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Proximal point algorithm for nonlinear complementarity problem based on the generalized Fischer-Burmeister merit function
A perturbation approach for an inverse linear second-order cone programming
1. | Department of Mathematics, School of Science, East China University of Science and Technology, Shanghai, 200237, China |
2. | School of Science, Shenyang Aerospace University, Shenyang, 110136, China |
3. | Institute of ORCT, School of Mathematical Sciences, Dalian University of Technology, Dalian, 116024, China |
4. | Division of Science and Technology, Beijing Normal University-Hong Kong, Baptist University United International College, Zhuhai, 519085, China |
References:
[1] |
R. Ahuja and J. Orlin, Inverse optimization,, Operations Research, 49 (2001), 771.
doi: 10.1287/opre.49.5.771.10607. |
[2] |
R. Ahuja and J. Orlin, Combinatorial algorithms for inverse network flow problems,, Networks, 40 (2002), 181.
doi: 10.1002/net.10048. |
[3] |
F. Alizadeh and D. Goldfarb, Second order cone programming,, Mathematical Programming, 95 (2003), 3.
doi: 10.1007/s10107-002-0339-5. |
[4] |
W. Burton and P. Toint, On an instance of the inverse shortest paths problem,, Mathematical Programming, 53 (1992), 45.
doi: 10.1007/BF01585693. |
[5] |
M. Cai, X. Yang and J. Zhang, The complexity analysis of the inverse center location problem,, Journal of Global Optimization, 15 (1999), 213.
doi: 10.1023/A:1008360312607. |
[6] |
J. Chen, D. Sun and J. Sun, The $SC^1$ property of the squared norm of the SOC Fischer-Burmeister function,, Operations Research Letters, 36 (2008), 385.
doi: 10.1016/j.orl.2007.08.005. |
[7] |
X. Chen and M. Fukushima, A smoothing method for a mathematical program with P-matrix linear complementarity constraints,, Computational Optimization and Applications, 27 (2004), 223.
doi: 10.1023/B:COAP.0000013057.54647.6d. |
[8] |
Ejiri Takeshi, "A Smoothing Method for Mathematical Programs with Second-Order Cone Complementarity Constraints,", Master thesis, (2007). Google Scholar |
[9] |
F. Facchinei, H. Jiang and L. Qi, A smoothing method for mathematical programs with equilibrium constraints,, Mathematical Programming, 85 (1999), 107.
doi: 10.1007/s101070050048. |
[10] |
M. Fukushima, Z. Luo and J. Pang, A globally convergent sequential quadratic programming algorithm for mathematical programs with linear complementarity constraints,, Computational Optimization and Applications, 10 (1998), 5.
doi: 10.1023/A:1018359900133. |
[11] |
M. Fukushima, Z. Luo and P. Tseng, Smoothing functions for second-order cone complimentarity problems,, SIAM Journal on Optimization, 12 (2001), 436.
doi: 10.1137/S1052623400380365. |
[12] |
M. Fukushima and J. Pang, Convergence of a smoothing continuation method for mathematical problems with complementarity constraints,, Lecture Notes in Economics and Mathematical Systems, 477 (1999), 105.
doi: 10.1007/978-3-642-45780-7_7. |
[13] |
M. Grant and S. Boyd, "CVX Users' Guide,", Available from: , (). Google Scholar |
[14] |
C. Heuberger, Inverse combinatorial optimization: a survey on problems, methods and results,, Journal of Combinatorial Optimization, 8 (2004), 329.
doi: 10.1023/B:JOCO.0000038914.26975.9b. |
[15] |
G. Iyengar and W. Kang, Inverse conic programming and applications,, Operations Research Letters, 33 (2005), 319.
doi: 10.1016/j.orl.2004.04.007. |
[16] |
H. Jiang and D. Ralph, Smooth SQP methods for mathematical programs with nonlinear complementarity constraints,, SIAM Journal on Optimization, 10 (2000), 779.
doi: 10.1137/S1052623497332329. |
[17] |
G. Lin and M. Fukushima, Some exact penalty results for nonlinear programs and their applications to mathematical programs with equilibrium constraints,, Journal of Optimization Theory and Applications, 118 (2003), 67.
doi: 10.1023/A:1024787424532. |
[18] |
G. Lin and M. Fukushima, A modified relaxation scheme for mathematical prgrams with complementarity constraints,, Annals of Operations Research, 133 (2005), 63.
doi: 10.1007/s10479-004-5024-z. |
[19] |
Z. Luo, J. Pang and D. Ralph, "Mathematical Programs with Equilibrium Constraints,", Cambridge University Press, (1996).
doi: 10.1017/CBO9780511983658. |
[20] |
L. Qi, Convergence analysis of some algorithms for solving nonsmooth equations,, Mathematics of Operations Research, 18 (1993), 227.
