• Bulletin of the Korean Mathematical Society
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• v.46 no.3
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• pp.521-534
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• 2009

In this paper we present new large-update primal-dual interior point algorithms for $P_*$ linear complementarity problems(LAPS) based on a class of kernel functions, ${\psi}(t)={\frac{t^{p+1}-1}{p+1}}+{\frac{1}{\sigma}}(e^{{\sigma}(1-t)}-1)$, p $\in$ [0, 1], ${\sigma}{\geq}1$. It is the first to use this class of kernel functions in the complexity analysis of interior point method(IPM) for $P_*$ LAPS. We showed that if a strictly feasible starting point is available, then new large-update primal-dual interior point algorithms for $P_*$ LAPS have $O((1+2+\kappa)n^{{\frac{1}{p+1}}}lognlog{\frac{n}{\varepsilon}})$ complexity bound. When p = 1, we have $O((1+2\kappa)\sqrt{n}lognlog\frac{n}{\varepsilon})$ complexity which is so far the best known complexity for large-update methods.

• Communications of the Korean Mathematical Society
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• v.25 no.4
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• pp.655-669
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• 2010

In this paper we propose new primal-dual interior point methods (IPMs) for $P_*(\kappa)$ linear complementarity problems (LCPs) and analyze the iteration complexity of the algorithm. New search directions and proximity measures are defined based on a class of kernel functions, $\psi(t)=\frac{t^2-1}{2}-{\int}^t_1e{^{q(\frac{1}{\xi}-1)}d{\xi}$, $q\;{\geq}\;1$. If a strictly feasible starting point is available and the parameter $q\;=\;\log\;\(1+a{\sqrt{\frac{2{\tau}+2{\sqrt{2n{\tau}}+{\theta}n}}{1-{\theta}}\)$, where $a\;=\;1\;+\;\frac{1}{\sqrt{1+2{\kappa}}}$, then new large-update primal-dual interior point algorithms have $O((1\;+\;2{\kappa})\sqrt{n}log\;n\;log\;{\frac{n}{\varepsilon}})$ iteration complexity which is the best known result for this method. For small-update methods, we have $O((1\;+\;2{\kappa})q{\sqrt{qn}}log\;{\frac{n}{\varepsilon}})$ iteration complexity.

• Journal of the Korean Society for Industrial and Applied Mathematics
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• v.13 no.3
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• pp.189-202
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• 2009

In this paper we propose new primal-dual interior point algorithms for $P_*(\kappa)$ linear complementarity problems based on a new class of kernel functions which contains the kernel function in [8] as a special case. We show that the iteration bounds are $O((1+2\kappa)n^{\frac{9}{14}}\;log\;\frac{n{\mu}^0}{\epsilon}$) for large-update and $O((1+2\kappa)\sqrt{n}log\frac{n{\mu}^0}{\epsilon}$) for small-update methods, respectively. This iteration complexity for large-update methods improves the iteration complexity with a factor $n^{\frac{5}{14}}$ when compared with the method based on the classical logarithmic kernel function. For small-update, the iteration complexity is the best known bound for such methods.

• East Asian mathematical journal
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• v.26 no.1
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• pp.9-23
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• 2010

In this paper we generalize large-update primal-dual interior point methods for linear optimization problems in [2] to the $P_*(\kappa)$ linear complementarity problems based on a new kernel function which includes the kernel function in [2] as a special case. The kernel function is neither self-regular nor eligible. Furthermore, we improve the complexity result in [2] from $O(\sqrt[]{n}(\log\;n)^2\;\log\;\frac{n{\mu}o}{\epsilon})$ to $O\sqrt[]{n}(\log\;n)\log(\log\;n)\log\;\frac{m{\mu}o}{\epsilon}$.

