• Previous Article
    Numerical solution of a pursuit-evasion differential game involving two spacecraft in low earth orbit
  • JIMO Home
  • This Issue
  • Next Article
    An interactive MOLP method for solving output-oriented DEA problems with undesirable factors
October  2015, 11(4): 1111-1125. doi: 10.3934/jimo.2015.11.1111

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

Received  September 2012 Revised  July 2014 Published  March 2015

When there is uncertainty in the lower level optimization problem of a bilevel programming, it can be formulated by a robust optimization method as a bilevel program with lower level second-order cone programming problem (SOCBLP). In this paper, we show that the Lagrange multiplier set mapping of the lower level problem of a class of the SOCBLPs is upper semicontinuous under suitable assumptions. Based on this fact, we detect the similarities and relationships between the SOCBLP and its KKT reformulation. Then we derive the specific expression of the critical cone at a feasible point, and show that the second order sufficient conditions are sufficient for the second order growth at an M-stationary point of the SOCBLP under suitable conditions.
Citation: Xiaoni Chi, Zhongping Wan, Zijun Hao. Second order sufficient conditions for a class of bilevel programs with lower level second-order cone programming problem. Journal of Industrial and Management Optimization, 2015, 11 (4) : 1111-1125. doi: 10.3934/jimo.2015.11.1111
References:
[1]

F. Alizadeh and D. Goldfarb, Second-order cone Programming, Math. Program., 95 (2003), 3-51. doi: 10.1007/s10107-002-0339-5.

[2]

A. Ben-Tal, L. El Ghaoui and A. Nemirovski, Robust Optimization, Princeton University Press, Princeton, 2009. doi: 10.1515/9781400831050.

[3]

J. F. Bonnans and A. Shapiro, Perturbation Analysis of Optimization Problems, Springer, New York, 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), 32-47. 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), 39-56. 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., Ser. A, 106 (2006), 513-525. 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), p168. doi: 10.1186/1029-242X-2014-168.

[8]

S. Dempe, Foundations of Bilevel Programming, Kluwer Academic Publishers, New York, 2002.

[9]

S. Dempe, Bilevel Optimization Problem: Existence of Optimal Solutions and Optimality Conditions, Report of CIMPA school, University of Delhi,, 2013., (). 

[10]

S. Dempe and J. Dutta, Is bilevel programming a special case of a mathematical programming with complementarity constraints?, Math. Program., 131 (2012), 37-48. 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., Ser. A, 147 (2014), 539-579. doi: 10.1007/s10107-013-0735-z.

[12]

T. Ejiri, A Smoothing Method for Mathematical Programs with Second-Order Cone Complementarity Constraints, Master thesis, Kyoto University in Kyoto, 2007.

[13]

U. Faraut and A. Korányi, Analysis on Symmetric Cones, Oxford University Press, New York, 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), 203-232. 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), 177-196. 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), 67-86. doi: 10.3934/jimo.2012.8.67.

[17]

Y. Jiang, Optimization Problems with Second-Order Cone Equilibrium Constraints, (Chinese) Ph.D. thesis, Dalian University of Technology in Dalian, 2011.

[18]

H. Th. Jongen and G.-W. Weber, Nonlinear optimization: Characterization of structural optimization, J. Glob. Optim., 1 (1991), 47-64. 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), Academic Press, 43 (1980), 93-138.

[20]

Y. J. Kuo and H. D. Mittelmann, Interior point methods for second-order cone programming and OR applications, Comput. Optim. Appl., 28 (2004), 255-285. 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), 363-380. 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), 1359-1373. 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), 193-228. doi: 10.1016/S0024-3795(98)10032-0.

[24]

B. S. Mordukhovich, Variational Analysis and Generalized Differentiation, vol.I : Basic Theory, Springer, Berlin, 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), 999-1014. doi: 10.1007/s11228-008-0092-x.

[26]

R. T. Rockafellar and R. J. -B. Wets, Variational Analysis, Springer, Berlin, 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), 359-385. 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), 345-372.

[29]

T. Yan and M. Fukushima, Smoothing method for mathematical programs with symmetric cone complementarity constraints, Optimization, 60 (2011), 113-128. 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), 609-646. 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), 3-51. doi: 10.1007/s10107-002-0339-5.

