An interior-point algorithm for P∗(κ)-LCP based on a new trigonometric kernel function with a double barrier term

In this paper, we present a new large-update interior-point algorithm for P ∗ ( κ ) -linear complementarity problem. The new algorithm is based on a trigonometric kernel function which differs from the existing kernel functions in which it has a double barrier term. By a simple analysis, we show tha...

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Published in	Journal of applied mathematics & computing Vol. 53; no. 1-2; pp. 487 - 506
Main Authors	Li, Xin, Zhang, Mingwang, Chen, Yan
Format	Journal Article
Language	English
Published	Berlin/Heidelberg Springer Berlin Heidelberg 01.02.2017
Subjects	Computational Mathematics and Numerical Analysis Mathematical and Computational Engineering Mathematics Mathematics and Statistics Mathematics of Computing Original Research Theory of Computation Iteration complexity Kernel function 90C51 90C33 Interior-point algorithm Large-update Linear complementarity problem
Online Access	Get full text
ISSN	1598-5865 1865-2085
DOI	10.1007/s12190-015-0978-3

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Abstract	In this paper, we present a new large-update interior-point algorithm for P ∗ ( κ ) -linear complementarity problem. The new algorithm is based on a trigonometric kernel function which differs from the existing kernel functions in which it has a double barrier term. By a simple analysis, we show that the new algorithm enjoys O ( ( 1 + 2 κ ) n 2 3 log n ε ) iteration complexity. This complexity estimate improves a result from El Ghami et al. (Optim Theory Decis Mak Oper Res Appl 31: 331–349, 2013 ) and matches the currently best known complexity result for P ∗ ( κ ) -linear complementarity problem based on trigonometric kernel functions.
AbstractList	In this paper, we present a new large-update interior-point algorithm for P ∗ ( κ ) -linear complementarity problem. The new algorithm is based on a trigonometric kernel function which differs from the existing kernel functions in which it has a double barrier term. By a simple analysis, we show that the new algorithm enjoys O ( ( 1 + 2 κ ) n 2 3 log n ε ) iteration complexity. This complexity estimate improves a result from El Ghami et al. (Optim Theory Decis Mak Oper Res Appl 31: 331–349, 2013 ) and matches the currently best known complexity result for P ∗ ( κ ) -linear complementarity problem based on trigonometric kernel functions.
Author	Zhang, Mingwang Chen, Yan Li, Xin
Author_xml	– sequence: 1 givenname: Xin surname: Li fullname: Li, Xin email: sxlixin12@126.com organization: Department of Mathematics and Computer Science, Guangxi Normal University for Nationalities – sequence: 2 givenname: Mingwang surname: Zhang fullname: Zhang, Mingwang organization: College of Science, China Three Gorges University – sequence: 3 givenname: Yan surname: Chen fullname: Chen, Yan organization: Department of Mathematics and Computer Science, Guangxi Normal University for Nationalities
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Keywords	Iteration complexity Kernel function 90C51 90C33 Interior-point algorithm Large-update Linear complementarity problem
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References	HafshejaniSFFatemiMPeyghamMRAn interior-point method for P(κ)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$P_\ast (\kappa )$$\end{document}-linear complementarity problem based on a trigonometric kernel functionJ. Appl. Math. Comput.201548111128334059810.1007/s12190-014-0794-11323.90070 PengJRoosCTerlakyTSelf-regular functions and new search directions for linear and semidefinite optimizationMath. Program.200293129171191227110.1007/s1010702002961007.90037 PeyghamiMRHafshejaniSFShirvaniLComplexity of interior-point methods for linear optimization based on a new trigonometric kernel functionJ. Comput. Appl. Math.20142557485309340510.1016/j.cam.2013.04.0391291.90313 Cai, X.Z., Wang, G.Q., El Ghami, M., Yue, Y.J.: Complexity analysis of primal-dual interior-point