• 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 & 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. doi: 10.1007/s10107-002-0339-5. Google Scholar

[2]

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

[3]

J. F. Bonnans and A. Shapiro, Perturbation Analysis of Optimization Problems,, Springer, (2000). doi: 10.1007/978-1-4612-1394-9. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[8]

S. Dempe, Foundations of Bilevel Programming,, Kluwer Academic Publishers, (2002). Google Scholar

[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. Google Scholar

[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. Google Scholar

[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). Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[19]

M. Kojima, Strongly stable stationary solutions in nonlinear programs,, in Analysis and Computation of Fixed Points (eds. S.M. Robinson), 43 (1980), 93. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[24]

B. S. Mordukhovich, Variational Analysis and Generalized Differentiation, vol.I : Basic Theory,, Springer, (2006). Google Scholar

[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. Google Scholar

[26]

R. T. Rockafellar and R. J. -B. Wets, Variational Analysis,, Springer, (1998). doi: 10.1007/978-3-642-02431-3. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

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. Google Scholar

[2]

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

[3]

J. F. Bonnans and A. Shapiro, Perturbation Analysis of Optimization Problems,, Springer, (2000). doi: 10.1007/978-1-4612-1394-9. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[8]

S. Dempe, Foundations of Bilevel Programming,, Kluwer Academic Publishers, (2002). Google Scholar

[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. Google Scholar

[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. Google Scholar

[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). Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[19]

M. Kojima, Strongly stable stationary solutions in nonlinear programs,, in Analysis and Computation of Fixed Points (eds. S.M. Robinson), 43 (1980), 93. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[24]

B. S. Mordukhovich, Variational Analysis and Generalized Differentiation, vol.I : Basic Theory,, Springer, (2006). Google Scholar

[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. Google Scholar

[26]

R. T. Rockafellar and R. J. -B. Wets, Variational Analysis,, Springer, (1998). doi: 10.1007/978-3-642-02431-3. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[1]

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

[2]

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 & Management Optimization, 2013, 9 (3) : 703-721. doi: 10.3934/jimo.2013.9.703

[3]

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

[4]

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

[5]

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

[6]

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

[7]

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

[8]

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

[9]

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

[10]

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 & Continuous Dynamical Systems - A, 2008, 21 (4) : 1047-1069. doi: 10.3934/dcds.2008.21.1047

[11]

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

[12]

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

[13]

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

[14]

Anurag Jayswala, Tadeusz Antczakb, Shalini Jha. Second order modified objective function method for twice differentiable vector optimization problems over cone constraints. Numerical Algebra, Control & Optimization, 2019, 9 (2) : 133-145. doi: 10.3934/naco.2019010

[15]

José F. Cariñena, Javier de Lucas Araujo. Superposition rules and second-order Riccati equations. Journal of Geometric Mechanics, 2011, 3 (1) : 1-22. doi: 10.3934/jgm.2011.3.1

[16]

Eugenii Shustin, Emilia Fridman, Leonid Fridman. Oscillations in a second-order discontinuous system with delay. Discrete & Continuous Dynamical Systems - A, 2003, 9 (2) : 339-358. doi: 10.3934/dcds.2003.9.339

[17]

Liping Tang, Xinmin Yang, Ying Gao. Higher-order symmetric duality for multiobjective programming with cone constraints. Journal of Industrial & Management Optimization, 2017, 13 (5) : 1-12. doi: 10.3934/jimo.2019033

[18]

B. Bonnard, J.-B. Caillau, E. Trélat. Second order optimality conditions with applications. Conference Publications, 2007, 2007 (Special) : 145-154. doi: 10.3934/proc.2007.2007.145

[19]

M. Soledad Aronna. Second order necessary and sufficient optimality conditions for singular solutions of partially-affine control problems. Discrete & Continuous Dynamical Systems - S, 2018, 11 (6) : 1233-1258. doi: 10.3934/dcdss.2018070

[20]

J.-P. Raymond, F. Tröltzsch. Second order sufficient optimality conditions for nonlinear parabolic control problems with state constraints. Discrete & Continuous Dynamical Systems - A, 2000, 6 (2) : 431-450. doi: 10.3934/dcds.2000.6.431

2018 Impact Factor: 1.025

Metrics

  • PDF downloads (16)
  • HTML views (0)
  • Cited by (1)

Other articles
by authors

[Back to Top]