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A variational problem and optimal control
A smoothing homotopy method based on Robinson's normal equation for mixed complementarity problems
1. | School of Mathematical Sciences, Dalian University of Technology, Dalian, Liaoning 116024, China |
2. | School of Mathematical Science, Dalian University of Technology, Dalian, Liaoning 116024, China |
References:
[1] |
K. Ahuja, L. T. Watson and S. C. Billups, Probability-one homotopy maps for mixed complementarity problems, Comput. Optim. Appl., 41 (2008), 363-375.
doi: 10.1007/s10589-007-9107-z. |
[2] |
E. L. Allgower and K. Georg, "Numerical Continuation Methods, An Introduction," Springer Series in Computational Mathematics, 13, Springer-Verlag, Berlin, 1990. |
[3] |
S. C. Billups, A homotopy-based algorithm for mixed complementarity problems, SIAM J. Optim., 12 (2002), 583-605.
doi: 10.1137/S1052623498337431. |
[4] |
S. C. Billups, S. P. Dirkse and M. C. Ferris, A comparison of large scale mixed complementarity problem solvers, Computational Issues in High Performance Software for Nonlinear Optimization (Capri, 1995), Comput. Optim. Appl., 7 (1997), 3-25.
doi: 10.1023/A:1008632215341. |
[5] |
S. C. Billups and L. T. Watson, A probability-one homotopy algorithm for nonsmooth equations and mixed complementarity problems, SIAM J. Optim., 12 (2002), 606-626.
doi: 10.1137/S105262340037758X. |
[6] |
C. H. Chen and O. L. Mangasarian, A class of smoothing functions for nonlinear and mixed complementarity problems, Comput. Optim. Appl., 5 (1996), 97-138.
doi: 10.1007/BF00249052. |
[7] |
X. Chen, L. Qi and D. Sun, Global and superlinear convergence of the smoothing Newton method and its application to general box constrained variational inequalities, Math. Comp., 67 (1998), 519-540.
doi: 10.1090/S0025-5718-98-00932-6. |
[8] |
X. Chen and Y. Ye, On homotopy-smoothing methods for box-constrained variational inequalities, SIAM J. Control Optim., 37 (1999), 589-616.
doi: 10.1137/S0363012997315907. |
[9] |
A. N. Daryina, A. F. Izmailov and M. V. Solodov, Numerical results for a globalized active-set Newton method for mixed complementarity problems, Comput. Appl. Math., 24 (2005), 293-316.
doi: 10.1590/S0101-82052005000200008. |
[10] |
S. P. Dirkse and M. C. Ferris, MCPLIB: A collection of nonlinear mixed complementarity problems, Optim. Methods Softw., 5 (1995), 319-345.
doi: 10.1080/10556789508805619. |
[11] |
F. Facchinei and J.-S. Pang, "Finite-Dimensional Variational Inequalities and Complementarity Problems," Vol. I, Springer Series in Operations Research, Springer-Verlag, New York, 2003. |
[12] |
F. Facchinei and J.-S. Pang, "Finite-Dimensional Variational Inequalities and Complementarity Problems," Vol. II, Springer Series in Operations Research, Springer-Verlag, New York, 2003. |
[13] |
X. Fan and B. Yu, Homotopy method for solving variational inequalities with bounded box constraints, Nonlinear Anal., 68 (2008), 2357-2361.
doi: 10.1016/j.na.2007.01.063. |
[14] |
M. C. Ferris and T. S. Munson, Interfaces to PATH 3.0: Design, implementation and usage, Computational Optimization-A tribute to Olvi Mangasarian, Part I, Comput. Optim. Appl., 12 (1999), 207-227.
doi: 10.1023/A:1008636318275. |
[15] |
M. C. Ferris and J.-S. Pang, Engineering and economic applications of complementarity problems, SIAM Rev., 39 (1997), 669-713.
doi: 10.1137/S0036144595285963. |
[16] |
S. A. Gabriel and J. J. Moré, Smoothing of mixed complementarity problems, in "Complementarity and Variational Problems" (Baltimore, MD, 1995), SIAM, Philadelphia, PA, 1997, 105-116. |
[17] |
C. B. García and W. I. Zangwill, "Pathways to Solutions, Fixed Points and Equilibria," Prentice-Hall, Englewood Cliffs, New Jersey, 1981. |
[18] |
P. T. Harker and J.-S. Pang, Finite-dimensional variational inequality and nonlinear complementarity problems: A survey of theory, algorithms and applications, Math. Programming, 48 (1990), 161-220.
doi: 10.1007/BF01582255. |
[19] |
C. Kanzow and H. Pieper, Jacobian smoothing methods for nonlinear complementarity problems, SIAM J. Optim., 9 (1999), 342-373.
