Two approaches for solving mathematical programs with second-order cone complementarity constraints

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July 2015, 11(3): 951-968. doi: 10.3934/jimo.2015.11.951

Two approaches for solving mathematical programs with second-order cone complementarity constraints

Xi-De Zhu ^1, , Li-Ping Pang ^2, and Gui-Hua Lin ^3,

1.	School of Mathematical Sciences, Dalian University of Technology, Dalian 116024, China
2.	School of Mathematical Sciences, Dalian University of Technology, Dalian 116024, Liaoning
3.	School of Management, Shanghai University, Shanghai 200444, China

Received August 2013 Revised June 2014 Published October 2014

Abstract
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This paper considers a mathematical program with second-order cone complementarity constrains (MPSOCC). We present two approximation methods for solving the MPSOCC. One employs some smoothing functions to approximate the MPSOCC and the other makes use of some techniques to relax the complementarity constrains in the MPSOCC. We investigate the limiting behavior of both methods. In particular, we show that, under mild conditions, any accumulation point of stationary points of the approximation problems must be a Clarke-type stationary point of the MPSOCC.

Keywords: convergence., Clarke-type stationarity, smoothing function, MPSOCC, relaxation.

Mathematics Subject Classification: Primary: 90C30, 90C33; Secondary: 90C4.

Citation: 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

References:

[1]	F. Alizadeh and D. Goldfarb, Second-order cone programming,, Mathematical Programming, 95 (2003), 3. doi: 10.1007/s10107-002-0339-5.
[2]	J. S. Chen, X. Chen and P. Tseng, Analysis of nonsmooth vector-valued functions associated with second-order cones,, Mathematical Programming, 101* (2004), 95. doi: 10.1007/s10107-004-0538-3.*
[3]	J. S. Chen and S. Pan, A survey on SOC complementarity functions and solution methods for SOCPs and SOCCPs,, Pacific Journal of Optimization, 8 (2012), 33.
[4]	X. D. Chen, D. Sun and J. Sun, Complementarity functions and numerical experiments on some smoothing Newton methods for second-order-cone complementarity problems,, Computational Optimization and Applications, 25 (2003), 39. doi: 10.1023/A:1022996819381.
[5]	Y. Chen and M. Florian, The nonlinear bilevel programming problem: Formulations, regularity and optimality conditions,, Optimization, 32 (1995), 193. doi: 10.1080/02331939508844048.
[6]	U. Faraut and A. Korányi, Analysis on Symmetric Cones,, Oxford Mathematical Monographs, (1994).
[7]	M. Fukushima and G. H. Lin, Smoothing methods for mathematical programs with equilibrium constraints,, Proceedings of the ICKS'04, 2004* (2004), 206. doi: 10.1109/ICKS.2004.1313426.*
[8]	M. Fukushima, Z. Q. Luo and P. Tseng, Smoothing functions for second-order cone complementarity problems,, SIAM Journal on Optimization, 12 (2001), 436. doi: 10.1137/S1052623400380365.
[9]	S. Hayashi, N. Yamashita and M. Fukushima, A combined smoothing and regularization method for monotone second-order cone complementarity problems,, SIAM Journal on Optimization, 15 (2005), 593. doi: 10.1137/S1052623403421516.
[10]	Y. C. Liang, X. D. Zhu and G. H. Lin, Necessary optimality conditions for mathematical programs with second-order cone complementarity constraints,, Set-Valued and Variational Analysis, 22 (2014), 59. doi: 10.1007/s11228-013-0250-7.
[11]	Z. Q. Luo, J. S. Pang and D. Ralph, Mathematical Programs with Equilibrium Constraints,, Cambridge University Press, (1996). doi: 10.1017/CBO9780511983658.
[12]	J. Outrata, M. Kocvara and J. Zowe, Nonsmooth Approach to Optimization Problems with Equlilibrium Constraints: Theory, Applications, and Numerical Results,, Kluwer Academic Publisher, (1998). doi: 10.1007/978-1-4757-2825-5.
[13]	J. V. Outrata and D. F. Sun, On the coderivative of the projection operator onto the second order cone,, Set-Valued Analysis, 16 (2008), 999. doi: 10.1007/s11228-008-0092-x.
[14]	T. Yan and M. Fukushima, Smoothing method for mathematical programs with symmetric cone complementarity,, Optimization, 60 (2011), 113. doi: 10.1080/02331934.2010.541458.
[15]	H. Yamamura, T. Okuno, S. Hayashi and M. Fukushima, A smoothing SQP method for mathematical programs with linear second-order cone complementarity constraints,, Pacific Journal of Optimization, 9 (2013), 345.
[16]	J. J. Ye, D. L. Zhu and Q. J. Zhu, Exact penalization and neccessary conditions for generalized bilevel programming problems,, SIAM Journal on Optimizaion, 7 (1997), 481. doi: 10.1137/S1052623493257344.
[17]	Y. Zhang, L. Zhang and J. Wu, Convergence properties of a smoothing approach for mathematical programs with second-order cone complementarity constraints,, Set-Valued Analysis, 19 (2011), 609. doi: 10.1007/s11228-011-0190-z.

