A new large-update interior point algorithm for P ( κ ) linear complementarity problems
In this paper we propose a new large-update primal-dual interior point algorithm for P * ( κ ) linear complementarity problems (LCPs). We generalize Bai et al.'s [A primal-dual interior-point method for linear optimization based on a new proximity function, Optim. Methods Software 17(2002) 985–...
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| Published in | Journal of computational and applied mathematics Vol. 216; no. 1; pp. 265 - 278 |
|---|---|
| Main Author | |
| Format | Journal Article |
| Language | English |
| Published |
Amsterdam
Elsevier B.V
15.06.2008
Elsevier |
| Subjects | |
| Online Access | Get full text |
| ISSN | 0377-0427 1879-1778 |
| DOI | 10.1016/j.cam.2007.05.007 |
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| Abstract | In this paper we propose a new large-update primal-dual interior point algorithm for
P
*
(
κ
)
linear complementarity problems (LCPs). We generalize Bai et al.'s [A primal-dual interior-point method for linear optimization based on a new proximity function, Optim. Methods Software 17(2002) 985–1008] primal-dual interior point algorithm for linear optimization (LO) problem to
P
*
(
κ
)
LCPs. New search directions and proximity measures are proposed based on a kernel function which is not logarithmic barrier nor self-regular for
P
*
(
κ
)
LCPs. We showed that if a strictly feasible starting point is available, then the new large-update primal-dual interior point algorithm for solving
P
*
(
κ
)
LCPs has the polynomial complexity
O
(
(
1
+
2
κ
)
n
3
/
4
log
(
n
/
ε
)
)
and gives a simple complexity analysis. This proximity function has not been used in the complexity analysis of interior point method (IPM) for
P
*
(
κ
)
LCPs before. |
|---|---|
| AbstractList | In this paper we propose a new large-update primal-dual interior point algorithm for
P
*
(
κ
)
linear complementarity problems (LCPs). We generalize Bai et al.'s [A primal-dual interior-point method for linear optimization based on a new proximity function, Optim. Methods Software 17(2002) 985–1008] primal-dual interior point algorithm for linear optimization (LO) problem to
P
*
(
κ
)
LCPs. New search directions and proximity measures are proposed based on a kernel function which is not logarithmic barrier nor self-regular for
P
*
(
κ
)
LCPs. We showed that if a strictly feasible starting point is available, then the new large-update primal-dual interior point algorithm for solving
P
*
(
κ
)
LCPs has the polynomial complexity
O
(
(
1
+
2
κ
)
n
3
/
4
log
(
n
/
ε
)
)
and gives a simple complexity analysis. This proximity function has not been used in the complexity analysis of interior point method (IPM) for
P
*
(
κ
)
LCPs before. |
| Author | Cho, Gyeong-Mi |
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| Cites_doi | 10.1080/1055678021000090024 10.1137/S1052623403423114 10.1007/BF01585565 10.1007/3-540-54509-3 10.1007/s101070200296 |
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| Keywords | Kernel function 65K05 90C33 49M15 Linear complementarity problem Polynomial algorithm Large-update interior point method Complexity Polynomial Logarithmic function Optimization method Interior point method Complementarity problem Algorithm 90C33,Large-update interior point method Numerical analysis Applied mathematics Software Mathematical programming |
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| References | Boyd, Vanderberghe (bib3) 2004 Wright (bib13) 1997 Cottle, Pang, Stone (bib5) 1992 Illes, Nagy (bib6) 2004 Kojima, Megiddo, Noma, Yoshise (bib7) 1989 Peng, Roos, Terlaky (bib11) 2002; 93 M. Kojima, N. Megiddo, T. Noma, A. Yoshise, A Unifided Approach to Interior Point Algorithms for Linear Complementarity Problems, Lecture Notes in Computer Science, vol. 538, Springer, Berlin, Germany, 1991. Miao (bib9) 1995; 69 Bai, El Ghami, Roos (bib1) 2002; 17 Bai, El Ghami, Roos (bib2) 2004; 15 Roos, Terlaky, Vial (bib12) 1997 Cho (bib4) 2005; 164 Peng, Roos, Terlaky (bib10) 2002 Kojima (10.1016/j.cam.2007.05.007_bib7) 1989 Roos (10.1016/j.cam.2007.05.007_bib12) 1997 Illes (10.1016/j.cam.2007.05.007_bib6) 2004 Boyd (10.1016/j.cam.2007.05.007_bib3) 2004 Bai (10.1016/j.cam.2007.05.007_bib1) 2002; 17 Cottle (10.1016/j.cam.2007.05.007_bib5) 1992 Bai (10.1016/j.cam.2007.05.007_bib2) 2004; 15 Peng (10.1016/j.cam.2007.05.007_bib10) 2002 Wright (10.1016/j.cam.2007.05.007_bib13) 1997 Cho (10.1016/j.cam.2007.05.007_bib4) 2005; 164 10.1016/j.cam.2007.05.007_bib8 Miao (10.1016/j.cam.2007.05.007_bib9) 1995; 69 Peng (10.1016/j.cam.2007.05.007_bib11) 2002; 93 |
