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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.
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