A Two-Dimensional Search Used with a Nonlinear Least-Squares Solver

This note describes a modified search strategy for use with a Gauss-Newton method for nonlinear least-squares problems. If a standard line search along the Gauss-Newton vector p is unable to make much progress, a new search direction is constructed which lies in the plane of p and the steepest-descent vector. Numerical experiments show that a quadratic model of the objective function in this plane can yield effective corrections when the basic Gauss-Newton technique experiences difficulty.     

Adaptive Algorithm for Constrained Least-Squares Problems

This paper is concerned with the implementation and testing of an algorithm for solving constrained least-squares problems. The algorithm is an adaptation to the least-squares case of sequential quadratic programming (SQP) trust-region methods for solving general constrained optimization problems. At each iteration, our local quadratic subproblem includes the use of the Gauss–Newton approximation but also encompasses a structured secant approximation along with tests of when to use this approximation. This method has been tested on a selection of standard problems. The results indicate that, for least-squares problems, the approach taken here is a viable alternative to standard general optimization methods such as the Byrd–Omojokun trust-region method and the Powell damped BFGS line search method.     

经济预测中的正交回归分析

本文介绍了一种新的线性模型参数回归分析方法即正交回归,并以建立经济模型为例,对正交回归和经典回归的结果进行了比较。     

A Class of Incomplete Orthogonal Factorization Methods. I: Methods and Theories

We present a class of incomplete orthogonal factorization methods based on Givens rotations for large sparse unsymmetric matrices. These methods include: Incomplete Givens Orthogonalization (IGO-method) and its generalisation (GIGO-method), which drop entries from the incomplete orthogonal and upper triangular factors by position; Threshold Incomplete Givens Orthogonalization (TIGO()-method), which drops entries dynamically by their magnitudes; and its generalisation (GTIGO(,p)-method), which drops entries dynamically by both their magnitudes and positions. Theoretical analyses show that these methods can produce a nonsingular sparse incomplete upper triangular factor and either a complete orthogonal factor or a sparse nonsingular incomplete orthogonal factor for a general nonsingular matrix. Therefore, these methods can potentially generate efficient preconditioners for Krylov subspace methods for solving large sparse systems of linear equations. Moreover, the upper triangular factor is an incomplete Cholesky factorization preconditioner for the normal equations matrix from least-squares problems.     

Scaling by Binormalization

We present an iterative algorithm (BIN) for scaling all the rows and columns of a real symmetric matrix to unit 2-norm. We study the theoretical convergence properties and its relation to optimal conditioning. Numerical experiments show that BIN requires 2–4 matrix–vector multiplications to obtain an adequate scaling, and in many cases significantly reduces the condition number, more than other scaling algorithms. We present generalizations to complex, nonsymmetric and rectangular matrices.     

Convergence Analysis of Gauss–Newton Methods for the Complementarity Problem

In a recent paper (Ref. 1), the first author presented a damped Gauss–Newton algorithm to solve the complementarity problem. Although the conclusions of the convergence theorems in the paper are valid, the proofs contain minor errors. The aim of this paper is to give correct proofs and also to show that they are valid under milder conditions.     

广义正交表和正交表的裂区试验设计法的比较

裂区试验设计方法是在正交表的基础上进行的.根据试验设计的数据分析结论要求具有再现性这一原理,将证明这种裂区试验设计法要有条件的使用才是合理的.由于广义正交表是保证设计表具有再现性的基本设计表,根据广义正交表来研究这种裂区试验设计方法的合理性.研究结果显示在裂区试验设计法对应的设计表是广义正交表,并且相应的数据分析方法采用广义正交表的数据分析方法时,才能保证其数据分析结论具有客观一致性和可重复再现性.     

