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

Previous Article
Optimization of capital structure in real estate enterprises
JIMO Home
This Issue
Next Article
Joint pricing and replenishment decisions for non-instantaneous deteriorating items with partial backlogging, inflation- and selling price-dependent demand and customer returns

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
Related Papers

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 and 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), 3-51. 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), 95-117. 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), 33-74.
[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), 39-56. doi: 10.1023/A:1022996819381.
[5]	Y. Chen and M. Florian, The nonlinear bilevel programming problem: Formulations, regularity and optimality conditions, Optimization, 32 (1995), 193-209. doi: 10.1080/02331939508844048.
[6]	U. Faraut and A. Korányi, Analysis on Symmetric Cones, Oxford Mathematical Monographs, Oxford University Press, New York, 1994.
[7]	M. Fukushima and G. H. Lin, Smoothing methods for mathematical programs with equilibrium constraints, Proceedings of the ICKS'04, IEEE Computer Society, 2004 (2004), 206-213. 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), 436-460. 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), 593-615. 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), 59-78. 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, Cambridge, United Kingdom, 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, Dordrect, The Netherlands, 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), 999-1014. 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), 113-128. 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), 345-372.
[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), 481-507. 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), 609-646. 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), 3-51. 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), 95-117. 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), 33-74.
[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), 39-56. doi: 10.1023/A:1022996819381.
[5]	Y. Chen and M. Florian, The nonlinear bilevel programming problem: Formulations, regularity and optimality conditions, Optimization, 32 (1995), 193-209. doi: 10.1080/02331939508844048.
[6]	U. Faraut and A. Korányi, Analysis on Symmetric Cones, Oxford Mathematical Monographs, Oxford University Press, New York, 1994.
[7]	M. Fukushima and G. H. Lin, Smoothing methods for mathematical programs with equilibrium constraints, Proceedings of the ICKS'04, IEEE Computer Society, 2004 (2004), 206-213. 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), 436-460. 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), 593-615. 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), 59-78. 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, Cambridge, United Kingdom, 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, Dordrect, The Netherlands, 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), 999-1014. 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), 113-128. 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), 345-372.
[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), 481-507. 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), 609-646. doi: 10.1007/s11228-011-0190-z.

