American Institute of Mathematical Sciences

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October  2011, 7(4): 1041-1055. doi: 10.3934/jimo.2011.7.1041

A trust-region filter-SQP method for mathematical programs with linear complementarity constraints

Chunlin Hao 1, and  Xinwei Liu 2,

1. 

Department of Applied Mathematics, Beijing University of Technology, Beijing 100124, China
2. 

Department of Applied Mathematics, Hebei University of Technology, Tianjin 300401, China

Received  October 2010 Revised  July 2011 Published  August 2011

A trust-region filter-SQP method for mathematical programs with linear complementarity constraints (MPLCCs) is presented. The method is similar to that proposed by Liu, Perakis and Sun [Computational Optimization and Applications, 34, 5-33, 2006] but it solves the trust-region quadratic programming subproblems at each iteration and uses the filter technique to promote the global convergence. As a result, the method here is more robust since it admits the use of Lagrangian Hessian information and its performance is not affected by any penalty parameter. The preliminary numerical results on test problems generated by the QPECgen generator show that the presented method is effective.
Keywords: filter-SQP, Mathematical program with linear complementarity constraints, trust-region method..
Mathematics Subject Classification: Primary: 90C30, 90C51; Secondary: 65K0.
Citation: Chunlin Hao, Xinwei Liu. A trust-region filter-SQP method for mathematical programs with linear complementarity constraints. Journal of Industrial and Management Optimization, 2011, 7 (4) : 1041-1055. doi: 10.3934/jimo.2011.7.1041
References:
[1]

H. Benson, A. Sen, D. F. Shanno and R. J. Vanderbei, Interior-point algorithms, penalty methods and equilibrium problems, Comput. Optim. Appl., 34 (2006), 155-182. doi: 10.1007/s10589-005-3908-8.

[2]

L. Chen and D. Goldfarb, An active set method for mathematical programs with linear complementarity constraints,, Available from: \url{http://www.corc.ieor.columbia.edu/reports/techreports/tr-2007-02.pdf}, (): 2007. 

[3]

R. Fletcher, N. I. M. Gould, S. Leyffer, P. L. Toint and A. Wächter, Global convergence of trust-region SQP-filter algorithms for general nonlinear programming, SIAM J. Optim., 13 (2002), 635-659. doi: 10.1137/S1052623499357258.

[4]

R. Fletcher and S. Leyffer, Nonlinear programming without a penalty function, Math. Program Ser. A, 91 (2002), 239-269. doi: 10.1007/s101070100244.

[5]

R. Fletcher, S. Leyffer and C. Shen, Nonmonotone filter method for nonlinear optimization,, Available from: \url{http://wiki.mcs.anl.gov/leyffer/images/archive/c/c4/20091014223041!Nfilter.pdf}, (). 

[6]

R. Fletcher, S. Leyffer and P. L. Toint, On the global convergence of a filter-SQP algorithm, SIAM J. Optim., 13 (2002), 44-59. doi: 10.1137/S105262340038081X.

[7]

M. Fukushima, Z. Q. Luo and J. S. Pang, A globally convergent sequential quadratic programming algorithm for mathematical programs with linear complementarity constraints, Comput. Optim. Appl., 10 (1998), 5-34. doi: 10.1023/A:1018359900133.

[8]

M. Fukushima and J. S. Pang, Some feasibility issues in mathematical programs with equilibrium constraints, SIAM J. Optim., 8 (1998), 673-681. doi: 10.1137/S105262349731577X.

[9]

M. Fukushima and P. Tseng, An implementable active-set algorithm for computing a B-stationary point of a mathematical program with linear complementarity constraints, SIAM J. Optim., 12 (2002), 724-739. doi: 10.1137/S1052623499363232.

[10]

N. I. M. Gould, S. Leyffer and P. L. Toint, A multidimensional filter algorithm for nonlinear equations and nonlinear least squares, SIAM J. Optim., 15 (2004), 17-38. doi: 10.1137/S1052623403422637.

[11]

Z. Huang and J. Sun, A smoothing Newton algorithm for mathematical programs with complementarity constraints, J. Ind. Man. Optim., 1 (2005), 153-170.

[12]

H. Jiang and D. Ralph, QPECgen: A MATLAB generator for mathematical programs with quadratic objectives and affine variational inequality constraints, Computational Optimization-A Tribute to Olvi Magasarian, Part II, Comput. Optim. Appl., 13 (1999), 25-59.

[13]

H. Y. Jiang and D. Ralph, Smooth SQP methods for mathematical programs with nonlinear complementarity constraints, SIAM J. Optim., 10 (2000), 779-808. doi: 10.1137/S1052623497332329.

[14]

A. Kadrani, J.-P. Dussault and A. Benchakroun, A new regularization scheme for mathematical programs with complementarity constraints, SIAM J. Optim., 20 (2009), 78-103. doi: 10.1137/070705490.

