A note on quadratic convergence of a smoothing Newton algorithm for the LCP
The linear complementarity problem (LCP) is to find such that ( x , s ) ≥ 0, s = Mx + q , x T s = 0 with and . The smoothing Newton algorithm is one of the most efficient methods for solving the LCP. To the best of our knowledge, the best local convergence results of the smoothing Newton algori...
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| Published in | Optimization letters Vol. 7; no. 3; pp. 519 - 531 |
|---|---|
| Main Authors | , |
| Format | Journal Article |
| Language | English |
| Published |
Berlin/Heidelberg
Springer-Verlag
01.03.2013
|
| Subjects | |
| Online Access | Get full text |
| ISSN | 1862-4472 1862-4480 |
| DOI | 10.1007/s11590-011-0436-6 |
Cover
| Abstract | The linear complementarity problem (LCP) is to find
such that (
x
,
s
) ≥ 0,
s
=
Mx
+
q
,
x
T
s
= 0 with
and
. The smoothing Newton algorithm is one of the most efficient methods for solving the LCP. To the best of our knowledge, the best local convergence results of the smoothing Newton algorithm for the LCP up to now were obtained by Huang et al. (Math Program 99:423–441,
2004
). In this note, by using a revised Chen–Harker–Kanzow–Smale smoothing function, we propose a variation of Huang–Qi–Sun’s algorithm and show that the algorithm possesses better local convergence properties than those given in Huang et al. (Math Program 99:423–441,
2004
). |
|---|---|
| AbstractList | The linear complementarity problem (LCP) is to find
such that (
x
,
s
) ≥ 0,
s
=
Mx
+
q
,
x
T
s
= 0 with
and
. The smoothing Newton algorithm is one of the most efficient methods for solving the LCP. To the best of our knowledge, the best local convergence results of the smoothing Newton algorithm for the LCP up to now were obtained by Huang et al. (Math Program 99:423–441,
2004
). In this note, by using a revised Chen–Harker–Kanzow–Smale smoothing function, we propose a variation of Huang–Qi–Sun’s algorithm and show that the algorithm possesses better local convergence properties than those given in Huang et al. (Math Program 99:423–441,
2004
). |
| Author | Hu, Sheng-Long Ni, Tie |
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| Keywords | Smoothing Newton algorithm Quadratic convergence Linear complementarity problem |
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| References | HuS.L.HuangZ.H.WangP.A nonmonotone smoothing Newton algorithm for solving nonlinear complementarity problemsOptim. Methods Softw.200924344746025331011173.9055210.1080/10556780902769862 ClarkeF.H.Optimization and Nonsmooth Analysis1983New YorkWiley0582.49001 PardalosP.M.YeY.Class of linear complementarity problems solvable in polynomial timeLinear Algebra Appl.199115231711075420742.6505410.1016/0024-3795(91)90267-Z PardalosP.M.YeY.HanC.-G.KalinskiJ.Solution of P-matrix linear complementarity problems using a potential reduction algorithmSIAM J. Matrix Anal. Appl.1993141048106012389190788.6507210.1137/0614069 QiL.SunD.ZhouG.A new look at smoothing Newton methods for nonlinear complementarity problems and box constrained variational inequality problemsMath. Program.20008713517346570989.90124 HuangZ.H.SunJ.A non-interior continuation algorithm for the P0 or P* LCP with strong global and local convergence propertiesAppl. Math. Optim.20055223726221572021112.9008310.1007/s00245-005-0827-0 PardalosP.M.RosenJ.B.Global optimization approach to the linear complementarity problemSIAM J. Sci. Stat. Comput.1988934135393005010.1137/0909022 HuangZ.H.XuS.W.Convergence properties of a non-interior-point smoothing algorithm for the P* NCPJ. Ind. Manag. Optim.2007356958423439831166.9037010.3934/jimo.2007.3.671 QiL.SunD.Improving the convergence of non-interior point algorithm for nonlinear complementarity problemsMath. Comput.20006928330416427660947.90117 HuangZ.H.QiL.SunD.Sub-quadratic convergence of a smoothing Newton algorithm for the P0- and monotone LCPMath. Program.20049942344120517051168.9064610.1007/s10107-003-0457-8 SunD.A regularization Newton method for solving nonlinear complementarity problemsAppl. Math. Optim.19993631533910.1007/s002459900128 ZhaoY.B.LiD.A globally and locally superlinearly convergent non-interior-point algorithm for P0 LCPsSIAM J. Optim.2003131196122110.1137/S1052623401384151 QiL.SunJ.A nonsmooth version of Newtons methodMath. Program.19935835336712167910780.9009010.1007/BF01581275 HuangZ.H.Locating a maximally complementary solution of the monotone NCP by using non-interior-point smoothing algorithmsMath. Methods Oper. Res.200561415521204001066.9012710.1007/s001860400384 Z.H. Huang (436_CR6) 2007; 3 S.L. Hu (436_CR2) 2009; 24 F.H. Clarke (436_CR1) 1983 Z.H. Huang (436_CR3) 2005; 61 P.M. Pardalos (436_CR7) 1988; 9 P.M. Pardalos (436_CR8) 1991; 152 Z.H. Huang (436_CR4) 2004; 99 L. Qi (436_CR10) 1993; 58 D. Sun (436_CR13) 1999; 36 Y.B. Zhao (436_CR14) 2003; 13 P.M. Pardalos (436_CR9) 1993; 14 Z.H. Huang (436_CR5) 2005; 52 L. Qi (436_CR11) 2000; 69 L. Qi (436_CR12) 2000; 87 |