[2]

A. Ben-Tal, L. El Ghaoui and A. Nemirovski, Robust Optimization, Princeton University Press, Princeton, 2009. doi: 10.1515/9781400831050.

[3]

J. F. Bonnans and A. Shapiro, Perturbation Analysis of Optimization Problems, Springer, New York, 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), 32-47. 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), 39-56. 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., Ser. A, 106 (2006), 513-525. 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), p168. doi: 10.1186/1029-242X-2014-168.

[8]

S. Dempe, Foundations of Bilevel Programming, Kluwer Academic Publishers, New York, 2002.

[9]

S. Dempe, Bilevel Optimization Problem: Existence of Optimal Solutions and Optimality Conditions, Report of CIMPA school, University of Delhi,, 2013., (). 

[10]

S. Dempe and J. Dutta, Is bilevel programming a special case of a mathematical programming with complementarity constraints?, Math. Program., 131 (2012), 37-48. 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., Ser. A, 147 (2014), 539-579. doi: 10.1007/s10107-013-0735-z.

[12]

T. Ejiri, A Smoothing Method for Mathematical Programs with Second-Order Cone Complementarity Constraints, Master thesis, Kyoto University in Kyoto, 2007.

[13]

U. Faraut and A. Korányi, Analysis on Symmetric Cones, Oxford University Press, New York, 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), 203-232. 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), 177-196. 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), 67-86. doi: 10.3934/jimo.2012.8.67.

[17]

Y. Jiang, Optimization Problems with Second-Order Cone Equilibrium Constraints, (Chinese) Ph.D. thesis, Dalian University of Technology in Dalian, 2011.

[18]

H. Th. Jongen and G.-W. Weber, Nonlinear optimization: Characterization of structural optimization, J. Glob. Optim., 1 (1991), 47-64. 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), Academic Press, 43 (1980), 93-138.

[20]

Y. J. Kuo and H. D. Mittelmann, Interior point methods for second-order cone programming and OR applications, Comput. Optim. Appl., 28 (2004), 255-285. 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), 363-380. 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), 1359-1373. 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), 193-228. doi: 10.1016/S0024-3795(98)10032-0.

[24]

B. S. Mordukhovich, Variational Analysis and Generalized Differentiation, vol.I : Basic Theory, Springer, Berlin, 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), 999-1014. doi: 10.1007/s11228-008-0092-x.

[26]

R. T. Rockafellar and R. J. -B. Wets, Variational Analysis, Springer, Berlin, 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), 359-385. 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), 345-372.

[29]

T. Yan and M. Fukushima, Smoothing method for mathematical programs with symmetric cone complementarity constraints, Optimization, 60 (2011), 113-128. 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), 609-646. doi: 10.1007/s11228-011-0190-z.

[1]

Yi Zhang, Yong Jiang, Liwei Zhang, Jiangzhong Zhang. A perturbation approach for an inverse linear second-order cone programming. Journal of Industrial and Management Optimization, 2013, 9 (1) : 171-189. doi: 10.3934/jimo.2013.9.171

[2]

Li Chu, Bo Wang, Jie Zhang, Hong-Wei Zhang. Convergence analysis of a smoothing SAA method for a stochastic mathematical program with second-order cone complementarity constraints. Journal of Industrial and Management Optimization, 2021, 17 (4) : 1863-1886. doi: 10.3934/jimo.2020050

[3]

Ye Tian, Shu-Cherng Fang, Zhibin Deng, Wenxun Xing. Computable representation of the cone of nonnegative quadratic forms over a general second-order cone and its application to completely positive programming. Journal of Industrial and Management Optimization, 2013, 9 (3) : 703-721. doi: 10.3934/jimo.2013.9.703

[4]

Liwei Zhang, Jihong Zhang, Yule Zhang. Second-order optimality conditions for cone constrained multi-objective optimization. Journal of Industrial and Management Optimization, 2018, 14 (3) : 1041-1054. doi: 10.3934/jimo.2017089

[5]

Shiyun Wang, Yong-Jin Liu, Yong Jiang. A majorized penalty approach to inverse linear second order cone programming problems. Journal of Industrial and Management Optimization, 2014, 10 (3) : 965-976. doi: 10.3934/jimo.2014.10.965

[6]

