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An interactive MOLP method for solving output-oriented DEA problems with undesirable factors
Second order sufficient conditions for a class of bilevel programs with lower level second-order cone programming problem
1. | School of Mathematics and Computing Science, Guilin University of Electronic Technology, Guilin 541004, China |
2. | School of Mathematics and Statistics, Wuhan University, Wuhan 430072, China, China |
References:
[1] |
F. Alizadeh and D. Goldfarb, Second-order cone Programming,, Math. Program., 95 (2003), 3.
doi: 10.1007/s10107-002-0339-5. |
[2] |
A. Ben-Tal, L. El Ghaoui and A. Nemirovski, Robust Optimization,, Princeton University Press, (2009).
doi: 10.1515/9781400831050. |
[3] |
J. F. Bonnans and A. Shapiro, Perturbation Analysis of Optimization Problems,, Springer, (2000).
doi: 10.1007/978-1-4612-1394-9. |
[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. -D. Chen, D. Sun and J. Sun, Complementarity functions and numerical experiments for second-order cone complementarity problems,, Comput. Optim. Appl., 25 (2003), 39.
doi: 10.1023/A:1022996819381. |
[6] |
X. J. Chen and S. H. Xiang, Computation of error bounds for P-matrix linear complementarity problems,, Math. Program., 106 (2006), 513.
doi: 10.1007/s10107-005-0645-9. |
[7] |
X. Chi, Z. Wan and Z. Hao, The models of bilevel programming with lower level second-order cone programs,, J. Inequal. Appl., 2014 (2014).
doi: 10.1186/1029-242X-2014-168. |
[8] |
S. Dempe, Foundations of Bilevel Programming,, Kluwer Academic Publishers, (2002).
|
[9] |
S. Dempe, Bilevel Optimization Problem: Existence of Optimal Solutions and Optimality Conditions, Report of CIMPA school, University of Delhi,, 2013., (). Google Scholar |
[10] |
S. Dempe and J. Dutta, Is bilevel programming a special case of a mathematical programming with complementarity constraints?,, Math. Program., 131 (2012), 37.
doi: 10.1007/s10107-010-0342-1. |
[11] |
C. Ding, D. Sun and J. Y. Ye, First order optimality conditions for mathematical programs with semidefinite cone complementarity constraints,, Math. Program., 147 (2014), 539.
doi: 10.1007/s10107-013-0735-z. |
[12] |
T. Ejiri, A Smoothing Method for Mathematical Programs with Second-Order Cone Complementarity Constraints,, Master thesis, (2007). Google Scholar |
[13] |
U. Faraut and A. Korányi, Analysis on Symmetric Cones,, Oxford University Press, (1994).
|
[14] |
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. |
[15] |
P. B. Hermanns and N. V. Thoai, Global optimization algorithm for solving bilevel programming problems with quadratic lower levels,, J. Ind. Manag Optim., 6 (2010), 177.
doi: 10.3934/jimo.2010.6.177. |
[16] |
Z. H. Huang and N. Lu, Global and global linear convergence of smoothing algorithm for the Cartesian $P_{*}(\kappa)$-SCLCP,, J. Ind. Manag Optim., 8 (2012), 67.
doi: 10.3934/jimo.2012.8.67. |
[17] |
Y. Jiang, Optimization Problems with Second-Order Cone Equilibrium Constraints, (Chinese), Ph.D. thesis, (2011). Google Scholar |
[18] |
H. Th. Jongen and G.-W. Weber, Nonlinear optimization: Characterization of structural optimization,, J. Glob. Optim., 1 (1991), 47.
doi: 10.1007/BF00120665. |
[19] |
M. Kojima, Strongly stable stationary solutions in nonlinear programs,, in Analysis and Computation of Fixed Points (eds. S.M. Robinson), 43 (1980), 93.
|
[20] |
Y. J. Kuo and H. D. Mittelmann, Interior point methods for second-order cone programming and OR applications,, Comput. Optim. Appl., 28 (2004), 255.
