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A penalty method for generalized Nash equilibrium problems
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 |
References:
[1] |
R. W. Cottle, J.-S. Pang and R. E. Stone, "The Linear Complementarity Problems,", Academic Press, (1992). Google Scholar |
[2] |
J. Faraut and A. Korányi, "Analysis on Symmetric Cones, Oxford Mathematical Monographs,", Oxford University Press, (1994). Google Scholar |
[3] |
L. Faybusovich, Euclidean Jordan algebras and interior-point algorithms,, Positivity, 1 (1997), 331. Google Scholar |
[4] |
L. Faybusovich, Linear systems in Jordan algebras and primal-dual interior-point algorithms,, J. Comput. Appl. Math., 86 (1997), 149. Google Scholar |
[5] |
L. Faybusovich and Y. Lu, Jordan-algebraic aspects of nonconvex optimization over symmetric cones,, Appl. Math. Optim., 53 (2006), 67. Google Scholar |
[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. Google Scholar |
[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. Google Scholar |
[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. Google Scholar |
[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, 52 (2009), 833. Google Scholar |
[10] |
Z. H. Huang and T. Ni, Smoothing algorithms for complementarity problems over symmetric cones,, Comput. Optim. Appl., 45 (2010), 557. Google Scholar |
[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. Google Scholar |
[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. Google Scholar |
[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. Google Scholar |
[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 (1991). Google Scholar |
[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. Google Scholar |
[16] |
L. C. Kong, L. Tunçel and N. H. Xiu, Fischer-Burmeister conplementarity function on Euclidean Jordan algebras,, Pacific J. Optim., 6 (2010), 423. Google Scholar |
[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. Google Scholar |
[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. Google Scholar |
[19] |
Z. Y. Luo and N. H. Xiu, Path-following interior point algorithms for the Cartesian $P_*(\kappa)$-LCP over symmetric cones,, Sci. China, 52 (2009), 1769. Google Scholar |
[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. Google Scholar |
[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. Google Scholar |
[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. Google Scholar |
[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. Google Scholar |
[24] |
S. H. Schimieta and F. Alizadeh, Extension of primal-dual interior-point algorithms to symmetric cones,, Math. Program., 96 (2003), 409. Google Scholar |
[25] |
H. Völiaho, $P_*(\kappa)$-matrices are just sufficient,, Linear Algebra Appl., 239 (1996), 103. Google Scholar |
[26] |
A. Yoshise, Interior point trajectories and a homogeneous model for nonlinear complementarity problems over symmetric cones,, SIAM J. Optim., 17 (2006), 1129. Google Scholar |
[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. Google Scholar |
[28] |
Y. B. Zhao and D. Li, A new path-following algorithm for nonlinear $P_*$ complementarity problems,, Comput. Optim. Appl., 34 (2005), 183. Google Scholar |
show all references
References:
[1] |
R. W. Cottle, J.-S. Pang and R. E. Stone, "The Linear Complementarity Problems,", Academic Press, (1992). Google Scholar |
[2] |
J. Faraut and A. Korányi, "Analysis on Symmetric Cones, Oxford Mathematical Monographs,", Oxford University Press, (1994). Google Scholar |
[3] |
L. Faybusovich, Euclidean Jordan algebras and interior-point algorithms,, Positivity, 1 (1997), 331. Google Scholar |
[4] |
L. Faybusovich, Linear systems in Jordan algebras and primal-dual interior-point algorithms,, J. Comput. Appl. Math., 86 (1997), 149. Google Scholar |
[5] |
L. Faybusovich and Y. Lu, Jordan-algebraic aspects of nonconvex optimization over symmetric cones,, Appl. Math. Optim., 53 (2006), 67. Google Scholar |
[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. Google Scholar |
[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. Google Scholar |
[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. Google Scholar |
[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, 52 (2009), 833. Google Scholar |
[10] |
Z. H. Huang and T. Ni, Smoothing algorithms for complementarity problems over symmetric cones,, Comput. Optim. Appl., 45 (2010), 557. Google Scholar |
[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. Google Scholar |
[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. Google Scholar |
[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. Google Scholar |
[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 (1991). Google Scholar |
[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. Google Scholar |
[16] |
L. C. Kong, L. Tunçel and N. H. Xiu, Fischer-Burmeister conplementarity function on Euclidean Jordan algebras,, Pacific J. Optim., 6 (2010), 423. Google Scholar |
[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. Google Scholar |
[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. Google Scholar |
[19] |
Z. Y. Luo and N. H. Xiu, Path-following interior point algorithms for the Cartesian $P_*(\kappa)$-LCP over symmetric cones,, Sci. China, 52 (2009), 1769. Google Scholar |
[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. Google Scholar |
[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. Google Scholar |
[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. Google Scholar |
[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. Google Scholar |
[24] |
S. H. Schimieta and F. Alizadeh, Extension of primal-dual interior-point algorithms to symmetric cones,, Math. Program., 96 (2003), 409. Google Scholar |
[25] |
H. Völiaho, $P_*(\kappa)$-matrices are just sufficient,, Linear Algebra Appl., 239 (1996), 103. Google Scholar |
[26] |
A. Yoshise, Interior point trajectories and a homogeneous model for nonlinear complementarity problems over symmetric cones,, SIAM J. Optim., 17 (2006), 1129. Google Scholar |
[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. Google Scholar |
[28] |
Y. B. Zhao and D. Li, A new path-following algorithm for nonlinear $P_*$ complementarity problems,, Comput. Optim. Appl., 34 (2005), 183. Google Scholar |
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