Xi-De Zhu, Li-Ping Pang, Gui-Hua Lin. Two approaches for solving mathematical programs with second-order cone complementarity constraints. Journal of Industrial and Management Optimization, 2015, 11 (3) : 951-968. doi: 10.3934/jimo.2015.11.951

[7]

Yanhong Yuan, Hongwei Zhang, Liwei Zhang. A smoothing Newton method for generalized Nash equilibrium problems with second-order cone constraints. Numerical Algebra, Control and Optimization, 2012, 2 (1) : 1-18. doi: 10.3934/naco.2012.2.1

[8]

Xin-He Miao, Kai Yao, Ching-Yu Yang, Jein-Shan Chen. Levenberg-Marquardt method for absolute value equation associated with second-order cone. Numerical Algebra, Control and Optimization, 2022, 12 (1) : 47-61. doi: 10.3934/naco.2021050

[9]

Lin Zhu, Xinzhen Zhang. Semidefinite relaxation method for polynomial optimization with second-order cone complementarity constraints. Journal of Industrial and Management Optimization, 2022, 18 (3) : 1505-1517. doi: 10.3934/jimo.2021030

[10]

Narges Torabi Golsefid, Maziar Salahi. Second order cone programming formulation of the fixed cost allocation in DEA based on Nash bargaining game. Numerical Algebra, Control and Optimization, 2021  doi: 10.3934/naco.2021032

[11]

Rubén Figueroa, Rodrigo López Pouso, Jorge Rodríguez–López. Existence and multiplicity results for second-order discontinuous problems via non-ordered lower and upper solutions. Discrete and Continuous Dynamical Systems - B, 2020, 25 (2) : 617-633. doi: 10.3934/dcdsb.2019257

[12]

Hongwei Lou. Second-order necessary/sufficient conditions for optimal control problems in the absence of linear structure. Discrete and Continuous Dynamical Systems - B, 2010, 14 (4) : 1445-1464. doi: 10.3934/dcdsb.2010.14.1445

[13]

Ahmed Y. Abdallah. Upper semicontinuity of the attractor for a second order lattice dynamical system. Discrete and Continuous Dynamical Systems - B, 2005, 5 (4) : 899-916. doi: 10.3934/dcdsb.2005.5.899

[14]

Qian Liu. The lower bounds on the second-order nonlinearity of three classes of Boolean functions. Advances in Mathematics of Communications, 2021  doi: 10.3934/amc.2020136

[15]

Alberto Boscaggin, Fabio Zanolin. Subharmonic solutions for nonlinear second order equations in presence of lower and upper solutions. Discrete and Continuous Dynamical Systems, 2013, 33 (1) : 89-110. doi: 10.3934/dcds.2013.33.89

[16]

Mariane Bourgoing. Viscosity solutions of fully nonlinear second order parabolic equations with $L^1$ dependence in time and Neumann boundary conditions. Existence and applications to the level-set approach. Discrete and Continuous Dynamical Systems, 2008, 21 (4) : 1047-1069. doi: 10.3934/dcds.2008.21.1047

[17]

Qingsong Duan, Mengwei Xu, Liwei Zhang, Sainan Zhang. Hadamard directional differentiability of the optimal value of a linear second-order conic programming problem. Journal of Industrial and Management Optimization, 2021, 17 (6) : 3085-3098. doi: 10.3934/jimo.2020108

[18]

Lihua Li, Yan Gao, Hongjie Wang. Second order sufficient optimality conditions for hybrid control problems with state jump. Journal of Industrial and Management Optimization, 2015, 11 (1) : 329-343. doi: 10.3934/jimo.2015.11.329

[19]

Qiong Meng, X. H. Tang. Solutions of a second-order Hamiltonian system with periodic boundary conditions. Communications on Pure and Applied Analysis, 2010, 9 (4) : 1053-1067. doi: 10.3934/cpaa.2010.9.1053

[20]

Qilin Wang, Xiao-Bing Li, Guolin Yu. Second-order weak composed epiderivatives and applications to optimality conditions. Journal of Industrial and Management Optimization, 2013, 9 (2) : 455-470. doi: 10.3934/jimo.2013.9.455

2020 Impact Factor: 1.801

Metrics

  • PDF downloads (235)
  • HTML views (0)
  • Cited by (1)

Other articles
by authors

[Back to Top]