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:

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[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. Google Scholar*
[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. Google Scholar
[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. Google Scholar
[5]	Y. Chen and M. Florian, The nonlinear bilevel programming problem: Formulations, regularity and optimality conditions,, Optimization, 32 (1995), 193. doi: 10.1080/02331939508844048. Google Scholar
[6]	U. Faraut and A. Korányi, Analysis on Symmetric Cones,, Oxford Mathematical Monographs, (1994). Google Scholar
[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. Google Scholar*
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[11]	Z. Q. Luo, J. S. Pang and D. Ralph, Mathematical Programs with Equilibrium Constraints,, Cambridge University Press, (1996). doi: 10.1017/CBO9780511983658. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar

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References:

[1]	F. Alizadeh and D. Goldfarb, Second-order cone programming,, Mathematical Programming, 95 (2003), 3. doi: 10.1007/s10107-002-0339-5. Google Scholar
[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. Google Scholar*
[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. Google Scholar
[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. Google Scholar
[5]	Y. Chen and M. Florian, The nonlinear bilevel programming problem: Formulations, regularity and optimality conditions,, Optimization, 32 (1995), 193. doi: 10.1080/02331939508844048. Google Scholar
[6]	U. Faraut and A. Korányi, Analysis on Symmetric Cones,, Oxford Mathematical Monographs, (1994). Google Scholar
[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. Google Scholar*
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[11]	Z. Q. Luo, J. S. Pang and D. Ralph, Mathematical Programs with Equilibrium Constraints,, Cambridge University Press, (1996). doi: 10.1017/CBO9780511983658. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar
[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. Google Scholar

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