[15]

S. Leyffer, Complementarity constraints as nonlinear equations: Theory and numerical experience, in "Optimization with Multivalued Mappings," 2, Springer, New York, (2006), 169-208.

[16]

S. Leyffer and T. S. Munson, A global convergent filter method for MPECs,, Available from: \url{http://www.mcs.anl.gov/~leyffer/papers/slpec.pdf}, (). 

[17]

G. Lin and M. Fukushima, New relaxation method for mathematical programs with complementarity constraints, J. Optim. Theory Appl., 118 (2003), 81-116. doi: 10.1023/A:1024739508603.

[18]

X.-W. Liu, G. Perakis and J. Sun, A robust SQP method for mathematical programs with linear complementarity constraints, Comput. Optim. Appl., 34 (2006), 5-33. doi: 10.1007/s10589-005-3075-y.

[19]

X.-W. Liu and J. Sun, Generalized stationary points and an interior-point method for mathematical programs with equilibrium constraints, Math. Program., 101 (2004), 231-261. doi: 10.1007/s10107-004-0543-6.

[20]

J. Long and S. Zeng, A projection-filter method for solving nonlinear complementarity problems, Appl. Math. Comput., 216 (2010), 300-307. doi: 10.1016/j.amc.2010.01.063.

[21]

Z. Q. Luo, J.-S. Pang and D. Ralph, "Mathematical Programs with Equilibrium Constraints," Cambridge University Press, Cambridge, 1996.

[22]

A. Raghunathan and L. T. Biegler, An interior point method for mathematical programs with complementarity constraints (MPCCs), SIAM J. Optim., 15 (2005), 720-750. doi: 10.1137/S1052623403429081.

[23]

D. Ralph, Sequential quadratic programming for mathematical programs with linear complementarity constraints, in "Computational Techniques and Applications: CTAC95" (eds. R. L. May and A. K. Easton), World Scientific Publ., River Edge, NJ, (1996), 663-668.

[24]

S. Schöltes, Convergence properties of regularization scheme for mathematical programs with complementarity constraints, SIAM J.Optim., 11 (2001), 918-936. doi: 10.1137/S1052623499361233.

[25]

S. Schöltes and M. Stöhr, Exact penalization of mathematical programs with equilibrium constraints, SIAM J. Control Optim., 37 (1999), 617-652. doi: 10.1137/S0363012996306121.

[26]

C. Shen, W. Xue and D. Pu, A globally convergent trust region multidimensional filter SQP algorithm for nonlinear programming, Int. J. Comput. Math., 86 (2009), 2201-2217. doi: 10.1080/00207160802702400.

[27]

A. Wächter and L. Biegler, Line search filter methods for nonlinear programming: Local convergence, SIAM J. Optim., 16 (2005), 32-48. doi: 10.1137/S1052623403426544.

[28]

A. Wächter and L. Biegler, Line search filter methods for nonlinear programming: Motivation and global convergence, SIAM J. Optim., 16 (2005), 1-31. doi: 10.1137/S1052623403426556.

[29]

J. Zhang, G. Liu and S. Wang, A globally convergent approximately active search algorithm for solving mathematical programs with linear complementarity constraints, Numer. Math., 98 (2004), 539-558. doi: 10.1007/s00211-004-0542-9.

show all references

References:
[1]

H. Benson, A. Sen, D. F. Shanno and R. J. Vanderbei, Interior-point algorithms, penalty methods and equilibrium problems, Comput. Optim. Appl., 34 (2006), 155-182. doi: 10.1007/s10589-005-3908-8.

[2]

L. Chen and D. Goldfarb, An active set method for mathematical programs with linear complementarity constraints,, Available from: \url{http://www.corc.ieor.columbia.edu/reports/techreports/tr-2007-02.pdf}, (): 2007. 

[3]

R. Fletcher, N. I. M. Gould, S. Leyffer, P. L. Toint and A. Wächter, Global convergence of trust-region SQP-filter algorithms for general nonlinear programming, SIAM J. Optim., 13 (2002), 635-659. doi: 10.1137/S1052623499357258.

[4]

R. Fletcher and S. Leyffer, Nonlinear programming without a penalty function, Math. Program Ser. A, 91 (2002), 239-269. doi: 10.1007/s101070100244.

[5]

R. Fletcher, S. Leyffer and C. Shen, Nonmonotone filter method for nonlinear optimization,, Available from: \url{http://wiki.mcs.anl.gov/leyffer/images/archive/c/c4/20091014223041!Nfilter.pdf}, (). 

[6]

R. Fletcher, S. Leyffer and P. L. Toint, On the global convergence of a filter-SQP algorithm, SIAM J. Optim., 13 (2002), 44-59. doi: 10.1137/S105262340038081X.

[7]

M. Fukushima, Z. Q. Luo and J. S. Pang, A globally convergent sequential quadratic programming algorithm for mathematical programs with linear complementarity constraints, Comput. Optim. Appl., 10 (1998), 5-34. doi: 10.1023/A:1018359900133.

