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Nonconvex quadratic reformulations and solvable conditions for mixed integer quadratic programming problems
A trust-region filter-SQP method for mathematical programs with linear complementarity constraints
1. | Department of Applied Mathematics, Beijing University of Technology, Beijing 100124, China |
2. | Department of Applied Mathematics, Hebei University of Technology, Tianjin 300401, China |
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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