A New Method for Computing Reproducing Kernels

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The Eigenvalue Method for Computing High-dimensional Varieties

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On the Kolmogorov Inequality for M Φ -Norm

In this paper, we prove the Kolmogorov inequality for M | -norms generated by concave functions (with the same constants as in the Kolmogorov inequality).     

A new iterative-doubling Greedy–Lookahead algorithm for the single container loading problem

The aim of the Single Container Loading Problem (SCLP) is to pack three-dimensional boxes into a three-dimensional container so as to maximize the volume utilization of the container. We propose a new block building approach that constructs packings by placing one block (of boxes) at a time until no more boxes can be loaded. The key to obtaining high quality solutions is to select the right block to place into the right free space cuboid (or residual space) in the container. We propose a new heuristic for evaluating the fitness of residual spaces, and use a tree search to decide the best residual space-block pair at each step. The resultant algorithm outperforms the best known algorithms based on the 1600 commonly used benchmark instances even when given fewer computational resources. We also adapted our approach to address the full support constraint. The computational results for the full support support variant on the 1600 instances similarly show a significant improvement over existing techniques even when given substantially fewer computational resources.     

A Simple Method for Computing the Homeomorphism Class of G． M．  and G． M． 

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Newton’s Method for Computing the Nearest Correlation Matrix with a Simple Upper Bound

The standard nearest correlation matrix can be efficiently computed by exploiting a recent development of Newton’s method (Qi and Sun in SIAM J. Matrix Anal. Appl. 28:360–385, 2006). Two key mathematical properties, that ensure the efficiency of the method, are the strong semismoothness of the projection operator onto the positive semidefinite cone and constraint nondegeneracy at every feasible point. In the case where a simple upper bound is enforced in the nearest correlation matrix in order to improve its condition number, it is shown, among other things, that constraint nondegeneracy does not always hold, meaning Newton’s method may lose its quadratic convergence. Despite this, the numerical results show that Newton’s method is still extremely efficient even for large scale problems. Through regularization, the developed method is applied to semidefinite programming problems with simple bounds.     

A General Version of the Retract Method for Discrete Equations

We study a problem concerning the compulsory behavior of the solutions of systems of discrete equations u（k ＋ 1） = F（k, u（k））, k ∈ N（a） = {a, a ＋ 1, a ＋ 2 }, a ∈ N,N= {0, 1,... } and F ： N（a） × R^n→R^n. A general principle for the existence of at least one solution with graph staying for every k ∈ N（a） in a previously prescribed domain is formulated. Such solutions are defined by means of the corresponding initial data and their existence is proved by means of retract type approach. For the development of this approach a notion of egress type points lying on the defined boundary of a given domain and with respect to the system considered is utilized. Unlike previous investigations, the boundary can contain points which are not points of egress type, too. Examples are inserted to illustrate the obtained result.     

A Note on Sin Θ Theorems for Singular Subspace Variations

New perturbation theorems for bases of singular subspaces are proved. These theorems complement the known sin theorems for singular subspace perturbations, taking into account a kind of sensitivity of singular vectors discarded by previous theorems. Furthermore these results guarantee that high relative accuracy algorithms for the SVD are able to compute reliably simultaneous bases of left and right singular subspaces.     

A Judging Method for Normal Family

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A Two-Grid Finite-Element Method for the Nonlinear Schrödinger Equation

In this paper, some two-grid finite element schemes are constructed for solving the nonlinear Schrödinger equation. With these schemes, the solution of the original problem is reduced to the solution of the same problem on a much coarser grid together with the solutions of two linear problems on a fine grid. We have shown, both theoretically and numerically, that our schemes are efficient and achieve asymptotically optimal accuracy.     

A Two-Level Method for Nonsymmetric Eigenvalue Problems

A two-level discretization method for eigenvalue problems is studied. Compared to the standard Galerkin finite element discretization technique performed on a fine grid this method discretizes the eigenvalue problem on a coarse grid and obtains an improved eigenvector (eigenvalue) approximation by solving only a linear problem on the fine grid (or two linear problems for the case of eigenvalue approximation of nonsymmetric problems). The improved solution has the asymptotic accuracy of the Galerkin discretization solution. The link between the method and the iterated Galerkin method is established. Error estimates for the general nonsymmetric case are derived.     

A Damped Gauss-Newton Method for the Second-Order Cone Complementarity Problem

We investigate some properties related to the generalized Newton method for the Fischer-Burmeister (FB) function over second-order cones, which allows us to reformulate the second-order cone complementarity problem (SOCCP) as a semismooth system of equations. Specifically, we characterize the B-subdifferential of the FB function at a general point and study the condition for every element of the B-subdifferential at a solution being nonsingular. In addition, for the induced FB merit function, we establish its coerciveness and provide a weaker condition than Chen and Tseng (Math. Program. 104:293–327, 2005) for each stationary point to be a solution, under suitable Cartesian P-properties of the involved mapping. By this, a damped Gauss-Newton method is proposed, and the global and superlinear convergence results are obtained. Numerical results are reported for the second-order cone programs from the DIMACS library, which verify the good theoretical properties of the method. S. Pan’s work is partially supported by the Doctoral Starting-up Foundation (B13B6050640) of GuangDong Province. J.-S. Chen is member of Mathematics Division, National Center for Theoretical Sciences, Taipei Office. J.-S. Chen’s work is partially supported by National Science Council of Taiwan.     

L1-Norm Estimation and Random Weighting Method in a Semiparametric Model

In this paper, the L_1-norm estimators and the random weighted statistic for a semiparametric regression model are constructed, the strong convergence rates of estimators are obtain under certain conditions, the strong efficiency of the random weighting method is shown. A simulation study is conducted to compare the L_1-norm estimator with the least square estimator in term of approximate accuracy, and simulation results are given for comparison between the random weighting method and normal approximation method.     

A Newton Method for Minimizing One-Order Lipschitz Functions

In this paper, we consider the following unconstrained optimization problem (P) min{f(x)|x∈R~n}, where f(x) is a one order Lipschitz function on R~n, i.e., g(x)—the gradient of f(x)—is Lipschitzian. We will represent a kind of Newton method for solving the problem (P).     

A Generalized Newton-Like Method for Solving Variational Inequalities

In this paper,we present a generalized Newton-like method for solving variational inequalities,and prove Q-superlinear and Q-quadratic convergence without the strict complementarity slackness condition     

A TVD Type Wavelet-Galerkin Method for Hamilton-Jacobi Equations

In this paper,we use Daubechies scaling functions as test functions for the Galerkin method,and discuss Wavelet-Galerkin solutions for the Hamilton-Jacobi equations.It can be proved that the schemesare TVD schemes.Numerical tests indicate that the schemes are suitable for the Hamilton-Jacobi equations.Furthermore,they have high-order accuracy in smooth regions and good resolution of singularities.     

A Filter-Based Pattern Search Method for Unconstrained Optimization

We discuss a filter-based pattern search method for unconstrained optimization in this paper. For the purpose to broaden the search range we use both filter technique and frames, which are fragments of grids, to provide a new criterion of iterate acceptance. The convergence can be ensured under some conditions. The numerical result shows that this method is practical and efficient.