doi: 10.1137/S1052623497328781. |
[20] |
D. H. Li and M. Fukushima, Smoothing Newton and quasi-Newton methods for mixed complementarity problems, Nonsmooth and Smoothing Methods (Hong Kong, 1998), Comput. Optim. Appl., 17 (2000), 203-230.
doi: 10.1023/A:1026502415830. |
[21] |
J.-S. Pang, A B-differentiable equation-based, globally and locally quadratically convergent algorithm for nonlinear programs, complementarity and variational inequality problems, Math. Programming, 51 (1991), 101-131.
doi: 10.1007/BF01586928. |
[22] |
L. Q. Qi, D. F. Sun and G. L. Zhou, A new look at smoothing Newton methods for nonlinear complementarity problems and box constrained variational inequalities, Math. Program., 87 (2000), 1-35. |
[23] |
S. M. Robinson, Normal maps induced by linear transformations, Math. Oper. Res., 17 (1992), 691-714.
doi: 10.1287/moor.17.3.691. |
[24] |
T. F. Rutherfold, "MILES: A Mixed Inequality and Nonlinear Equation Solver," Working paper, Department of Economics, University of Colorado, Boulder, 1993. |
[25] |
D. F. Sun and R. S. Womersley, A new unconstrained differentiable merit function for box constrained variational inequality problems and a damped Gauss-Newton method, SIAM J. Optim., 9 (1999), 388-413.
doi: 10.1137/S1052623496314173. |
[26] |
D. F. Sun, R. S. Womersley and H. D. Qi, A feasible semismooth asymptotically Newton method for mixed complementarity problems, Math. Program., 94 (2002), 167-187.
doi: 10.1007/s10107-002-0305-2. |
[27] |
Q. Xu, B. Yu, G. C. Feng and C. Y. Dang, Condition for global convergence of a homotopy method for variational inequality problems on unbounded sets, Optim. Methods Softw., 22 (2007), 587-599.
doi: 10.1080/10556780600887883. |
[28] |
Q. Xu and C. Y. Dang, A new homotopy method for solving non-linear complementarity problems, Optimization, 57 (2008), 681-689.
doi: 10.1080/02331930802355317. |
[29] |
G. L. Zhou, D. F. Sun and L. Q. Qi, "Numerical Experiments for a Class of Squared Smoothing Newton Methods for Box Constrained Variational Inequality Problems," Reformation: Nonsmooth, Piecewise Smooth, Semismooth and Smoothing Methods (Lausanne, 1997), Appl. Optim., 22, Kluwer Acad. Publ., Dordrecht, (1999), 421-441. |
show all references
References:
[1] |
K. Ahuja, L. T. Watson and S. C. Billups, Probability-one homotopy maps for mixed complementarity problems, Comput. Optim. Appl., 41 (2008), 363-375.
doi: 10.1007/s10589-007-9107-z. |
[2] |
E. L. Allgower and K. Georg, "Numerical Continuation Methods, An Introduction," Springer Series in Computational Mathematics, 13, Springer-Verlag, Berlin, 1990. |
[3] |
S. C. Billups, A homotopy-based algorithm for mixed complementarity problems, SIAM J. Optim., 12 (2002), 583-605.
doi: 10.1137/S1052623498337431. |
[4] |
S. C. Billups, S. P. Dirkse and M. C. Ferris, A comparison of large scale mixed complementarity problem solvers, Computational Issues in High Performance Software for Nonlinear Optimization (Capri, 1995), Comput. Optim. Appl., 7 (1997), 3-25.
doi: 10.1023/A:1008632215341. |
[5] |
S. C. Billups and L. T. Watson, A probability-one homotopy algorithm for nonsmooth equations and mixed complementarity problems, SIAM J. Optim., 12 (2002), 606-626.
doi: 10.1137/S105262340037758X. |
[6] |
C. H. Chen and O. L. Mangasarian, A class of smoothing functions for nonlinear and mixed complementarity problems, Comput. Optim. Appl., 5 (1996), 97-138.
doi: 10.1007/BF00249052. |
[7] |
X. Chen, L. Qi and D. Sun, Global and superlinear convergence of the smoothing Newton method and its application to general box constrained variational inequalities, Math. Comp., 67 (1998), 519-540.
doi: 10.1090/S0025-5718-98-00932-6. |
[8] |
X. Chen and Y. Ye, On homotopy-smoothing methods for box-constrained variational inequalities, SIAM J. Control Optim., 37 (1999), 589-616.
doi: 10.1137/S0363012997315907. |
[9] |
A. N. Daryina, A. F. Izmailov and M. V. Solodov, Numerical results for a globalized active-set Newton method for mixed complementarity problems, Comput. Appl. Math., 24 (2005), 293-316.
