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A class of nonlinear Lagrangian algorithms for minimax problems
1. | School of Science, Wuhan University of Technology, Wuhan Hubei, 430070, China, China |
References:
[1] |
A. Ben-Tal and A. Nemirovski, "Lectures on Modern Convex Optimization: Analysis, Algorithms, and Engineering Applications,", MPS/SIAM Ser. Optim. 2, (2001).
|
[2] |
D. P. Bertsekas, "Constrained Optimization and Lagrange Multiplier Methods,", Academic Press, (1982).
|
[3] |
C. Charalambous, Acceleration of the least $p$th algorithm for minimax optimization with engineering applications,, Math. Program., 19 (1979), 270.
|
[4] |
G. Dipillo, L. Grippo and S. Lucidi, A smooth method for the finite minimax problem,, Math. Program., 60 (1993), 187.
|
[5] |
J. P. Dussault, Augmented non-quadratic penalty algorithms,, Math. Program., 99 (2004), 467.
|
[6] |
G. D. Erdmann, "A New Minimax Algorithm and Its Application to Optics Problems,", Ph. D. Thesis, (2003).
|
[7] |
S. X. He and L. W. Zhang, Convergence of a dual algorithm for minimax problems,, Arch. Control Sci., 10 (2000), 47.
|
[8] |
Q. J. Hu, Y. Chen, N. P. Chen and X. Q. Li, A modified SQP algorithm for minimax problems,, J. Math. Anal. Appl., 360 (2009), 211.
|
[9] |
J. B. Jian, R. Quan and X. L. Zhang, Generalized monotone line search algorithm for degenerate nonlinear minimax problems,, Bull. Austral. Math. Soc., 73 (2006), 117.
doi: 10.1017/S0004972700038673. |
[10] |
J. B. Jian, R. Quan and X. L. Zhang, Feasible generalized monotone line search SQP algorithm for nonlinear minimax problems with inequality constraints,, J. Comput. Appl. Math., 205 (2007), 406.
|
[11] |
X. S. Li, An entropy-based aggregate method for minimax optimization,, Eng. Optim., 18 (1992), 277. Google Scholar |
[12] |
L. Lukšan, C. Matonoha and J. Vlček, Primal interior-point method for large sparse minimax optimization,, Tech. Rep. V-941, (2005). Google Scholar |
[13] |
E. Y. Pee and J. O. Royset, On solving large-scale finite minimax problems using exponential smoothing,, J. Optimiz. Theory App., 148 (2011), 390.
doi: 10.1007/s10957-010-9759-1. |
[14] |
E. Polak, On the mathematical foundations of nondifferentiable optimization in engineering design,, SIAM Rev., 29 (1987), 21.
doi: 10.1137/1029002. |
[15] |
E. Polak, S. Salcudean and D. Q. Mayne, Adaptive control of ARMA plants using worst-case design by semi-infinite optimization,, IEEE Trans. Autom. Control, 32 (1987), 388.
|
[16] |
E. Polak, "Optimization: Algorithms and Consistent Approximations,", Springer-Verlag, (1997).
|
[17] |
E. Polak, J. E. Higgins and D. Q. Mayne, A barrier function method for minimax problems,, Math. Program., 54 (1992), 155.
|
[18] |
E. Polak, J. O. Royset and R. S. Womersley, Algorithms with adaptive smoothing for finite minimax problems,, J. Optimiz. Theory App., 119 (2003), 459.
doi: 10.1023/B:JOTA.0000006685.60019.3e. |
[19] |
R. A. Polyak, Smooth optimization methods for minimax problems,, SIAM J. Control Optim., 26 (1988), 1274.
|
[20] |
R. A. Polyak, Nonlinear rescaling in discrete minimax,, in, (2000). Google Scholar |
[21] |
R. A. Polyak, Modified barrier function: Theory and mehtods,, Math. Program., 54 (1992), 177.
|
[22] |
R. A. Polyak, Log-Sigmoid multipliers method in constrained optimization,, Ann. Oper. Res., 101 (2001), 427.
|
[23] |
R. A. Polyak, Nonlinear rescaling vs. smoothing technique in convex optimization,, Math. Program., 92 (2002), 197.
|
[24] |
S. Xu, Smoothing method for minimax problems,, Comput. Optim. Appl., 20 (2001), 267.
|
[25] |
S. E. Sussman-Fort, Approximate direct-search minimax circuit optimization,, Int. J. Numer. Methods Eng., 28 (1989), 359.
|
[26] |
F. S. Wang and Y. P. Wang, Nonmontone algorithm for minimax optimization problems,, Appl. Math. Comput., 217 (2011), 6296.
|
[27] |
A. D. Warren, L. S. Lasdon and D. F. Suchman, Optimization in engineering design,, Proc. IEEE, 55 (1967), 1885. Google Scholar |
[28] |
F. Ye, H. W. Liu, S. S. Zhou and S. Y. Liu, A smoothing trust-region Newton-CG method for minimax problem,, Appl. Math. Comput., 199 (2008), 581.
doi: 10.1016/j.amc.2007.10.070. |
[29] |
L. W. Zhang, Y. H. Ren, Y. Wu and X. T. Xiao, A class of nonlinear Lagrangians: Theory and algorithm,, Asia-Pac. J. Oper. Res., 25 (2008), 327.
doi: 10.1142/S021759590800178X. |
[30] |
L. W. Zhang and H. W. Tang, A maximum entropy algorithm with parameters for solving minimax problem,, Arch. Control Sci., 6 (1997), 47.