doi: 10.1023/B:COAP.0000033964.95511.23. |
[21] |
X. H. Liu and W. Z. Gu, Smoothing Newton algorithm based on a regularized one-parametric class of smoothing functions for generalized complementarity problems over symmetric cones,, J. Ind. Manag Optim., 6 (2010), 363.
doi: 10.3934/jimo.2010.6.363. |
[22] |
Y. Liu and L. Zhang, Convergence analysis of the augmented Lagrangian method for nonlinear second-order cone optimization problems,, Nonlinear Anal., 67 (2007), 1359.
doi: 10.1016/j.na.2006.07.022. |
[23] |
M. S. Lobo, L. Vandenberghe, S. Boyd and H. Lebret, Applications of second-order cone programming,, Linear Algebra Appl., 284 (1998), 193.
doi: 10.1016/S0024-3795(98)10032-0. |
[24] |
B. S. Mordukhovich, Variational Analysis and Generalized Differentiation, vol.I : Basic Theory,, Springer, (2006).
|
[25] |
J. V. Outrata and D. Sun, On the coderivative of the projection operator onto the second-order cone,, Set-Valued Anal., 16 (2008), 999.
doi: 10.1007/s11228-008-0092-x. |
[26] |
R. T. Rockafellar and R. J. -B. Wets, Variational Analysis,, Springer, (1998).
doi: 10.1007/978-3-642-02431-3. |
[27] |
J. Wu, L. Zhang and Y. Zhang, A smoothing Newton method for mathematical programs governed by second-order cone constrained generalized equations,, J. Glob. Optim., 55 (2013), 359.
doi: 10.1007/s10898-012-9880-9. |
[28] |
H. Yamamura, T. Okuno, S. Hayashi and M. Fukushima, A smoothing SQP method for mathematical programms with linear second-order cone complementarity constraints,, Pac. J. Optim., 9 (2013), 345.
|
[29] |
T. Yan and M. Fukushima, Smoothing method for mathematical programs with symmetric cone complementarity constraints,, Optimization, 60 (2011), 113.
doi: 10.1080/02331934.2010.541458. |
[30] |
Y. Zhang, L. Zhang and J. Wu, Convergence properties of a smoothing approach for mathematical programs with second-order cone complementarity constraints,, Set-Valued Anal., 19 (2011), 609.
doi: 10.1007/s11228-011-0190-z. |
show all references
References:
[1] |
F. Alizadeh and D. Goldfarb, Second-order cone Programming,, Math. Program., 95 (2003), 3.
doi: 10.1007/s10107-002-0339-5. |
[2] |
A. Ben-Tal, L. El Ghaoui and A. Nemirovski, Robust Optimization,, Princeton University Press, (2009).
doi: 10.1515/9781400831050. |
[3] |
J. F. Bonnans and A. Shapiro, Perturbation Analysis of Optimization Problems,, Springer, (2000).
doi: 10.1007/978-1-4612-1394-9. |
[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. -D. Chen, D. Sun and J. Sun, Complementarity functions and numerical experiments for second-order cone complementarity problems,, Comput. Optim. Appl., 25 (2003), 39.
doi: 10.1023/A:1022996819381. |
[6] |
X. J. Chen and S. H. Xiang, Computation of error bounds for P-matrix linear complementarity problems,, Math. Program., 106 (2006), 513.
doi: 10.1007/s10107-005-0645-9. |
[7] |
X. Chi, Z. Wan and Z. Hao, The models of bilevel programming with lower level second-order cone programs,, J. Inequal. Appl., 2014 (2014).
doi: 10.1186/1029-242X-2014-168. |
[8] |
S. Dempe, Foundations of Bilevel Programming,, Kluwer Academic Publishers, (2002).
|
[9] |
S. Dempe, Bilevel Optimization Problem: Existence of Optimal Solutions and Optimality Conditions, Report of CIMPA school, University of Delhi,, 2013., (). Google Scholar |
[10] |
S. Dempe and J. Dutta, Is bilevel programming a special case of a mathematical programming with complementarity constraints?,, Math. Program., 131 (2012), 37.
doi: 10.1007/s10107-010-0342-1. |
[11] |
C. Ding, D. Sun and J. Y. Ye, First order optimality conditions for mathematical programs with semidefinite cone complementarity constraints,, Math. Program., 147 (2014), 539.