methods for linear optimization based on a new parametric kernel function with a trigonometric barrier term. Abstr. Appl. Anal. ID 710158 (2014) HarkerPTPangJSSimulation and Optimization of Large Systems1990Providence, RIAMS265284 KheirfamBPrimal-dual interior-point algorithm for semidefinite optimization based on a new kernel function with trigonometric barrier termNumer. Algorithms201261659680299522610.1007/s11075-012-9557-y1259.65091 BaiYQLesajaGRoosCA new class of polynomial interior-point algorithms for P(κ)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$P_\ast (\kappa )$$\end{document} linear complementarity problemsPac. J. Optim.20084119411161.90507 GyeongMCLarge-update interior point algorithm for P(κ)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$P_\ast (\kappa )$$\end{document} linear complementarity problemJ. Inequal. Appl.201436311123347684 BaiYQEl GhamiMRoosCA comparative study of kernel functions for primal-dual interior-point algorithms in linear optimizationSIAM J. Optim.2004151101128211297810.1137/S10526234034231141077.90038 El GhamiMGuennounZABoulaSSteihaugTInterior-point methods for linear optimization based on a kernel function with a trigonometric barrier termJ. Comput. Appl. Math.201223636133623292349410.1016/j.cam.2011.05.0361242.90292 WangGQFanXJZhuDTWangDZNew complexity analysis of a full-Newton step feasible interior-point algorithm for P(κ)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$P_\ast (\kappa )$$\end{document}-LCPOptim. Lett.2015911051119337367110.1007/s11590-014-0800-41331.90085 WangGQBaiYQA new primal-dual path-following interior-point algorithm for semi-definite optimizationJ. Math. Anal. Appl.200835333934910.1016/j.jmaa.2008.12.0161172.90011 Lee, Y.H., Cho, Y.Y., Cho, G.M.: Interior-point algorithms for P(κ)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$P_\ast (\kappa )$$\end{document}-LCP based on a new class of kernel functions. J. Glob. Optim. (2013). doi:10.1007/s10898-013-0072-z ChoGMA new large-update interior point algorithm for P(κ)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$P_\ast (\kappa )$$\end{document} linear complementarity problemsJ. Comput. Appl. Math.2008216265278242185510.1016/j.cam.2007.05.007 ChoGMKimMKA new large-update interior point algorithm for P(κ)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$P_\ast (\kappa )$$\end{document} LCPs based on kernel functionsAppl. Math. Comput.200618221169118322825601108.65061 AminiKPeyghamiMRExploring complexity of large update interior-point methods for P(κ)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$P_\ast (\kappa )$$\end{document} linear complementarity problem based on kernel functionAppl. Math. Comput.200920750151324891041160.90009 ZhuDHZhangMWA full-Newton step infeasible interior-point algorithm for P(κ)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$P_\ast (\kappa )$$\end{document} linear complementarity problemJ. Syst. Sci. Complex.20142710271044326484410.1007/s11424-014-1273-31326.90091 AminiKHaseliAA new proximity function generating the best known iteration bounds for both large-update and small-update interior-point methodsANZIAM J.200749259270240851910.1017/S14461811000128271142.90039 WangGQBaiYQPolynomial interior-point algorithms for P(κ)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$P_\ast (\kappa )$$\end{document} horizontal linear complementarity problemJ. Comput. Appl. Math.2009233224826310.1016/j.cam.2009.07.0141183.65072 PengJRoosCTerlakyTSelf-Regularity: A New Paradigm for Primal-Dual Interior-Point Algorithms2002PrincetonPrinceton University Press1136.90045 El GhamiMPrimal dual interior-point methods for P(κ)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$P_\ast (\kappa )$$\end{document}-linear complementarity problem based on a kernel function with a trigonometric barrier termOptim. Theory Decis. Mak. Oper. Res. Appl.2013313313493068945 KojimaMMegiddoNNomaTYoshiseAA Unified Approach to Interior Point Algorithms for Linear Complementarity Problems1991BerlinSpringer0745.90069 