doi: 10.1590/S0101-82052005000200008. |
[10] |
S. P. Dirkse and M. C. Ferris, MCPLIB: A collection of nonlinear mixed complementarity problems, Optim. Methods Softw., 5 (1995), 319-345.
doi: 10.1080/10556789508805619. |
[11] |
F. Facchinei and J.-S. Pang, "Finite-Dimensional Variational Inequalities and Complementarity Problems," Vol. I, Springer Series in Operations Research, Springer-Verlag, New York, 2003. |
[12] |
F. Facchinei and J.-S. Pang, "Finite-Dimensional Variational Inequalities and Complementarity Problems," Vol. II, Springer Series in Operations Research, Springer-Verlag, New York, 2003. |
[13] |
X. Fan and B. Yu, Homotopy method for solving variational inequalities with bounded box constraints, Nonlinear Anal., 68 (2008), 2357-2361.
doi: 10.1016/j.na.2007.01.063. |
[14] |
M. C. Ferris and T. S. Munson, Interfaces to PATH 3.0: Design, implementation and usage, Computational Optimization-A tribute to Olvi Mangasarian, Part I, Comput. Optim. Appl., 12 (1999), 207-227.
doi: 10.1023/A:1008636318275. |
[15] |
M. C. Ferris and J.-S. Pang, Engineering and economic applications of complementarity problems, SIAM Rev., 39 (1997), 669-713.
doi: 10.1137/S0036144595285963. |
[16] |
S. A. Gabriel and J. J. Moré, Smoothing of mixed complementarity problems, in "Complementarity and Variational Problems" (Baltimore, MD, 1995), SIAM, Philadelphia, PA, 1997, 105-116. |
[17] |
C. B. García and W. I. Zangwill, "Pathways to Solutions, Fixed Points and Equilibria," Prentice-Hall, Englewood Cliffs, New Jersey, 1981. |
[18] |
P. T. Harker and J.-S. Pang, Finite-dimensional variational inequality and nonlinear complementarity problems: A survey of theory, algorithms and applications, Math. Programming, 48 (1990), 161-220.
doi: 10.1007/BF01582255. |
[19] |
C. Kanzow and H. Pieper, Jacobian smoothing methods for nonlinear complementarity problems, SIAM J. Optim., 9 (1999), 342-373.
doi: 10.1137/S1052623497328781. |
[20] |
D. H. Li and M. Fukushima, Smoothing Newton and quasi-Newton methods for mixed complementarity problems, Nonsmooth and Smoothing Methods (Hong Kong, 1998), Comput. Optim. Appl., 17 (2000), 203-230.
doi: 10.1023/A:1026502415830. |
[21] |
J.-S. Pang, A B-differentiable equation-based, globally and locally quadratically convergent algorithm for nonlinear programs, complementarity and variational inequality problems, Math. Programming, 51 (1991), 101-131.
doi: 10.1007/BF01586928. |
[22] |
L. Q. Qi, D. F. Sun and G. L. Zhou, A new look at smoothing Newton methods for nonlinear complementarity problems and box constrained variational inequalities, Math. Program., 87 (2000), 1-35. |
[23] |
S. M. Robinson, Normal maps induced by linear transformations, Math. Oper. Res., 17 (1992), 691-714.
doi: 10.1287/moor.17.3.691. |
[24] |
T. F. Rutherfold, "MILES: A Mixed Inequality and Nonlinear Equation Solver," Working paper, Department of Economics, University of Colorado, Boulder, 1993. |
[25] |
D. F. Sun and R. S. Womersley, A new unconstrained differentiable merit function for box constrained variational inequality problems and a damped Gauss-Newton method, SIAM J. Optim., 9 (1999), 388-413.
doi: 10.1137/S1052623496314173. |
[26] |
D. F. Sun, R. S. Womersley and H. D. Qi, A feasible semismooth asymptotically Newton method for mixed complementarity problems, Math. Program., 94 (2002), 167-187.
doi: 10.1007/s10107-002-0305-2. |
[27] |
Q. Xu, B. Yu, G. C. Feng and C. Y. Dang, Condition for global convergence of a homotopy method for variational inequality problems on unbounded sets, Optim. Methods Softw., 22 (2007), 587-599.
doi: 10.1080/10556780600887883. |
[28] |
Q. Xu and C. Y. Dang, A new homotopy method for solving non-linear complementarity problems, Optimization, 57 (2008), 681-689.
doi: 10.1080/02331930802355317. |
[29] |
G. L. Zhou, D. F. Sun and L. Q. Qi, "Numerical Experiments for a Class of Squared Smoothing Newton Methods for Box Constrained Variational Inequality Problems," Reformation: Nonsmooth, Piecewise Smooth, Semismooth and Smoothing Methods (Lausanne, 1997), Appl. Optim., 22, Kluwer Acad. Publ., Dordrecht, (1999), 421-441. |
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