doi: 10.1007/s10107-013-0735-z. |
[12] |
T. Ejiri, A Smoothing Method for Mathematical Programs with Second-Order Cone Complementarity Constraints,, Master thesis, (2007). Google Scholar |
[13] |
U. Faraut and A. Korányi, Analysis on Symmetric Cones,, Oxford University Press, (1994).
|
[14] |
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. |
[15] |
P. B. Hermanns and N. V. Thoai, Global optimization algorithm for solving bilevel programming problems with quadratic lower levels,, J. Ind. Manag Optim., 6 (2010), 177.
doi: 10.3934/jimo.2010.6.177. |
[16] |
Z. H. Huang and N. Lu, Global and global linear convergence of smoothing algorithm for the Cartesian $P_{*}(\kappa)$-SCLCP,, J. Ind. Manag Optim., 8 (2012), 67.
doi: 10.3934/jimo.2012.8.67. |
[17] |
Y. Jiang, Optimization Problems with Second-Order Cone Equilibrium Constraints, (Chinese), Ph.D. thesis, (2011). Google Scholar |
[18] |
H. Th. Jongen and G.-W. Weber, Nonlinear optimization: Characterization of structural optimization,, J. Glob. Optim., 1 (1991), 47.
doi: 10.1007/BF00120665. |
[19] |
M. Kojima, Strongly stable stationary solutions in nonlinear programs,, in Analysis and Computation of Fixed Points (eds. S.M. Robinson), 43 (1980), 93.
|
[20] |
Y. J. Kuo and H. D. Mittelmann, Interior point methods for second-order cone programming and OR applications,, Comput. Optim. Appl., 28 (2004), 255.
doi: 10.1023/B:COAP.0000033964.95511.23. |
[21] |
X. H. Liu and W. Z. Gu, Smoothing Newton algorithm based on a regularized one-parametric class of smoothing functions for generalized complementarity problems over symmetric cones,, J. Ind. Manag Optim., 6 (2010), 363.
doi: 10.3934/jimo.2010.6.363. |
[22] |
Y. Liu and L. Zhang, Convergence analysis of the augmented Lagrangian method for nonlinear second-order cone optimization problems,, Nonlinear Anal., 67 (2007), 1359.
doi: 10.1016/j.na.2006.07.022. |
[23] |
M. S. Lobo, L. Vandenberghe, S. Boyd and H. Lebret, Applications of second-order cone programming,, Linear Algebra Appl., 284 (1998), 193.
doi: 10.1016/S0024-3795(98)10032-0. |
[24] |
B. S. Mordukhovich, Variational Analysis and Generalized Differentiation, vol.I : Basic Theory,, Springer, (2006).
|
[25] |
J. V. Outrata and D. Sun, On the coderivative of the projection operator onto the second-order cone,, Set-Valued Anal., 16 (2008), 999.
doi: 10.1007/s11228-008-0092-x. |
[26] |
R. T. Rockafellar and R. J. -B. Wets, Variational Analysis,, Springer, (1998).
doi: 10.1007/978-3-642-02431-3. |
[27] |
J. Wu, L. Zhang and Y. Zhang, A smoothing Newton method for mathematical programs governed by second-order cone constrained generalized equations,, J. Glob. Optim., 55 (2013), 359.
doi: 10.1007/s10898-012-9880-9. |
[28] |
H. Yamamura, T. Okuno, S. Hayashi and M. Fukushima, A smoothing SQP method for mathematical programms with linear second-order cone complementarity constraints,, Pac. J. Optim., 9 (2013), 345.
|
[29] |
T. Yan and M. Fukushima, Smoothing method for mathematical programs with symmetric cone complementarity constraints,, Optimization, 60 (2011), 113.
doi: 10.1080/02331934.2010.541458. |
[30] |
Y. Zhang, L. Zhang and J. Wu, Convergence properties of a smoothing approach for mathematical programs with second-order cone complementarity constraints,, Set-Valued Anal., 19 (2011), 609.
doi: 10.1007/s11228-011-0190-z. |
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