• East Asian mathematical journal
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• v.25 no.1
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• pp.55-68
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• 2009

In this paper we propose new large-update primal-dual inte-rior point algorithms for $P_*(\kappa)$ linear complementarity problems(LCPs). New search directions and proximity measures are proposed based on the kernel function$\psi(t)=\frac{t^{p+1}-1}{p+1}+\frac{e^{\frac{1}{t}}-e}{e}$,$p{\in}$[0,1]. We showed that if a strictly feasible starting point is available, then the algorithm has $O((1+2\kappa)(logn)^{2}n^{\frac{1}{p+1}}log\frac{n}{\varepsilon}$ complexity bound.

• Journal of the Korean Society for Industrial and Applied Mathematics
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• v.13 no.1
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• pp.41-53
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• 2009

A primal-dual interior point method(IPM) not only is the most efficient method for a computational point of view but also has polynomial complexity. Most of polynomialtime interior point methods(IPMs) are based on the logarithmic barrier functions. Peng et al.([14, 15]) and Roos et al.([3]-[9]) proposed new variants of IPMs based on kernel functions which are called self-regular and eligible functions, respectively. In this paper we define a new kernel function and propose a new IPM based on this kernel function which has $O(n^{\frac{2}{3}}log\frac{n}{\epsilon})$ and $O(\sqrt{n}log\frac{n}{\epsilon})$ iteration bounds for large-update and small-update methods, respectively.

• Journal of the Korean Society for Industrial and Applied Mathematics
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• v.12 no.4
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• pp.227-238
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• 2008

In this paper we extend primal-dual interior point algorithm for linear optimization (LO) problems to $P_*({\kappa})$ linear complementarity problems(LCPs) ([1]). We define proximity functions and search directions based on kernel functions, ${\psi}(t)=\frac{t^{p+1}-1}{p+1}-{\log}\;t$, $p{\in}$[0, 1], which is a generalized form of the one in [16]. It is the first to use this class of kernel functions in the complexity analysis of interior point method(IPM) for $P_*({\kappa})$ LCPs. We show that if a strictly feasible starting point is available, then new large-update primal-dual interior point algorithms for $P_*({\kappa})$ LCPs have $O((1+2{\kappa})nlog{\frac{n}{\varepsilon}})$ complexity which is similar to the one in [16]. For small-update methods, we have $O((1+2{\kappa})\sqrt{n}{\log}{\frac{n}{\varepsilon}})$ which is the best known complexity so far.

• East Asian mathematical journal
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• v.29 no.3
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• pp.279-292
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• 2013

It is well known that each kernel function defines a primal-dual interior-point method(IPM). Most of polynomial-time interior-point algorithms for linear optimization(LO) are based on the logarithmic kernel function([2, 11]). In this paper we define a new eligible kernel function and propose a new search direction and proximity function based on this function for LO problems. We show that the new algorithm has ${\mathcal{O}}((log\;p){\sqrt{n}}\;log\;n\;log\;{\frac{n}{\epsilon}})$ and ${\mathcal{O}}((q\;log\;p)^{\frac{3}{2}}{\sqrt{n}}\;log\;{\frac{n}{\epsilon}})$ iteration bound for large- and small-update methods, respectively. These are currently the best known complexity results.

• Honam Mathematical Journal
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• v.35 no.2
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• pp.235-249
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• 2013

It is well known that each kernel function defines primal-dual interior-point method (IPM). Most of polynomial-time interior-point algorithms for linear optimization (LO) are based on the logarithmic kernel function ([9]). In this paper we define new eligible kernel function and propose a new search direction and proximity function based on this function for LO problems. We show that the new algorithm has $\mathcal{O}(({\log}\;p)^{\frac{5}{2}}\sqrt{n}{\log}\;n\;{\log}\frac{n}{\epsilon})$ and $\mathcal{O}(q^{\frac{3}{2}}({\log}\;p)^3\sqrt{n}{\log}\;\frac{n}{\epsilon})$ iteration complexity for large- and small-update methods, respectively. These are currently the best known complexity results for such methods.