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A class of nonlinear Lagrangian algorithms for minimax problems
Solution properties and error bounds for semi-infinite complementarity problems
1. | Department of Mathematics, Shandong University of Technology, Zibo 255049, China |
2. | Department of Mathematics, Beijing Jiaotong University, Beijing 100044, China |
3. | Department of Mathematics, National Taiwan Normal University, Taipei 11677, Taiwan |
References:
[1] |
F. Alizadeh and D. Goldfarb, Second-order cone programming, Math. Programming, Ser. B, 95 (2003), 3-51. |
[2] |
A. Baken, F. Deutsch and W. Li, Strong CHIP, normality, and linear regularity of convex sets, Trans. Amer. Math. Soc., 357 (2005), 3831-3863.
doi: 10.1090/S0002-9947-05-03945-0. |
[3] |
H. H. Bauschke, J. M. Borwein and W. Li, Strong conical hull intersection property, bounded linear regularity, Jameson's property (G), and error bounds in convex optimization, Math. Programming, 86 (1999), 135-160.
doi: 10.1007/s101070050083. |
[4] |
H. H. Bauschke, J. M. Borwein and P. Tseng, Bounded linear regularity, strong CHIP, and CHIP are distinct properties, Journal of Convex Analysis, 7 (2000), 395-412. |
[5] |
Q. Chen, D. Chu and R. Tan, Optimal control of obstacle for quasi-linear elliptic variational bilateral problems, SIAM J. Control Optim., 44 (2005), 1067-1080.
doi: 10.1137/S0363012904443075. |
[6] |
X. Chen and M. Fukushima, Expected residual minimization method for stochastic linear complementarity problems, Math. Oper. Res., 30 (2005), 1022-1038.
doi: 10.1287/moor.1050.0160. |
[7] |
X. Chen and S. Xiang, Computation of error bounds for $P$-matrix linear complementarity problems, Math. Programming, 106 (2006), 513-525.
doi: 10.1007/s10107-005-0645-9. |
[8] |
X. Chen, C. Zhang and M. Fukushima, Robust solution of monotone stochastic linear complementarity problems, Math. Programming, 117 (2009), 51-80.
doi: 10.1007/s10107-007-0163-z. |
[9] |
F. H. Clarke, "Optimization and Nonsmooth Analysis," Wiley, New York, 1983. |
[10] |
R. W. Cottle, J. S. Pang and R. E. Stone, "The Linear Complementarity Problem," Academic Press, New York, 1992. |
[11] |
F. Facchinei, A. Fischer and V. Piccialli, Generalized Nash equilibrium problems and Newton methods, Math. Programming, 117 (2009), 163-194.
doi: 10.1007/s10107-007-0160-2. |
[12] |
F. Facchinei and J.-S. Pang, "Finite-Dimensional Variational Inequalities and Complementarity Problems, I and II," Springer Verlag, New York, 2003. |
[13] |
H. Fang, X. Chen and M. Fukushima, Stochastic $R_0$ matrix linear complementarity problems, SIAM J. Optim., 18 (2007), 482-506.
doi: 10.1137/050630805. |
[14] |
S. A. Gabriel and J. J. More, Smoothing of mixed complementarity problems, in "Complementarity and Variational Problems" (ed. M. C. Ferris and J. S. Pang), SIAM Publications, Philadelphia, (1997). |
[15] |
P. T. Harker and J.-S. Pang, Finite-dimensional variational inequality and nonlinear complementarity problems: A survey of theory, algorithms and applications, Math. Programming, Ser. B, 48 (1990), 161-220. |
[16] |
E. Klerk, "Aspects of Semidefinite Programming: Interior Point Algorithms and Selected Applications," Dordrecht: Kluwer Academic Publishers, 2002. |
[17] |
G. Lin, X. Chen and M. Fukushima, New restricted NCP functions and their applications to stochastic NCP and stochastic MPEC, Optimization, 56 (2007), 641-653.
doi: 10.1080/02331930701617320. |
[18] |
G. Lin and M. Fukushima, New reformulations for stochastic nonlinear complementarity problems, Optim. Methods Softw., 21 (2006), 551-564.
