., 239 (1996), 103-108. [26] A. Yoshise, Interior point trajectories and a homogeneous model for nonlinear complementarity problems over symmetric cones, SIAM J. Optim., 17 (2006), 1129-1153. [27] Y. B. Zhao and D. Li, A globally and locally superlinearly convergent non-interior-point algorithm for $P_0$ LCPs, SIAM J Optim., 13 (2003), 1196-1221. [28] Y. B. Zhao and D. Li, A new path-following algorithm for nonlinear $P_*$ complementarity problems, Comput. Optim. Appl., 34 (2005), 183-214.
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