[8]

M. Fukushima and J. S. Pang, Some feasibility issues in mathematical programs with equilibrium constraints, SIAM J. Optim., 8 (1998), 673-681. doi: 10.1137/S105262349731577X.

[9]

M. Fukushima and P. Tseng, An implementable active-set algorithm for computing a B-stationary point of a mathematical program with linear complementarity constraints, SIAM J. Optim., 12 (2002), 724-739. doi: 10.1137/S1052623499363232.

[10]

N. I. M. Gould, S. Leyffer and P. L. Toint, A multidimensional filter algorithm for nonlinear equations and nonlinear least squares, SIAM J. Optim., 15 (2004), 17-38. doi: 10.1137/S1052623403422637.

[11]

Z. Huang and J. Sun, A smoothing Newton algorithm for mathematical programs with complementarity constraints, J. Ind. Man. Optim., 1 (2005), 153-170.

[12]

H. Jiang and D. Ralph, QPECgen: A MATLAB generator for mathematical programs with quadratic objectives and affine variational inequality constraints, Computational Optimization-A Tribute to Olvi Magasarian, Part II, Comput. Optim. Appl., 13 (1999), 25-59.

[13]

H. Y. Jiang and D. Ralph, Smooth SQP methods for mathematical programs with nonlinear complementarity constraints, SIAM J. Optim., 10 (2000), 779-808. doi: 10.1137/S1052623497332329.

[14]

A. Kadrani, J.-P. Dussault and A. Benchakroun, A new regularization scheme for mathematical programs with complementarity constraints, SIAM J. Optim., 20 (2009), 78-103. doi: 10.1137/070705490.

[15]

S. Leyffer, Complementarity constraints as nonlinear equations: Theory and numerical experience, in "Optimization with Multivalued Mappings," 2, Springer, New York, (2006), 169-208.

[16]

S. Leyffer and T. S. Munson, A global convergent filter method for MPECs,, Available from: \url{http://www.mcs.anl.gov/~leyffer/papers/slpec.pdf}, (). 

[17]

G. Lin and M. Fukushima, New relaxation method for mathematical programs with complementarity constraints, J. Optim. Theory Appl., 118 (2003), 81-116. doi: 10.1023/A:1024739508603.

[18]

X.-W. Liu, G. Perakis and J. Sun, A robust SQP method for mathematical programs with linear complementarity constraints, Comput. Optim. Appl., 34 (2006), 5-33. doi: 10.1007/s10589-005-3075-y.

[19]

X.-W. Liu and J. Sun, Generalized stationary points and an interior-point method for mathematical programs with equilibrium constraints, Math. Program., 101 (2004), 231-261. doi: 10.1007/s10107-004-0543-6.

[20]

J. Long and S. Zeng, A projection-filter method for solving nonlinear complementarity problems, Appl. Math. Comput., 216 (2010), 300-307. doi: 10.1016/j.amc.2010.01.063.

[21]

Z. Q. Luo, J.-S. Pang and D. Ralph, "Mathematical Programs with Equilibrium Constraints," Cambridge University Press, Cambridge, 1996.

[22]

A. Raghunathan and L. T. Biegler, An interior point method for mathematical programs with complementarity constraints (MPCCs), SIAM J. Optim., 15 (2005), 720-750. doi: 10.1137/S1052623403429081.

[23]

D. Ralph, Sequential quadratic programming for mathematical programs with linear complementarity constraints, in "Computational Techniques and Applications: CTAC95" (eds. R. L. May and A. K. Easton), World Scientific Publ., River Edge, NJ, (1996), 663-668.

[24]

S. Schöltes, Convergence properties of regularization scheme for mathematical programs with complementarity constraints, SIAM J.Optim., 11 (2001), 918-936. doi: 10.1137/S1052623499361233.

[25]

S. Schöltes and M. Stöhr, Exact penalization of mathematical programs with equilibrium constraints, SIAM J. Control Optim., 37 (1999), 617-652. doi: 10.1137/S0363012996306121.

[26]

C. Shen, W. Xue and D. Pu, A globally convergent trust region multidimensional filter SQP algorithm for nonlinear programming, Int. J. Comput. Math., 86 (2009), 2201-2217. doi: 10.1080/00207160802702400.

[27]

A. Wächter and L. Biegler, Line search filter methods for nonlinear programming: Local convergence, SIAM J. Optim., 16 (2005), 32-48. doi: 10.1137/S1052623403426544.

[28]

A. Wächter and L. Biegler, Line search filter methods for nonlinear programming: Motivation and global convergence, SIAM J. Optim., 16 (2005), 1-31. doi: 10.1137/S1052623403426556.

[29]

J. Zhang, G. Liu and S. Wang, A globally convergent approximately active search algorithm for solving mathematical programs with linear complementarity constraints, Numer. Math., 98 (2004), 539-558. doi: 10.1007/s00211-004-0542-9.

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