doi: 10.1080/10556780600627610. |
[19] |
C. Ling, L. Qi, G. Zhou and L. Caccettac, The SC1 property of an expected residual function arising from stochastic complementarity problems, Oper. Res. Lett., 36 (2008), 456-460.
doi: 10.1016/j.orl.2008.01.010. |
[20] |
O. L. Mangasarian and J. Ren, New improved error bounds for the linear complementarity problem, Math. Programming, 66 (1994), 241-255.
doi: 10.1007/BF01581148. |
[21] |
R. Mifflin, Semismooth and semiconvex functions in constrained optimization, SIAM J. Control Optim., 15 (1977), 957-972.
doi: 10.1137/0315061. |
[22] |
K. F. Ng and W. H. Yang, Regularities and their relations to error bounds, Math. Programming, 99 (2004), 521-538.
doi: 10.1007/s10107-003-0464-9. |
[23] |
J.-S. Pang, Error bounds in mathematical programming, Math. Programming, 79 (1997), 299-332.
doi: 10.1007/BF02614322. |
[24] |
E. Polak, "Optimization: Algorithms and Consistent Approximation," Springer-Verlag, New York, 1997. |
[25] |
R. T. Rockafellar, "Convex Analysis," Princeton University Press, 1970. |
[26] |
R. T. Rockafellar and R. J. Wets, "Variational Analysis," Springer, New York, 1998.
doi: 10.1007/978-3-642-02431-3. |
[27] |
S. H. Schmieta and F. Alizadeh, Associative and Jordan algebras, and polynomial time interior-point algorithms for symmetric cones, Math. Oper. Res., 26 (2001), 543-564.
doi: 10.1287/moor.26.3.543.10582. |
[28] |
S. H. Schmieta and F. Alizadeh, Extension of primal-dual interior point algorithms to symmetric cones, Math. Programming, 96 (2003), 409-438.
doi: 10.1007/s10107-003-0380-z. |
[29] |
D. Sun and L. Qi, On NCP-functions, Comp. Optim. Appl., 13 (1999), 201-220.
doi: 10.1023/A:1008669226453. |
[30] |
X. Y. Zheng and K. F. Ng, Linear regularity for a collection of subsmooth sets in Banach spaces, SIAM J. Optim., 19 (2008), 62-76.
doi: 10.1137/060659132. |
show all references
References:
[1] |
F. Alizadeh and D. Goldfarb, Second-order cone programming, Math. Programming, Ser. B, 95 (2003), 3-51. |
[2] |
A. Baken, F. Deutsch and W. Li, Strong CHIP, normality, and linear regularity of convex sets, Trans. Amer. Math. Soc., 357 (2005), 3831-3863.
doi: 10.1090/S0002-9947-05-03945-0. |
[3] |
H. H. Bauschke, J. M. Borwein and W. Li, Strong conical hull intersection property, bounded linear regularity, Jameson's property (G), and error bounds in convex optimization, Math. Programming, 86 (1999), 135-160.
doi: 10.1007/s101070050083. |
[4] |
H. H. Bauschke, J. M. Borwein and P. Tseng, Bounded linear regularity, strong CHIP, and CHIP are distinct properties, Journal of Convex Analysis, 7 (2000), 395-412. |
[5] |
Q. Chen, D. Chu and R. Tan, Optimal control of obstacle for quasi-linear elliptic variational bilateral problems, SIAM J. Control Optim., 44 (2005), 1067-1080.
doi: 10.1137/S0363012904443075. |
[6] |
X. Chen and M. Fukushima, Expected residual minimization method for stochastic linear complementarity problems, Math. Oper. Res., 30 (2005), 1022-1038.
doi: 10.1287/moor.1050.0160. |
[7] |
X. Chen and S. Xiang, Computation of error bounds for $P$-matrix linear complementarity problems, Math. Programming, 106 (2006), 513-525.
doi: 10.1007/s10107-005-0645-9. |
[8] |
X. Chen, C. Zhang and M. Fukushima, Robust solution of monotone stochastic linear complementarity problems, Math. Programming, 117 (2009), 51-80.