Ferris, M.C., Pang, J.S.: Complementarity and variational problems state of the art. In: Proceedings of the International Conference on Complementarity Problems. SIAM, Philadelphia (1997) PeyghamiMRAminiKA kernel function based interior-point methods for solving P(κ)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$P_\ast (\kappa )$$\end{document}-linear complementarity problemActa Math. Sin. (Engl. Ser.)201026917611778267281610.1007/s10114-010-7529-51206.90195 WrightSJPrimal-Dual Interior-Point Methods1997PhiladelphiaSIAM10.1137/1.97816119714530863.65031 CottleRWPangJSStoneREThe Linear Complementarity Problem1992San Diego, CAAcademic Press Inc.0757.90078 MohamedAComplexity analysis of an interior point algorithm for the semidefinite optimization based on a kernel function with a double barrier termActa Math. Sin. (Engl. Ser.)201533154355633066641308.65086 WangGQYuCJTeoKLA full-Newton step feasible interior-point algorithm for P*(κ)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$P_\ast (\kappa )$$\end{document}-linear complementarity problemJ. Glob. Optim.20145918199318252110.1007/s10898-013-0090-x1300.90055
References_xml	– reference: AminiKHaseliAA new proximity function generating the best known iteration bounds for both large-update and small-update interior-point methodsANZIAM J.200749259270240851910.1017/S14461811000128271142.90039 – reference: WangGQFanXJZhuDTWangDZNew complexity analysis of a full-Newton step feasible interior-point algorithm for P(κ)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$P_\ast (\kappa )$$\end{document}-LCPOptim. Lett.2015911051119337367110.1007/s11590-014-0800-41331.90085 – reference: Lee, Y.H., Cho, Y.Y., Cho, G.M.: Interior-point algorithms for P(κ)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$P_\ast (\kappa )$$\end{document}-LCP based on a new class of kernel functions. J. Glob. Optim. (2013). doi:10.1007/s10898-013-0072-z – reference: PeyghamiMRAminiKA kernel function based interior-point methods for solving P(κ)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$P_\ast (\kappa )$$\end{document}-linear complementarity problemActa Math. Sin. (Engl. Ser.)201026917611778267281610.1007/s10114-010-7529-51206.90195 – reference: ZhuDHZhangMWA full-Newton step infeasible interior-point algorithm for P(κ)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$P_\ast (\kappa )$$\end{document} linear complementarity problemJ. Syst. Sci. Complex.20142710271044326484410.1007/s11424-014-1273-31326.90091 – reference: BaiYQLesajaGRoosCA new class of polynomial interior-point algorithms for P(κ)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$P_\ast (\kappa )$$\end{document} linear complementarity problemsPac. J. Optim.20084119411161.90507 – reference: WangGQYuCJTeoKLA full-Newton step feasible interior-point algorithm for P(κ)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$P_\ast (\kappa )$$\end{document}-linear complementarity problemJ. Glob. Optim.20145918199318252110.1007/s10898-013-0090-x1300.90055 – reference: ChoGMA new large-update interior point algorithm for P(κ)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$P_\ast (\kappa )$$\end{document} linear complementarity problemsJ. Comput. Appl. Math.2008216265278242185510.1016/j.cam.2007.05.007 – reference: WangGQBaiYQA new primal-dual path-following interior-point algorithm for semi-definite optimizationJ. Math. Anal. Appl.200835333934910.1016/j.jmaa.2008.12.0161172.90011 – reference: Cai, X.Z., Wang, G.Q., El Ghami, M., Yue, Y.J.: Complexity analysis of primal-dual interior-point methods for linear optimization based on a new parametric kernel function with a trigonometric barrier term. Abstr. Appl. Anal. ID 710158 (2014) – reference: HarkerPTPangJSSimulation and Optimization of Large Systems1990Providence, RIAMS265284 – reference: El GhamiMPrimal dual interior-point methods for P(κ)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$P_\ast (\kappa )$$\end{document}-linear complementarity problem based on a kernel function with a trigonometric barrier termOptim. Theory