doi: 10.1287/moor.18.1.227. |
[21] |
R. Rockafellar and R. Wets, "Variational Analysis,", Springer-Verlag, (1998).
doi: 10.1007/978-3-642-02431-3. |
[22] |
S. Scholtes and M. Stöhr, Exact penalization of mathematical programs with equilibrium constraints,, SIAM Journal on Control and Optimization, 37 (1999), 617.
doi: 10.1137/S0363012996306121. |
[23] |
S. Scholtes, Convergence properties of a regularization scheme for mathematical programs with complementarity constraints,, SIAM Journal on Optimization, 25 (2001), 1.
doi: 10.1137/S1052623499361233. |
[24] |
X. Xiao, L. Zhang and J. Zhang, A smoothing Newton method for a type of inverse semi-definite quadratic programming problem,, Journal of Computational and Applied Mathematics, 223 (2009), 485.
doi: 10.1016/j.cam.2008.01.028. |
[25] |
J. Zhang and Z. Liu, Calculating some inverse linear programming problems,, Journal of Computational and Applied Mathematics, 72 (1996), 261.
doi: 10.1016/0377-0427(95)00277-4. |
[26] |
J. Zhang and Z. Liu, A further study on inverse linear programming problems,, Journal of Computational and Applied Mathematics, 106 (1999), 345.
doi: 10.1016/S0377-0427(99)00080-1. |
[27] |
J. Zhang, Z. Liu and Z. Ma, Some reverse location problems,, European Journal of Operations Research, 124 (2000), 77.
doi: 10.1016/S0377-2217(99)00122-8. |
[28] |
J. Zhang and Z. Ma, Solution structure of some inverse combinatorial optimization problems,, Journal of Combinatorial Optimization, 3 (1999), 127.
doi: 10.1023/A:1009829525096. |
[29] |
J. Zhang and L. Zhang, An augmented Lagrangian method for a class of inverse quadratic programming problems,, Applied Mathematics and Optimization, 61 (2010), 57.
doi: 10.1007/s00245-009-9075-z. |
[30] |
J. Zhang, L. Zhang and X. Xiao, A Perturbation approach for an inverse quadratic programming problem,, Mathematical Methods of Operations Research, 72 (2010), 379.
doi: 10.1007/s00186-010-0323-4. |
[31] |
Y. Zhang, L. Zhang and J. Wu, Convergence properties of a smoothing approach for mathematical programs with second-order cone complementarity constraints,, Set-Valued and Variational Analysis, 19 (2011), 609.
doi: 10.1007/s11228-011-0190-z. |
show all references
References:
[1] |
R. Ahuja and J. Orlin, Inverse optimization,, Operations Research, 49 (2001), 771.
doi: 10.1287/opre.49.5.771.10607. |
[2] |
R. Ahuja and J. Orlin, Combinatorial algorithms for inverse network flow problems,, Networks, 40 (2002), 181.
doi: 10.1002/net.10048. |
[3] |
F. Alizadeh and D. Goldfarb, Second order cone programming,, Mathematical Programming, 95 (2003), 3.
doi: 10.1007/s10107-002-0339-5. |
[4] |
W. Burton and P. Toint, On an instance of the inverse shortest paths problem,, Mathematical Programming, 53 (1992), 45.
doi: 10.1007/BF01585693. |
[5] |
M. Cai, X. Yang and J. Zhang, The complexity analysis of the inverse center location problem,, Journal of Global Optimization, 15 (1999), 213.
doi: 10.1023/A:1008360312607. |
[6] |
J. Chen, D. Sun and J. Sun, The $SC^1$ property of the squared norm of the SOC Fischer-Burmeister function,, Operations Research Letters, 36 (2008), 385.
doi: 10.1016/j.orl.2007.08.005. |
[7] |
X. Chen and M. Fukushima, A smoothing method for a mathematical program with P-matrix linear complementarity constraints,, Computational Optimization and Applications, 27 (2004), 223.
doi: 10.1023/B:COAP.0000013057.54647.6d. |
[8] |
Ejiri Takeshi, "A Smoothing Method for Mathematical Programs with Second-Order Cone Complementarity Constraints,", Master thesis, (2007). Google Scholar |
[9] |
F. Facchinei, H. Jiang and L. Qi, A smoothing method for mathematical programs with equilibrium constraints,, Mathematical Programming, 85 (1999), 107.
doi: 10.1007/s101070050048. |
[10] |
M. Fukushima, Z. Luo and J. Pang, A globally convergent sequential quadratic programming algorithm for mathematical programs with linear complementarity constraints,, Computational Optimization and Applications, 10 (1998), 5.