| References_xml | – reference: HuangZ.H.XuS.W.Convergence properties of a non-interior-point smoothing algorithm for the P* NCPJ. Ind. Manag. Optim.2007356958423439831166.9037010.3934/jimo.2007.3.671 – reference: QiL.SunD.Improving the convergence of non-interior point algorithm for nonlinear complementarity problemsMath. Comput.20006928330416427660947.90117 – reference: ClarkeF.H.Optimization and Nonsmooth Analysis1983New YorkWiley0582.49001 – reference: ZhaoY.B.LiD.A globally and locally superlinearly convergent non-interior-point algorithm for P0 LCPsSIAM J. Optim.2003131196122110.1137/S1052623401384151 – reference: HuS.L.HuangZ.H.WangP.A nonmonotone smoothing Newton algorithm for solving nonlinear complementarity problemsOptim. Methods Softw.200924344746025331011173.9055210.1080/10556780902769862 – reference: QiL.SunJ.A nonsmooth version of Newtons methodMath. Program.19935835336712167910780.9009010.1007/BF01581275 – reference: PardalosP.M.RosenJ.B.Global optimization approach to the linear complementarity problemSIAM J. Sci. Stat. Comput.1988934135393005010.1137/0909022 – reference: QiL.SunD.ZhouG.A new look at smoothing Newton methods for nonlinear complementarity problems and box constrained variational inequality problemsMath. Program.20008713517346570989.90124 – reference: PardalosP.M.YeY.Class of linear complementarity problems solvable in polynomial timeLinear Algebra Appl.199115231711075420742.6505410.1016/0024-3795(91)90267-Z – reference: HuangZ.H.QiL.SunD.Sub-quadratic convergence of a smoothing Newton algorithm for the P0- and monotone LCPMath. Program.20049942344120517051168.9064610.1007/s10107-003-0457-8 – reference: SunD.A regularization Newton method for solving nonlinear complementarity problemsAppl. Math. Optim.19993631533910.1007/s002459900128 – reference: HuangZ.H.SunJ.A non-interior continuation algorithm for the P0 or P* LCP with strong global and local convergence propertiesAppl. Math. Optim.20055223726221572021112.9008310.1007/s00245-005-0827-0 – reference: HuangZ.H.Locating a maximally complementary solution of the monotone NCP by using non-interior-point smoothing algorithmsMath. Methods Oper. Res.200561415521204001066.9012710.1007/s001860400384 – reference: PardalosP.M.YeY.HanC.-G.KalinskiJ.Solution of P-matrix linear complementarity problems using a potential reduction algorithmSIAM J. Matrix Anal. Appl.1993141048106012389190788.6507210.1137/0614069 – volume: 52 start-page: 237 year: 2005 ident: 436_CR5 publication-title: Appl. Math. Optim. doi: 10.1007/s00245-005-0827-0 – volume: 61 start-page: 41 year: 2005 ident: 436_CR3 publication-title: Math. Methods Oper. Res. doi: 10.1007/s001860400384 – volume: 36 start-page: 315 year: 1999 ident: 436_CR13 publication-title: Appl. Math. Optim. doi: 10.1007/s002459900128 – volume: 3 start-page: 569 year: 2007 ident: 436_CR6 publication-title: J. Ind. Manag. Optim. doi: 10.3934/jimo.2007.3.671 – volume: 152 start-page: 3 year: 1991 ident: 436_CR8 publication-title: Linear Algebra Appl. doi: 10.1016/0024-3795(91)90267-Z – volume: 9 start-page: 341 year: 1988 ident: 436_CR7 publication-title: SIAM J. Sci. Stat. Comput. doi: 10.1137/0909022 – volume: 14 start-page: 1048 year: 1993 ident: 436_CR9 publication-title: SIAM J. Matrix Anal. Appl. doi: 10.1137/0614069 – volume-title: Optimization and Nonsmooth Analysis year: 1983 ident: 436_CR1 – volume: 69 start-page: 283 year: 2000 ident: 436_CR11 publication-title: Math. Comput. doi: 10.1090/S0025-5718-99-01082-0 – volume: 58 start-page: 353 year: 1993 ident: 436_CR10 publication-title: Math. Program. doi: 10.1007/BF01581275 – volume: 13 start-page: 1196 year: 2003 ident: 436_CR14 publication-title: SIAM J. Optim. – volume: 99 start-page: 423 year: 2004 ident: 436_CR4 publication-title: Math. Program. doi: 10.1007/s10107-003-0457-8 – volume: 24 start-page: 447 issue: 3 year: 2009 ident: 436_CR2 publication-title: Optim. Methods Softw. doi: 10.1080/10556780902769862 – volume: 87 start-page: 1 year: 2000 ident: 436_CR12 publication-title: Math. Program. doi: 10.1007/s101079900127 |
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| Snippet | The linear complementarity problem (LCP) is to find
such that (
x
,
s
) ≥ 0,
s
=
Mx
+
q
,
x
T
s
= 0 with
and
. The smoothing Newton algorithm is one of... |
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| StartPage | 519 |
| SubjectTerms | Computational Intelligence Mathematics Mathematics and Statistics Numerical and Computational Physics Operations Research/Decision Theory Optimization Original Paper Simulation |
| Title | A note on quadratic convergence of a smoothing Newton algorithm for the LCP |
| URI | https://link.springer.com/article/10.1007/s11590-011-0436-6 |
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