2002年, 第20卷, 第3期　刊出日期：2002-05-15

  

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  AN INTERIOR TRUST REGION ALGORITHM FOR NONLINEAR MINIMIZATION WITH LINEAR CONSTRAINTS

  Jian Guo LIU

  Journal of Computational Mathematics. 2002, 20(3): 225-244.

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  An interior trust-region-based algorithm for linearly constained minimization problems is proposed and analyzed. This algorithm is similar to trust region algorithms for unconstrained minimization: a trust region subproblem on a subspace is solved in each iteration.We establish that the proposed algorithm has convergence properties analogous point of the generated sequence satisfies the Krush-Kuhn-Tucker (KKT)conditions and at least one limit point satisfies second order necessary optimatity conditions. In addition, if one limit point is a strong local minimizer and the Hessian is Lipschitz continuous in a neighborbood of that point, then the generated sequence converges globally to that point in the rate of at least 2-step quadratic. We are mainly concerned with the theoretical properties of the algorithm in this paper. Implementation issues and adaptation to large-scale problems will be addressed in a future report.
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  THE SOLVABILITY CONDITIONS FOR INVERSE EIGENVALUE PROBLEM OF ANTI-BISYMMETRIC MATRICES

  Dong Xiu XIE(1),Xi Yan HU(2),Lei ZHANG(3)

  Journal of Computational Mathematics. 2002, 20(3): 245-256.

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  This paper is mainly concerned with solving the following two problems: \par Problem I. Given $X\in C^{n\times m},\Lambda=\textmd{diag}(\lambda_1,\lambda_2,\cdots,\lambda_m)\in C^{m\times m}.$Find $A\in ABSR^{n\times n}$such that$$AX = X\Lambda$$where $ABSR^{n\times n}$ is the set of all real $n\times n$ anti-bisymmetric matrices. \par Problem II.
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  A NUMERICAL EMBEDDING METHOD FOR SOLVING THE NONLINEAR COMPLEMENTARITY PROBLEM(I) ——THEORY

  Jian Jun ZHANG,De Ren WANG

  Journal of Computational Mathematics. 2002, 20(3): 257-266.

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  In this paper, we extend the numerical embedding method for solving the smooth equations to the nonlinear complementarity problem. By using the nonsmooth theory, we prove the existence and the continuation of the following path for the corresponding method for solving the nonlinear complementarity problem is established. In part II of this paper, we will further study the implementation of the method and give some numerical exapmles.
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  SYMPLECTIC COMPUTATION OF HAMILTONIAN SYSTEMS (I)

  Yi Fa TANG

  Journal of Computational Mathematics. 2002, 20(3): 267-276.

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  We get $\tao^6$-terms of the formal energy of the mid-point rule, and use the mathematical pendulum to test the convergence of the formal energy.
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  SUPERAPPROXIMATION PROPERTIES OF THE INTERPOLATION OPERATOR OF PROJECTION TYPE AND APPLICATIONS

  Tie ZHANG(1),Yan Ping LIN(2),R.J.Tait(2)

  Journal of Computational Mathematics. 2002, 20(3): 277-288.

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  Some superapproximation and ultra-approximation properties in function, gradient and two-order derivative approximations are shown for the interpolation operator of projection type on two-dimensional domain. Then, we consider the Ritz projection and Ritz-Volterra projection on finite element spaces, and by means of the superapproximation elementary estimates and Green function methods, derive the superconvergence and ultraconvergence error estimates for both prjections, which are also the finite slement approximation solutions of the elliptic problems and the Sobolev equations, respectively.
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  GLOBALLY CONVERGENT INEXACT GENERALIZED NEWTON METHODS WITH DECREASING NORM OF THE GRADIENT

  Ding Guo PU

  Journal of Computational Mathematics. 2002, 20(3): 289-300.

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  In this paper, motivated by the Martinez and Qi methods[l], we propose one type of globally convergent inexact generalized Newton methods to solve unconstrained optimization problems in which the objective functions are not twice differentiable,but have LC gradient. They make the norm of the gradient decreasing.These methods are implementable and globally convergent. We prove that the algorithms have superlinear convergence rates under some mile conditions.\par The methods may also be used to solve nonsmooth epuations.
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  SEQUENTIAL SYSTEMS OF LINEAR EQUATIONS ALGORITHM FOR NONLINEAR OPTIMIZATION PROBLEMS-INEQUALITY CONSTRAINED PROBLEMS

  Zi You GAO(1),Tian De GUO(2),Guo Ping HE(3),Fang WU(3)

  Journal of Computational Mathematics. 2002, 20(3): 301-312.

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  In this paper, a new superlinearly convergent algorithm of sequential systems of linear equations (SSLE) for nonlinear optimization problems with inequality constraints is proposed. Since the new algorithm only needs to solve several systems of linear equations gaving a same coefficient matrix per iteration, the computation amount of the algorithm is much less than that of the existing SQP algorithms per iteration.Moreover, for the SQP type algorithms, there exist so-called inconsistent problems, i.e., quadratic qrogramming subproblems of the SQP algorithms may not have a solution at some iterations, but this phenomenon will not occur with the SSLE algorithms because the relatrd systems of linear equations always have solutions. Some numerical results are reported.
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  THE MORTAR ELEMENT METHOD FOR ROTATED Q1 ELEMENT

  Jin Ru CHEN(1),Xue Jun XU(2)

  Journal of Computational Mathematics. 2002, 20(3): 313-324.

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  In this paper, a mortar element version for rotated Q1 element is proposed. The optimal error estimate is proven for the rotated Q1 mortar element method.
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  CONVERGENCE RESULTS OF RUNGE-KUTTA METHODS FOR MULTIPLY-STIFF SINGULAR PERTURBATION PROBLEMS

  Ai Guo XIAO

  Journal of Computational Mathematics. 2002, 20(3): 325-336.

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  The main purpose of this paper is to present some convergence results for algebraically stable Runge-Kutta methods applied to some classes of one- and two-parameter multiply-stiff singular perturbation problems whose stiffness is caused by small parameters and some other factors. A numerical example confirms our results.