show all references

References:

[1]	F. Alizadeh and D. Goldfarb, Second-order cone programming,, Mathematical Programming, 95 (2003), 3. doi: 10.1007/s10107-002-0339-5.
[2]	J. S. Chen, X. Chen and P. Tseng, Analysis of nonsmooth vector-valued functions associated with second-order cones,, Mathematical Programming, 101* (2004), 95. doi: 10.1007/s10107-004-0538-3.*
[3]	J. S. Chen and S. Pan, A survey on SOC complementarity functions and solution methods for SOCPs and SOCCPs,, Pacific Journal of Optimization, 8 (2012), 33.
[4]	X. D. Chen, D. Sun and J. Sun, Complementarity functions and numerical experiments on some smoothing Newton methods for second-order-cone complementarity problems,, Computational Optimization and Applications, 25 (2003), 39. doi: 10.1023/A:1022996819381.
[5]	Y. Chen and M. Florian, The nonlinear bilevel programming problem: Formulations, regularity and optimality conditions,, Optimization, 32 (1995), 193. doi: 10.1080/02331939508844048.
[6]	U. Faraut and A. Korányi, Analysis on Symmetric Cones,, Oxford Mathematical Monographs, (1994).
[7]	M. Fukushima and G. H. Lin, Smoothing methods for mathematical programs with equilibrium constraints,, Proceedings of the ICKS'04, 2004* (2004), 206. doi: 10.1109/ICKS.2004.1313426.*
[8]	M. Fukushima, Z. Q. Luo and P. Tseng, Smoothing functions for second-order cone complementarity problems,, SIAM Journal on Optimization, 12 (2001), 436. doi: 10.1137/S1052623400380365.
[9]	S. Hayashi, N. Yamashita and M. Fukushima, A combined smoothing and regularization method for monotone second-order cone complementarity problems,, SIAM Journal on Optimization, 15 (2005), 593. doi: 10.1137/S1052623403421516.
[10]	Y. C. Liang, X. D. Zhu and G. H. Lin, Necessary optimality conditions for mathematical programs with second-order cone complementarity constraints,, Set-Valued and Variational Analysis, 22 (2014), 59. doi: 10.1007/s11228-013-0250-7.
[11]	Z. Q. Luo, J. S. Pang and D. Ralph, Mathematical Programs with Equilibrium Constraints,, Cambridge University Press, (1996). doi: 10.1017/CBO9780511983658.
[12]	J. Outrata, M. Kocvara and J. Zowe, Nonsmooth Approach to Optimization Problems with Equlilibrium Constraints: Theory, Applications, and Numerical Results,, Kluwer Academic Publisher, (1998). doi: 10.1007/978-1-4757-2825-5.
[13]	J. V. Outrata and D. F. Sun, On the coderivative of the projection operator onto the second order cone,, Set-Valued Analysis, 16 (2008), 999. doi: 10.1007/s11228-008-0092-x.
[14]	T. Yan and M. Fukushima, Smoothing method for mathematical programs with symmetric cone complementarity,, Optimization, 60 (2011), 113. doi: 10.1080/02331934.2010.541458.
[15]	H. Yamamura, T. Okuno, S. Hayashi and M. Fukushima, A smoothing SQP method for mathematical programs with linear second-order cone complementarity constraints,, Pacific Journal of Optimization, 9 (2013), 345.
[16]	J. J. Ye, D. L. Zhu and Q. J. Zhu, Exact penalization and neccessary conditions for generalized bilevel programming problems,, SIAM Journal on Optimizaion, 7 (1997), 481. doi: 10.1137/S1052623493257344.
[17]	Y. Zhang, L. Zhang and J. Wu, Convergence properties of a smoothing approach for mathematical programs with second-order cone complementarity constraints,, Set-Valued Analysis, 19 (2011), 609. doi: 10.1007/s11228-011-0190-z.

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