[1]	Liping Zhang, Soon-Yi Wu, Shu-Cherng Fang. Convergence and error bound of a D-gap function based Newton-type algorithm for equilibrium problems. Journal of Industrial and Management Optimization, 2010, 6 (2) : 333-346. doi: 10.3934/jimo.2010.6.333
[2]	Xiaoqin Jiang, Ying Zhang. A smoothing-type algorithm for absolute value equations. Journal of Industrial and Management Optimization, 2013, 9 (4) : 789-798. doi: 10.3934/jimo.2013.9.789
[3]	Zheng-Hai Huang, Shang-Wen Xu. Convergence properties of a non-interior-point smoothing algorithm for the PNCP. Journal of Industrial and Management Optimization*, 2007, 3 (3) : 569-584. doi: 10.3934/jimo.2007.3.569
[4]	Yi Zhang, Liwei Zhang, Jia Wu. On the convergence properties of a smoothing approach for mathematical programs with symmetric cone complementarity constraints. Journal of Industrial and Management Optimization, 2018, 14 (3) : 981-1005. doi: 10.3934/jimo.2017086
[5]	Regina S. Burachik, C. Yalçın Kaya. An update rule and a convergence result for a penalty function method. Journal of Industrial and Management Optimization, 2007, 3 (2) : 381-398. doi: 10.3934/jimo.2007.3.381
[6]	Xiaojiao Tong, Shuzi Zhou. A smoothing projected Newton-type method for semismooth equations with bound constraints. Journal of Industrial and Management Optimization, 2005, 1 (2) : 235-250. doi: 10.3934/jimo.2005.1.235
[7]	Filippo Dell'Oro, Vittorino Pata. Memory relaxation of type III thermoelastic extensible beams and Berger plates. Evolution Equations and Control Theory, 2012, 1 (2) : 251-270. doi: 10.3934/eect.2012.1.251
[8]	Na Zhao, Zheng-Hai Huang. A nonmonotone smoothing Newton algorithm for solving box constrained variational inequalities with a $P_0$ function. Journal of Industrial and Management Optimization, 2011, 7 (2) : 467-482. doi: 10.3934/jimo.2011.7.467
[9]	Z.Y. Wu, H.W.J. Lee, F.S. Bai, L.S. Zhang. Quadratic smoothing approximation to $l_1$ exact penalty function in global optimization. Journal of Industrial and Management Optimization, 2005, 1 (4) : 533-547. doi: 10.3934/jimo.2005.1.533
[10]	Sandra Carillo. Some remarks on the model of rigid heat conductor with memory: Unbounded heat relaxation function. Evolution Equations and Control Theory, 2019, 8 (1) : 31-42. doi: 10.3934/eect.2019002
[11]	Zheng-Hai Huang, Nan Lu. Global and global linear convergence of smoothing algorithm for the Cartesian $P_(\kappa)$-SCLCP. Journal of Industrial and Management Optimization*, 2012, 8 (1) : 67-86. doi: 10.3934/jimo.2012.8.67
[12]	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
[13]	Huijiang Zhao, Yinchuan Zhao. Convergence to strong nonlinear rarefaction waves for global smooth solutions of $p-$system with relaxation. Discrete and Continuous Dynamical Systems, 2003, 9 (5) : 1243-1262. doi: 10.3934/dcds.2003.9.1243
[14]	Stefan Kindermann, Antonio Leitão. Convergence rates for Kaczmarz-type regularization methods. Inverse Problems and Imaging, 2014, 8 (1) : 149-172. doi: 10.3934/ipi.2014.8.149
[15]	Qun Liu, Daqing Jiang, Ningzhong Shi, Tasawar Hayat, Ahmed Alsaedi. Stationarity and periodicity of positive solutions to stochastic SEIR epidemic models with distributed delay. Discrete and Continuous Dynamical Systems - B, 2017, 22 (6) : 2479-2500. doi: 10.3934/dcdsb.2017127
[16]	Zhi Guo Feng, Kok Lay Teo, Volker Rehbock. A smoothing approach for semi-infinite programming with projected Newton-type algorithm. Journal of Industrial and Management Optimization, 2009, 5 (1) : 141-151. doi: 10.3934/jimo.2009.5.141
[17]	Jean-Marie Barbaroux, Dirk Hundertmark, Tobias Ried, Semjon Vugalter. Strong smoothing for the non-cutoff homogeneous Boltzmann equation for Maxwellian molecules with Debye-Yukawa type interaction. Kinetic and Related Models, 2017, 10 (4) : 901-924. doi: 10.3934/krm.2017036
[18]	Steve Hofmann, Dorina Mitrea, Marius Mitrea, Andrew J. Morris. Square function estimates in spaces of homogeneous type and on uniformly rectifiable Euclidean sets. Electronic Research Announcements, 2014, 21: 8-18. doi: 10.3934/era.2014.21.8
[19]	Jeremy Levesley, Xinping Sun, Fahd Jarad, Alexander Kushpel. Interpolation of exponential-type functions on a uniform grid by shifts of a basis function. Discrete and Continuous Dynamical Systems - S, 2021, 14 (7) : 2399-2416. doi: 10.3934/dcdss.2020403
[20]	Xiaoqin P. Wu, Liancheng Wang. Hopf bifurcation of a class of two coupled relaxation oscillators of the van der Pol type with delay. Discrete and Continuous Dynamical Systems - B, 2010, 13 (2) : 503-516. doi: 10.3934/dcdsb.2010.13.503

2020 Impact Factor: 1.801

Tools

Metrics

Other articles
by authors

on AIMS
Xi-De Zhu Li-Ping Pang Gui-Hua Lin
on Google Scholar
Xi-De Zhu Li-Ping Pang Gui-Hua Lin

[Back to Top]