Smoothing Newton methods for symmetric cone linear complementarity problem with the Cartesian P/P

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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

Li-Xia Liu ^1, , Sanyang Liu ^2, and Chun-Feng Wang ^3,

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

Abstract
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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.

Keywords: smoothing Newton method., linear complementarity problems, Cartesian $P$/$P_0$-property, Symmetric cone.

Mathematics Subject Classification: Primary: 90C25; Secondary: 90C3.

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 & 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. 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., 106 (2006), 177.
[3]	X. J. Chen and Y. Y. Ye, On homotopy-smoothing methods for box-constrained variational inequalities,, SIAM J. Control. Optim., 37 (1999), 589. 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. 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., 106 (2006), 513.
[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. doi: 10.1016/j.nonrwa.2008.02.010.
[7]	F. H. Clarke, "Optimization and Nonsmooth Analysis,", Wiley Press, (1983).
[8]	J. Faraut and A. Koranyi, "Analysis on Symmetric Cones,", Clarendon Press, (1994).
[9]	F. Facchinei and F. Pang, "Finite-dimensional Variational Ineqaulities and Complementarity Problems, vol. I and II,", Springer-Verlag, (2003).
[10]	F. Facchinei and C. Kanzow, Beyond monotonicity in regularization methods for nonlinear complementarity problems,, SIAM J. Control Optim., 37 (1999), 1150. 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. 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. 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. 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., 99 (2004), 423.
[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. 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.
[17]	M. S. Lobo, L. Vandenberghe, S. Boyd and H. Lebret, Application of second-order cone programming,, Linear Algebra Appl., 284 (1998), 193. 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.
[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. 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. 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.
[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. doi: 10.1007/s11425-008-0174-0.
[23]	R. Mifflin, Semismooth and semiconvex functions in constrained optimization,, SIAM J. Control Optim., 15 (1977), 959. 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.
[25]	L. Q. Qi and J. Sun, A nonsmooth version of Newton's method,, Math. Program. Ser. A, 58 (1993), 353. 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. 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. 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. 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. 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., 106 (2006), 177.
[3]	X. J. Chen and Y. Y. Ye, On homotopy-smoothing methods for box-constrained variational inequalities,, SIAM J. Control. Optim., 37 (1999), 589. 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. 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., 106 (2006), 513.
[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. doi: 10.1016/j.nonrwa.2008.02.010.
[7]	F. H. Clarke, "Optimization and Nonsmooth Analysis,", Wiley Press, (1983).
[8]	J. Faraut and A. Koranyi, "Analysis on Symmetric Cones,", Clarendon Press, (1994).
[9]	F. Facchinei and F. Pang, "Finite-dimensional Variational Ineqaulities and Complementarity Problems, vol. I and II,", Springer-Verlag, (2003).
[10]	F. Facchinei and C. Kanzow, Beyond monotonicity in regularization methods for nonlinear complementarity problems,, SIAM J. Control Optim., 37 (1999), 1150. 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. 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. 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. 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., 99 (2004), 423.
[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. 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.
[17]	M. S. Lobo, L. Vandenberghe, S. Boyd and H. Lebret, Application of second-order cone programming,, Linear Algebra Appl., 284 (1998), 193. 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.
[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. 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. 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.
[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. doi: 10.1007/s11425-008-0174-0.
[23]	R. Mifflin, Semismooth and semiconvex functions in constrained optimization,, SIAM J. Control Optim., 15 (1977), 959. 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.
[25]	L. Q. Qi and J. Sun, A nonsmooth version of Newton's method,, Math. Program. Ser. A, 58 (1993), 353. 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. 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. 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. doi: 10.1023/A:1022528320719.

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