• Journal of applied mathematics & informatics
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• 제29권5_6호
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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.

• Journal of applied mathematics & informatics
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• 제29권5_6호
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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.

• 대한수학회논문집
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• 제25권4호
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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 applied mathematics & informatics
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• 제32권1_2호
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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.

• 대한수학회보
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• 제50권2호
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• pp.589-599
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• 2013

In this paper, we propose a sufficient condition for the existence of solutions to general variational inequality problems (GVI(K, F, $g$)). The condition is also necessary when F is a $g-P^M_*$ function. We also investigate the boundedness of the solution set of (GVI(K, F, $g$)). Furthermore, we show that when F is norm-coercive, the general complementarity problems (GCP(K, F, $g$)) has a nonempty compact solution set. Finally, we establish some existence theorems for (GNCP(K, F, $g$)).

• Structural Engineering and Mechanics
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• 제6권8호
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• pp.857-870
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• 1998

This paper deals with a special class of inverse problems in discrete structural plasticity involving the identification of elastic limits and hardening moduli on the basis of information on displacements. The governing equations lead naturally to a special and challenging optimization problem known as a Mathematical Program with Equilibrium Constraints (MPEC), a key feature of which is the orthogonality of two sign-constrained vectors or so-called "complementarity" condition. We investigate numerically the application of two simple algorithms, both based on the use of the general purpose nonlinear programming code CONOPT accessed via the GAMS modeling language, for solving the suitably reformulated problem. Application is illustrated by means of two numerical examples.

• 호남수학학술지
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• 제19권1호
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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.

• East Asian mathematical journal
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• 제13권1호
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• pp.71-78
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• 1997

• 한국경영과학회지
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• 제4권1호
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• pp.79-86
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• 1979

• Journal of applied mathematics & informatics
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• 제17권1_2_3호
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• pp.59-72
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• 2005

The auxiliary principle is used to suggest and analyze some iterative methods for solving solving hemivariational inequalities under mild conditions. The results obtained in this paper can be considered as a novel application of the auxiliary principle technique. Since hemivariational in­equalities include variational inequalities and nonlinear optimization problems as special cases, our results continue to hold for these problems.