doi: 10.1023/A:1018359900133. |
[11] |
M. Fukushima, Z. Luo and P. Tseng, Smoothing functions for second-order cone complimentarity problems,, SIAM Journal on Optimization, 12 (2001), 436.
doi: 10.1137/S1052623400380365. |
[12] |
M. Fukushima and J. Pang, Convergence of a smoothing continuation method for mathematical problems with complementarity constraints,, Lecture Notes in Economics and Mathematical Systems, 477 (1999), 105.
doi: 10.1007/978-3-642-45780-7_7. |
[13] |
M. Grant and S. Boyd, "CVX Users' Guide,", Available from: , (). Google Scholar |
[14] |
C. Heuberger, Inverse combinatorial optimization: a survey on problems, methods and results,, Journal of Combinatorial Optimization, 8 (2004), 329.
doi: 10.1023/B:JOCO.0000038914.26975.9b. |
[15] |
G. Iyengar and W. Kang, Inverse conic programming and applications,, Operations Research Letters, 33 (2005), 319.
doi: 10.1016/j.orl.2004.04.007. |
[16] |
H. Jiang and D. Ralph, Smooth SQP methods for mathematical programs with nonlinear complementarity constraints,, SIAM Journal on Optimization, 10 (2000), 779.
doi: 10.1137/S1052623497332329. |
[17] |
G. Lin and M. Fukushima, Some exact penalty results for nonlinear programs and their applications to mathematical programs with equilibrium constraints,, Journal of Optimization Theory and Applications, 118 (2003), 67.
doi: 10.1023/A:1024787424532. |
[18] |
G. Lin and M. Fukushima, A modified relaxation scheme for mathematical prgrams with complementarity constraints,, Annals of Operations Research, 133 (2005), 63.
doi: 10.1007/s10479-004-5024-z. |
[19] |
Z. Luo, J. Pang and D. Ralph, "Mathematical Programs with Equilibrium Constraints,", Cambridge University Press, (1996).
doi: 10.1017/CBO9780511983658. |
[20] |
L. Qi, Convergence analysis of some algorithms for solving nonsmooth equations,, Mathematics of Operations Research, 18 (1993), 227.
doi: 10.1287/moor.18.1.227. |
[21] |
R. Rockafellar and R. Wets, "Variational Analysis,", Springer-Verlag, (1998).
doi: 10.1007/978-3-642-02431-3. |
[22] |
S. Scholtes and M. Stöhr, Exact penalization of mathematical programs with equilibrium constraints,, SIAM Journal on Control and Optimization, 37 (1999), 617.
doi: 10.1137/S0363012996306121. |
[23] |
S. Scholtes, Convergence properties of a regularization scheme for mathematical programs with complementarity constraints,, SIAM Journal on Optimization, 25 (2001), 1.
doi: 10.1137/S1052623499361233. |
[24] |
X. Xiao, L. Zhang and J. Zhang, A smoothing Newton method for a type of inverse semi-definite quadratic programming problem,, Journal of Computational and Applied Mathematics, 223 (2009), 485.
doi: 10.1016/j.cam.2008.01.028. |
[25] |
J. Zhang and Z. Liu, Calculating some inverse linear programming problems,, Journal of Computational and Applied Mathematics, 72 (1996), 261.
doi: 10.1016/0377-0427(95)00277-4. |
[26] |
J. Zhang and Z. Liu, A further study on inverse linear programming problems,, Journal of Computational and Applied Mathematics, 106 (1999), 345.
doi: 10.1016/S0377-0427(99)00080-1. |
[27] |
J. Zhang, Z. Liu and Z. Ma, Some reverse location problems,, European Journal of Operations Research, 124 (2000), 77.
doi: 10.1016/S0377-2217(99)00122-8. |
[28] |
J. Zhang and Z. Ma, Solution structure of some inverse combinatorial optimization problems,, Journal of Combinatorial Optimization, 3 (1999), 127.
doi: 10.1023/A:1009829525096. |
[29] |
J. Zhang and L. Zhang, An augmented Lagrangian method for a class of inverse quadratic programming problems,, Applied Mathematics and Optimization, 61 (2010), 57.
doi: 10.1007/s00245-009-9075-z. |
[30] |
J. Zhang, L. Zhang and X. Xiao, A Perturbation approach for an inverse quadratic programming problem,, Mathematical Methods of Operations Research, 72 (2010), 379.
doi: 10.1007/s00186-010-0323-4. |
[31] |
Y. Zhang, L. Zhang and J. Wu, Convergence properties of a smoothing approach for mathematical programs with second-order cone complementarity constraints,, Set-Valued and Variational Analysis, 19 (2011), 609.
doi: 10.1007/s11228-011-0190-z. |
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