• Communications of the Korean Mathematical Society
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• v.34 no.2
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• pp.657-670
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• 2019

In this paper, we consider the preconditioned iterative methods for solving linear complementarity problem associated with an M-matrix. Based on the generalized Gunawardena's preconditioner, two preconditioned SSOR methods for solving the linear complementarity problem are proposed. The convergence of the proposed methods are analyzed, and the comparison results are derived. The comparison results showed that preconditioned SSOR methods accelerate the convergent rate of the original SSOR method. Numerical examples are used to illustrate the theoretical results.

• Journal of applied mathematics & informatics
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• v.31 no.5_6
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• pp.769-784
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• 2013

We consider the modulus-based successive overrelaxation method for the linear complementarity problems from the discretization of Black-Scholes American options model. The $H_+$-matrix property of the system matrix discretized from American option pricing which guarantees the convergence of the proposed method for the linear complementarity problem is analyzed. Numerical experiments confirm the theoretical analysis, and further show that the modulus-based successive overrelaxation method is superior to the classical projected successive overrelaxation method with optimal parameter.

• Journal of applied mathematics & informatics
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• v.29 no.5_6
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• pp.1285-1293
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• 2011

We establishes the polynomial convergence of a new class of path-following methods for semidefinite linear complementarity problems (SDLCP) whose search directions belong to the class of directions introduced by Monteiro [9]. Namely, we show that the polynomial iteration-complexity bound of the well known algorithms for linear programming, namely the predictor-corrector algorithm of Mizuno and Ye, carry over to the context of SDLCP.

• 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.

• Honam Mathematical Journal
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• v.19 no.1
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• pp.157-164
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• 1997

We are concerned with three classes of matrices that are relevant to the linear complementary problem. We prove that within the class of $P_0$-matrices, the Q-matrices are precisely the regular matrices and we show that the same characterizations hold for an L-matrix as well, and that the symmetric copositive-plus Q-matrices are precisely those which are strictly copositive.

• Journal of applied mathematics & informatics
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• v.32 no.1_2
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• pp.285-295
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• 2014

In the setting of semidenite linear complementarity problems on $S^n$, we focus on the Stein Transformation $S_A(X)\;:=X-AXA^T$, and show that $S_A$ is (strictly) monotone if and only if ${\nu}_r(UAU^T{\circ}\;UAU^T)$(<)${\leq}1$, for all orthogonal matrices U where ${\circ}$ is the Hadamard product and ${\nu}_r$ is the real numerical radius. In particular, we show that if ${\rho}(A)$ < 1 and ${\nu}_r(UAU^T{\circ}\;UAU^T){\leq}1$, then SDLCP($S_A$, Q) has a unique solution for all $Q{\in}S^n$. In an attempt to characterize the GUS-property of a nonmonotone $S_A$, we give an instance of a nonnormal $2{\times}2$ matrix A such that SDLCP($S_A$, Q) has a unique solution for Q either a diagonal or a symmetric positive or negative semidenite matrix. We show that this particular $S_A$ has the $P^{\prime}_2$-property.

• Journal of the Korean Operations Research and Management Science Society
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• v.4 no.1
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• pp.79-86
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• 1979

• 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.

• Journal of applied mathematics & informatics
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• v.29 no.5_6
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• pp.1511-1523
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• 2011

In this paper, we consider the smoothing Newton method for the nonlinear complementarity problems with $P_0$-function. The proposed algorithm is based on a new smoothing function and it needs only to solve one linear system of equations and perform one line search per iteration. Under the condition that the solution set is nonempty and bounded, the proposed algorithm is proved to be convergent globally. Furthermore, the local superlinearly(quadratic) convergence is established under suitable conditions. Preliminary numerical results show that the proposed algorithm is very promising.

• Communications of the Korean Mathematical Society
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• v.32 no.3
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• pp.765-778
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• 2017

Finding an initial feasible solution on the central path is the main difficulty of feasible interior-point methods. Although, some algorithms have been suggested to remedy this difficulty, many practical implementations often do not use perfectly centered starting points. Therefore, it is worth to analyze the case that the starting point is not exactly on the central path. In this paper, we propose a weighted-path following interior-point algorithm for solving the Cartesian $P_{\ast}({\kappa})$-linear complementarity problems (LCPs) over symmetric cones. The convergence analysis of the algorithm is shown and it is proved that the algorithm terminates after at most $O\((1+4{\kappa}){\sqrt{r}}{\log}{\frac{x^0{\diamond}s^0}{\varepsilon}}\)$ iterations.