| References_xml | – year: 1997 ident: bib13 article-title: Prima-dual interior-point methods – year: 1997 ident: bib12 article-title: Theory and Algorithms for Linear Optimization – year: 2002 ident: bib10 article-title: Self-Regularity: A New Paradigm for Primal-Dual Interior-Point Algorithms – volume: 69 start-page: 355 year: 1995 end-page: 368 ident: bib9 article-title: A quadratically convergent publication-title: Math. Programming – year: 2004 ident: bib3 article-title: Convex Optimizaton – volume: 164 start-page: 45 year: 2005 end-page: 69 ident: bib4 article-title: Log-barrier method for two-stage quadratic stochastic programming publication-title: Appl. Math. Comput. – year: 2004 ident: bib6 article-title: The Mizuno–Todd-Ye predictor–corrector algorithm for sufficient matrix linear complementarity problem – volume: 93 start-page: 129 year: 2002 end-page: 171 ident: bib11 article-title: Self-regular functions and new search directions for linear and semidefinite optimization publication-title: Math. Programming – volume: 17 start-page: 985 year: 2002 end-page: 1008 ident: bib1 article-title: A primal-dual interior-point method for linear optimization based on a new proximity function publication-title: Optim. Methods Software – year: 1992 ident: bib5 article-title: The Linear Complementarity Problem – start-page: 29 year: 1989 end-page: 47 ident: bib7 article-title: A primal-dual interior point algorithm for linear programming publication-title: Progress in Mathematical Programming; Interior Point Related Methods – volume: 15 start-page: 101 year: 2004 end-page: 128 ident: bib2 article-title: A comparative study of kernel functions for primal-dual interior-point algorithms in linear optimization publication-title: SIAM J. Optim. – reference: M. Kojima, N. Megiddo, T. Noma, A. Yoshise, A Unifided Approach to Interior Point Algorithms for Linear Complementarity Problems, Lecture Notes in Computer Science, vol. 538, Springer, Berlin, Germany, 1991. – volume: 17 start-page: 985 year: 2002 ident: 10.1016/j.cam.2007.05.007_bib1 article-title: A primal-dual interior-point method for linear optimization based on a new proximity function publication-title: Optim. Methods Software doi: 10.1080/1055678021000090024 – year: 2004 ident: 10.1016/j.cam.2007.05.007_bib6 – volume: 15 start-page: 101 year: 2004 ident: 10.1016/j.cam.2007.05.007_bib2 article-title: A comparative study of kernel functions for primal-dual interior-point algorithms in linear optimization publication-title: SIAM J. Optim. doi: 10.1137/S1052623403423114 – year: 2004 ident: 10.1016/j.cam.2007.05.007_bib3 – year: 1997 ident: 10.1016/j.cam.2007.05.007_bib12 – year: 1997 ident: 10.1016/j.cam.2007.05.007_bib13 – volume: 69 start-page: 355 year: 1995 ident: 10.1016/j.cam.2007.05.007_bib9 article-title: A quadratically convergent O((1+κ)nL)-iteration algorithm for the P*(κ)-matrix linear complementarity problem publication-title: Math. Programming doi: 10.1007/BF01585565 – year: 1992 ident: 10.1016/j.cam.2007.05.007_bib5 – volume: 164 start-page: 45 year: 2005 ident: 10.1016/j.cam.2007.05.007_bib4 article-title: Log-barrier method for two-stage quadratic stochastic programming publication-title: Appl. Math. Comput. – start-page: 29 year: 1989 ident: 10.1016/j.cam.2007.05.007_bib7 article-title: A primal-dual interior point algorithm for linear programming – ident: 10.1016/j.cam.2007.05.007_bib8 doi: 10.1007/3-540-54509-3 – year: 2002 ident: 10.1016/j.cam.2007.05.007_bib10 – volume: 93 start-page: 129 year: 2002 ident: 10.1016/j.cam.2007.05.007_bib11 article-title: Self-regular functions and new search directions for linear and semidefinite optimization publication-title: Math. Programming doi: 10.1007/s101070200296 |
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| Snippet | In this paper we propose a new large-update primal-dual interior point algorithm for
P
*
(
κ
)
linear complementarity problems (LCPs). We generalize Bai et... |
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| SubjectTerms | Calculus of variations and optimal control Complexity Exact sciences and technology Kernel function Large-update interior point method Linear complementarity problem Mathematical analysis Mathematics Numerical analysis Numerical analysis. Scientific computation Numerical methods in mathematical programming, optimization and calculus of variations Polynomial algorithm Sciences and techniques of general use |
| Title | A new large-update interior point algorithm for P ( κ ) linear complementarity problems |
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