Decis. Mak. Oper. Res. Appl.2013313313493068945 – reference: AminiKPeyghamiMRExploring complexity of large update interior-point methods for P(κ)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$P_\ast (\kappa )$$\end{document} linear complementarity problem based on kernel functionAppl. Math. Comput.200920750151324891041160.90009 – reference: KheirfamBPrimal-dual interior-point algorithm for semidefinite optimization based on a new kernel function with trigonometric barrier termNumer. Algorithms201261659680299522610.1007/s11075-012-9557-y1259.65091 – reference: BaiYQEl GhamiMRoosCA comparative study of kernel functions for primal-dual interior-point algorithms in linear optimizationSIAM J. Optim.2004151101128211297810.1137/S10526234034231141077.90038 – reference: El GhamiMGuennounZABoulaSSteihaugTInterior-point methods for linear optimization based on a kernel function with a trigonometric barrier termJ. Comput. Appl. Math.201223636133623292349410.1016/j.cam.2011.05.0361242.90292 – reference: Ferris, M.C., Pang, J.S.: Complementarity and variational problems state of the art. In: Proceedings of the International Conference on Complementarity Problems. SIAM, Philadelphia (1997) – reference: ChoGMKimMKA new large-update interior point algorithm for P(κ)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$P_\ast (\kappa )$$\end{document} LCPs based on kernel functionsAppl. Math. Comput.200618221169118322825601108.65061 – reference: GyeongMCLarge-update interior point algorithm for P(κ)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$P_\ast (\kappa )$$\end{document} linear complementarity problemJ. Inequal. Appl.201436311123347684 – reference: HafshejaniSFFatemiMPeyghamMRAn interior-point method for P(κ)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$P_\ast (\kappa )$$\end{document}-linear complementarity problem based on a trigonometric kernel functionJ. Appl. Math. Comput.201548111128334059810.1007/s12190-014-0794-11323.90070 – reference: KojimaMMegiddoNNomaTYoshiseAA Unified Approach to Interior Point Algorithms for Linear Complementarity Problems1991BerlinSpringer0745.90069 – reference: MohamedAComplexity analysis of an interior point algorithm for the semidefinite optimization based on a kernel function with a double barrier termActa Math. Sin. (Engl. Ser.)201533154355633066641308.65086 – reference: WrightSJPrimal-Dual Interior-Point Methods1997PhiladelphiaSIAM10.1137/1.97816119714530863.65031 – reference: PengJRoosCTerlakyTSelf-regular functions and new search directions for linear and semidefinite optimizationMath. Program.200293129171191227110.1007/s1010702002961007.90037 – reference: PengJRoosCTerlakyTSelf-Regularity: A New Paradigm for Primal-Dual Interior-Point Algorithms2002PrincetonPrinceton University Press1136.90045 – reference: CottleRWPangJSStoneREThe Linear Complementarity Problem1992San Diego, CAAcademic Press Inc.0757.90078 – reference: WangGQBaiYQPolynomial interior-point algorithms for P*(κ)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$P_\ast (\kappa )$$\end{document} horizontal linear complementarity problemJ. Comput. Appl. Math.2009233224826310.1016/j.cam.2009.07.0141183.65072 – reference: PeyghamiMRHafshejaniSFShirvaniLComplexity of interior-point methods for linear optimization based on a new trigonometric kernel functionJ. Comput. Appl. Math.20142557485309340510.1016/j.cam.2013.04.0391291.90313
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Snippet	In this paper, we present a new large-update interior-point algorithm for P ∗ ( κ ) -linear complementarity problem. The new algorithm is based on a...
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SubjectTerms	Computational Mathematics and Numerical Analysis Mathematical and Computational Engineering Mathematics Mathematics and Statistics Mathematics of Computing Original Research Theory of Computation
Title	An interior-point algorithm for P∗(κ)-LCP based on a new trigonometric kernel function with a double barrier term
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