|
show all references
References:
[1] |
A. Ben-Tal and A. Nemirovski, "Lectures on Modern Convex Optimization: Analysis, Algorithms, and Engineering Applications,", MPS/SIAM Ser. Optim. 2, (2001).
|
[2] |
D. P. Bertsekas, "Constrained Optimization and Lagrange Multiplier Methods,", Academic Press, (1982).
|
[3] |
C. Charalambous, Acceleration of the least $p$th algorithm for minimax optimization with engineering applications,, Math. Program., 19 (1979), 270.
|
[4] |
G. Dipillo, L. Grippo and S. Lucidi, A smooth method for the finite minimax problem,, Math. Program., 60 (1993), 187.
|
[5] |
J. P. Dussault, Augmented non-quadratic penalty algorithms,, Math. Program., 99 (2004), 467.
|
[6] |
G. D. Erdmann, "A New Minimax Algorithm and Its Application to Optics Problems,", Ph. D. Thesis, (2003).
|
[7] |
S. X. He and L. W. Zhang, Convergence of a dual algorithm for minimax problems,, Arch. Control Sci., 10 (2000), 47.
|
[8] |
Q. J. Hu, Y. Chen, N. P. Chen and X. Q. Li, A modified SQP algorithm for minimax problems,, J. Math. Anal. Appl., 360 (2009), 211.
|
[9] |
J. B. Jian, R. Quan and X. L. Zhang, Generalized monotone line search algorithm for degenerate nonlinear minimax problems,, Bull. Austral. Math. Soc., 73 (2006), 117.
doi: 10.1017/S0004972700038673. |
[10] |
J. B. Jian, R. Quan and X. L. Zhang, Feasible generalized monotone line search SQP algorithm for nonlinear minimax problems with inequality constraints,, J. Comput. Appl. Math., 205 (2007), 406.
|
[11] |
X. S. Li, An entropy-based aggregate method for minimax optimization,, Eng. Optim., 18 (1992), 277. Google Scholar |
[12] |
L. Lukšan, C. Matonoha and J. Vlček, Primal interior-point method for large sparse minimax optimization,, Tech. Rep. V-941, (2005). Google Scholar |
[13] |
E. Y. Pee and J. O. Royset, On solving large-scale finite minimax problems using exponential smoothing,, J. Optimiz. Theory App., 148 (2011), 390.
doi: 10.1007/s10957-010-9759-1. |
[14] |
E. Polak, On the mathematical foundations of nondifferentiable optimization in engineering design,, SIAM Rev., 29 (1987), 21.
doi: 10.1137/1029002. |
[15] |
E. Polak, S. Salcudean and D. Q. Mayne, Adaptive control of ARMA plants using worst-case design by semi-infinite optimization,, IEEE Trans. Autom. Control, 32 (1987), 388.
|
[16] |
E. Polak, "Optimization: Algorithms and Consistent Approximations,", Springer-Verlag, (1997).
|
[17] |
E. Polak, J. E. Higgins and D. Q. Mayne, A barrier function method for minimax problems,, Math. Program., 54 (1992), 155.
|
[18] |
E. Polak, J. O. Royset and R. S. Womersley, Algorithms with adaptive smoothing for finite minimax problems,, J. Optimiz. Theory App., 119 (2003), 459.
doi: 10.1023/B:JOTA.0000006685.60019.3e. |
[19] |
R. A. Polyak, Smooth optimization methods for minimax problems,, SIAM J. Control Optim., 26 (1988), 1274.
|
[20] |
R. A. Polyak, Nonlinear rescaling in discrete minimax,, in, (2000). Google Scholar |
[21] |
R. A. Polyak, Modified barrier function: Theory and mehtods,, Math. Program., 54 (1992), 177.
|
[22] |
R. A. Polyak, Log-Sigmoid multipliers method in constrained optimization,, Ann. Oper. Res., 101 (2001), 427.
|
[23] |
R. A. Polyak, Nonlinear rescaling vs. smoothing technique in convex optimization,, Math. Program., 92 (2002), 197.
|
[24] |
S. Xu, Smoothing method for minimax problems,, Comput. Optim. Appl., 20 (2001), 267.
|
[25] |
S. E. Sussman-Fort, Approximate direct-search minimax circuit optimization,, Int. J. Numer. Methods Eng., 28 (1989), 359.
|
[26] |
F. S. Wang and Y. P. Wang, Nonmontone algorithm for minimax optimization problems,, Appl. Math. Comput., 217 (2011), 6296.
|
[27] |
A. D. Warren, L. S. Lasdon and D. F. Suchman, Optimization in engineering design,, Proc. IEEE, 55 (1967), 1885. Google Scholar |
[28] |
F. Ye, H. W. Liu, S. S. Zhou and S. Y. Liu, A smoothing trust-region Newton-CG method for minimax problem,, Appl. Math. Comput., 199 (2008), 581.
doi: 10.1016/j.amc.2007.10.070. |
[29] |
L. W. Zhang, Y. H. Ren, Y. Wu and X. T. Xiao, A class of nonlinear Lagrangians: Theory and algorithm,, Asia-Pac. J. Oper. Res., 25 (2008), 327.
doi: 10.1142/S021759590800178X. |
[30] |
L. W. Zhang and H. W. Tang, A maximum entropy algorithm with parameters for solving minimax problem,, Arch. Control Sci., 6 (1997), 47.
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