Adaptive Strategy for the Damping Parameters in an Iteratively Regularized Gauss–Newton Method

In this paper, a convergence analysis of an adaptive choice of the sequence of damping parameters in the iteratively regularized Gauss–Newton method for solving nonlinear ill-posed operator equations is presented. The selection criterion is motivated from the damping parameter choice criteria, which are used for the efficient solution of nonlinear least-square problems. The performance of this selection criterion is tested for the solution of nonlinear ill-posed model problems.     

Quadratic approximation of penalty functions for solving large-scale linear programs

Using the least squares, modified Lagrangian function, and some other methods as examples, the capabilities of the new optimization technique based on the quadratic approximation of penalty functions that has been recently proposed by O. Mangasarian for a special class of linear programming problems are demonstrated. The application of this technique makes it possible to use unified matrix operations and standard linear algebra packages (including parallel ones) for solving large-scale problems with sparse strongly structured constraint matrices. With this technique, the computational schemes of some well-known algorithms can take an unexpected form.     

The convergence analysis of inexact Gauss–Newton methods for nonlinear problems

In this paper, inexact Gauss–Newton methods for nonlinear least squares problems are studied. Under the hypothesis that derivative satisfies some kinds of weak Lipschitz conditions, the local convergence properties of inexact Gauss–Newton and inexact Gauss–Newton like methods for nonlinear problems are established with the modified relative residual control. The obtained results can provide an estimate of convergence ball for inexact Gauss–Newton methods.     

Multiple Penalty Regression: Fitting and Extrapolating a Discrete Incomplete Multi-way Layout

The d iscrete multi-way layout is an abstract data type associated with regression, experimental designs, digital images or videos, spatial statistics, gene or protein chips, and more. The factors influencing response can be nominal or ordinal. The observed factor level combinations are finitely discrete and often incomplete or irregularly spaced. This paper develops low risk biased estimators of the means at the observed factor level combinations; and extrapolates the estimated means to larger discrete complete layouts. Candidate penalized least squares (PLS) estimators with multiple quadratic penalties express competing conjectures about each of the main effects and interactions in the analysis of variance decomposition of the means. The candidate PLS estimator with smallest estimated quadratic risk attains, asymptotically, the smallest risk over all candidate PLS estimators. In the theoretical analysis, the dimension of the regression space tends to infinity. No assumptions are made about the unknown means or about replication.     

Formal Orthogonal Polynomials and Newton–Padé Approximants

We proved in [8] that the denominators of Newton–Padé approximants for a formal Newton series are formal orthogonal with respect to linear functionals. The same functional is used along an antidiagonal of the Newton–Padé denominator table. The two linear functionals, corresponding to two adjacent antidiagonals, are linked with a very simple relation. Recurrence relations between denominators are given along an antidiagonal or two adjacent antidiagonals in the normal and non-normal case. The same recurrence relations are also satisfied by the Newton–Padé numerators, which implies another formal orthogonality.     

用LSD法再探泥石流地貌要素之隐式关系

藉助MATLAB工具,对泥石流沟的六个地貌要素之流域面积、主沟长、切割度、沟口高程、流域高差和沟床比降等,一起用最小平方距离法(LSD)进行了初步处理分析之探讨:建立了线性相关的回归方程,并与用最小二乘法(LS)得到的线性回归方程进行比较,一些参数的估计值得到了改善。     

用最小平方距离法对泥石流地貌要素之初探

藉助MATLAB工具 ,对泥石流沟的六个地貌要素之沟床比降与切割度、流域面积与主沟长 ,用最小平方距离法进行了初步处理分析的探讨 :建立了线性相关的回归方程 ,并与用最小二乘法得到的线性回归方程进行比较 ,一些参数的估计值得到了改善     

Four Mutually Orthogonal Latin Squares of Order 14

The paper gives example of orthogonal array OA(6, 14) obtained from a difference matrix . The construction is equivalent to four mutually orthogonal Latin squares (MOLS) of order 14. © 2012 Wiley Periodicals, Inc. J. Combin. Designs 20: 363–367, 2012