• 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

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

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.
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.  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

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.  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

[1]

Yi-Ming Tai, Zhengyang Zhang. Relaxation oscillations in a spruce-budworm interaction model with Holling's type II functional response. Discrete & Continuous Dynamical Systems - B, 2021  doi: 10.3934/dcdsb.2021027

[2]

Yifan Chen, Thomas Y. Hou. Function approximation via the subsampled Poincaré inequality. Discrete & Continuous Dynamical Systems - A, 2021, 41 (1) : 169-199. doi: 10.3934/dcds.2020296

[3]

George W. Patrick. The geometry of convergence in numerical analysis. Journal of Computational Dynamics, 2021, 8 (1) : 33-58. doi: 10.3934/jcd.2021003

[4]

Matania Ben–Artzi, Joseph Falcovitz, Jiequan Li. The convergence of the GRP scheme. Discrete & Continuous Dynamical Systems - A, 2009, 23 (1&2) : 1-27. doi: 10.3934/dcds.2009.23.1

[5]

Jing Zhou, Cheng Lu, Ye Tian, Xiaoying Tang. A SOCP relaxation based branch-and-bound method for generalized trust-region subproblem. Journal of Industrial & Management Optimization, 2021, 17 (1) : 151-168. doi: 10.3934/jimo.2019104

[6]

Bahaaeldin Abdalla, Thabet Abdeljawad. Oscillation criteria for kernel function dependent fractional dynamic equations. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020443

[7]

Liping Tang, Ying Gao. Some properties of nonconvex oriented distance function and applications to vector optimization problems. Journal of Industrial & Management Optimization, 2021, 17 (1) : 485-500. doi: 10.3934/jimo.2020117

[8]

Thierry Horsin, Mohamed Ali Jendoubi. On the convergence to equilibria of a sequence defined by an implicit scheme. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020465

[9]

Philipp Harms. Strong convergence rates for markovian representations of fractional processes. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020367

[10]

Alberto Bressan, Carlotta Donadello. On the convergence of viscous approximations after shock interactions. Discrete & Continuous Dynamical Systems - A, 2009, 23 (1&2) : 29-48. doi: 10.3934/dcds.2009.23.29

[11]

Mohammed Abdulrazaq Kahya, Suhaib Abduljabbar Altamir, Zakariya Yahya Algamal. Improving whale optimization algorithm for feature selection with a time-varying transfer function. Numerical Algebra, Control & Optimization, 2021, 11 (1) : 87-98. doi: 10.3934/naco.2020017

[12]

Lingfeng Li, Shousheng Luo, Xue-Cheng Tai, Jiang Yang. A new variational approach based on level-set function for convex hull problem with outliers. Inverse Problems & Imaging, , () : -. doi: 10.3934/ipi.2020070

[13]

Juntao Sun, Tsung-fang Wu. The number of nodal solutions for the Schrödinger–Poisson system under the effect of the weight function. Discrete & Continuous Dynamical Systems - A, 2021  doi: 10.3934/dcds.2021011

[14]

Madhurima Mukhopadhyay, Palash Sarkar, Shashank Singh, Emmanuel Thomé. New discrete logarithm computation for the medium prime case using the function field sieve. Advances in Mathematics of Communications, 2020  doi: 10.3934/amc.2020119

[15]

Kateřina Škardová, Tomáš Oberhuber, Jaroslav Tintěra, Radomír Chabiniok. Signed-distance function based non-rigid registration of image series with varying image intensity. Discrete & Continuous Dynamical Systems - S, 2021, 14 (3) : 1145-1160. doi: 10.3934/dcdss.2020386

[16]

Adrian Constantin, Darren G. Crowdy, Vikas S. Krishnamurthy, Miles H. Wheeler. Stuart-type polar vortices on a rotating sphere. Discrete & Continuous Dynamical Systems - A, 2021, 41 (1) : 201-215. doi: 10.3934/dcds.2020263

[17]

Parikshit Upadhyaya, Elias Jarlebring, Emanuel H. Rubensson. A density matrix approach to the convergence of the self-consistent field iteration. Numerical Algebra, Control & Optimization, 2021, 11 (1) : 99-115. doi: 10.3934/naco.2020018

[18]

Gang Luo, Qingzhi Yang. The point-wise convergence of shifted symmetric higher order power method. Journal of Industrial & Management Optimization, 2021, 17 (1) : 357-368. doi: 10.3934/jimo.2019115

[19]

Toshiko Ogiwara, Danielle Hilhorst, Hiroshi Matano. Convergence and structure theorems for order-preserving dynamical systems with mass conservation. Discrete & Continuous Dynamical Systems - A, 2020, 40 (6) : 3883-3907. doi: 10.3934/dcds.2020129

[20]

Xiuli Xu, Xueke Pu. Optimal convergence rates of the magnetohydrodynamic model for quantum plasmas with potential force. Discrete & Continuous Dynamical Systems - B, 2021, 26 (2) : 987-1010. doi: 10.3934/dcdsb.2020150

2019 Impact Factor: 1.366

Metrics

  • PDF downloads (62)
  • HTML views (0)
  • Cited by (1)

Other articles
by authors

[Back to Top]