doi: 10.1007/s10107-007-0163-z. |
[9] |
F. H. Clarke, "Optimization and Nonsmooth Analysis," Wiley, New York, 1983. |
[10] |
R. W. Cottle, J. S. Pang and R. E. Stone, "The Linear Complementarity Problem," Academic Press, New York, 1992. |
[11] |
F. Facchinei, A. Fischer and V. Piccialli, Generalized Nash equilibrium problems and Newton methods, Math. Programming, 117 (2009), 163-194.
doi: 10.1007/s10107-007-0160-2. |
[12] |
F. Facchinei and J.-S. Pang, "Finite-Dimensional Variational Inequalities and Complementarity Problems, I and II," Springer Verlag, New York, 2003. |
[13] |
H. Fang, X. Chen and M. Fukushima, Stochastic $R_0$ matrix linear complementarity problems, SIAM J. Optim., 18 (2007), 482-506.
doi: 10.1137/050630805. |
[14] |
S. A. Gabriel and J. J. More, Smoothing of mixed complementarity problems, in "Complementarity and Variational Problems" (ed. M. C. Ferris and J. S. Pang), SIAM Publications, Philadelphia, (1997). |
[15] |
P. T. Harker and J.-S. Pang, Finite-dimensional variational inequality and nonlinear complementarity problems: A survey of theory, algorithms and applications, Math. Programming, Ser. B, 48 (1990), 161-220. |
[16] |
E. Klerk, "Aspects of Semidefinite Programming: Interior Point Algorithms and Selected Applications," Dordrecht: Kluwer Academic Publishers, 2002. |
[17] |
G. Lin, X. Chen and M. Fukushima, New restricted NCP functions and their applications to stochastic NCP and stochastic MPEC, Optimization, 56 (2007), 641-653.
doi: 10.1080/02331930701617320. |
[18] |
G. Lin and M. Fukushima, New reformulations for stochastic nonlinear complementarity problems, Optim. Methods Softw., 21 (2006), 551-564.
doi: 10.1080/10556780600627610. |
[19] |
C. Ling, L. Qi, G. Zhou and L. Caccettac, The SC1 property of an expected residual function arising from stochastic complementarity problems, Oper. Res. Lett., 36 (2008), 456-460.
doi: 10.1016/j.orl.2008.01.010. |
[20] |
O. L. Mangasarian and J. Ren, New improved error bounds for the linear complementarity problem, Math. Programming, 66 (1994), 241-255.
doi: 10.1007/BF01581148. |
[21] |
R. Mifflin, Semismooth and semiconvex functions in constrained optimization, SIAM J. Control Optim., 15 (1977), 957-972.
doi: 10.1137/0315061. |
[22] |
K. F. Ng and W. H. Yang, Regularities and their relations to error bounds, Math. Programming, 99 (2004), 521-538.
doi: 10.1007/s10107-003-0464-9. |
[23] |
J.-S. Pang, Error bounds in mathematical programming, Math. Programming, 79 (1997), 299-332.
doi: 10.1007/BF02614322. |
[24] |
E. Polak, "Optimization: Algorithms and Consistent Approximation," Springer-Verlag, New York, 1997. |
[25] |
R. T. Rockafellar, "Convex Analysis," Princeton University Press, 1970. |
[26] |
R. T. Rockafellar and R. J. Wets, "Variational Analysis," Springer, New York, 1998.
doi: 10.1007/978-3-642-02431-3. |
[27] |
S. H. Schmieta and F. Alizadeh, Associative and Jordan algebras, and polynomial time interior-point algorithms for symmetric cones, Math. Oper. Res., 26 (2001), 543-564.
doi: 10.1287/moor.26.3.543.10582. |
[28] |
S. H. Schmieta and F. Alizadeh, Extension of primal-dual interior point algorithms to symmetric cones, Math. Programming, 96 (2003), 409-438.
doi: 10.1007/s10107-003-0380-z. |
[29] |
D. Sun and L. Qi, On NCP-functions, Comp. Optim. Appl., 13 (1999), 201-220.
doi: 10.1023/A:1008669226453. |
[30] |
X. Y. Zheng and K. F. Ng, Linear regularity for a collection of subsmooth sets in Banach spaces, SIAM J. Optim., 19 (2008), 62-76.
doi: 10.1137/060659132. |
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