求解非光滑方程组的三次正则化方法

考虑求解非光滑方程组的三次正则化方法及其收敛性分析.利用信赖域方法的技巧,保证该方法是全局收敛的.在子问题非精确求解和BD正则性条件成立的前提下,分析了非光滑三次正则化方法的局部收敛速度.最后,数值实验结果验证了该算法的有效性.     

The cubic semilocal convergence on two variants of Newton's method

In this paper, we discuss two variants of Newton's method without using any second derivative for solving nonlinear equations. By using the majorant function and confirming the majorant sequences, we obtain the cubic semilocal convergence and the error estimation in the Kantorovich-type theorems. The numerical examples are presented to support the usefulness and significance.     

On the convergence of the midpoint method

The midpoint method is an iterative method for the solution of nonlinear equations in a Banach space. Convergence results for this method have been studied in [3, 4, 9, 12]. Here we show how to improve and extend these results. In particular, we use hypotheses on the second Fréchet derivative of the nonlinear operator instead of the third-derivative hypotheses employed in the previous results and we obtain Banach space versions of some results that were derived in [9, 12] only in the real or complex space. We also provide various examples that validate our results.      

On the convergence of an inexact Newton-type method

In this paper we give local convergence results of an inexact Newton-type method for monotone equations under a local error bound condition. This condition may hold even for problems with non-isolated solutions, and it therefore is weaker than the standard non-singularity condition.     

On the local convergence of a deformed Newton's method under Argyros-type condition

For the iteration which was independently proposed by King [R.F. King, Tangent method for nonlinear equations, Numer. Math. 18 (1972) 298-304] and Werner [W. Werner, Über ein Verfarhren der Ordnung zur Nullstellenbestimmung, Numer. Math. 32 (1979) 333-342] for solving a nonlinear operator equation in Banach space, we established a local convergence theorem under the condition which was introduced recently by Argyros [I.K. Argyros, A unifying local-semilocal convergence analysis and application for two-point Newton-like methods in Banach space, J. Math. Anal. Appl. 298 (2004) 374-397].     

On cubic Hermite coalescence hidden variable fractal interpolation functions

Hermite interpolation is a very important tool in approximation theory and numerical analysis, and provides a popular method for modeling in the area of computer aided geometric design. However, the classical Hermite interpolant is unique for a prescribed data set,and hence lacks freedom for the choice of an interpolating curve, which is a crucial requirement in design environment. Even though there is a rather well developed fractal theory for Hermite interpolation that offers a large flexibility in the choice of interpolants, it also has the shortcoming that the functions that can be well approximated are highly restricted to the class of self-affine functions. The primary objective of this paper is to suggest a C1-cubic Hermite interpolation scheme using a fractal methodology, namely, the coalescence hidden variable fractal interpolation, which works equally well for the approximation of a self-affine and non-self-affine data generating functions. The uniform error bound for the proposed fractal interpolant is established to demonstrate that the convergence properties are similar to that of the classical Hermite interpolant. For the Hermite interpolation problem, if the derivative values are not actually prescribed at the knots, then we assign these values so that the interpolant gains global C2-continuity. Consequently, the procedure culminates with the construction of cubic spline coalescence hidden variable fractal interpolants. Thus, the present article also provides an alternative to the construction of cubic spline coalescence hidden variable fractal interpolation functions through moments proposed by Chand and Kapoor [Fractals, 15(1)(2007), pp. 41-53].     

HOMOCENTRIC CONVERGENCE BALL OF THE SECANT METHOD

A local convergence theorem and five semi-local convergence theorems of the secant method are listed in this paper.For every convergence theorem,a convergence ball is respectively introduced,where the hypothesis conditions of the corresponding theorem can be satisfied.Since all of these convergence balls have the same center x~*,they can be viewed as a homocentric ball. Convergence theorems are sorted by the different sizes of various radii of this homocentric ball, and the sorted sequence represents the degree of weakness on the conditions of convergence theorems.     

On the convergence of the secant method under the gamma condition

We provide sufficient convergence conditions for the Secant method of approximating a locally unique solution of an operator equation in a Banach space. The main hypothesis is the gamma condition first introduced in [10] for the study of Newton’s method. Our sufficient convergence condition reduces to the one obtained in [10] for Newton’s method. A numerical example is also provided.      

Exact order of convergence of the secant method

We study the exact order of convergence of the secant method when applied to the problem of finding a zero of a nonlinear function defined from into . Under the standard assumptions for which Newton's method has the exact Q-order of convergencep, wherep is some positive integer, we establish that the secant method has the Q-order and the exact R-order of convergence . We prove also that, forp=2 andp=3, the secant method has the exact Q-order of convergenceS(p). Moreover, we present a counterexample to show that, forp4, it may not have an exact Q-order of convergence.The author wishes to thank Florian Potra, Richard Tapia, and the referees for helpful comments and suggestions.This paper was prepared while the author was Visiting Professor, Department of Mathematics, University of Kentucky, Lexington, Kentucky.     

On the convergence of the projected gradient method

We study the projected gradient algorithm for linearly constrained optimization. Wolfe (Ref. 1) has produced a counterexample to show that this algorithm can jam. However, his counterexample is only ¹( ⁿ ), and it is conjectured that the algorithm is convergent for ²-functions. We show that this conjecture is partly right. We also show that one needs more assumptions to prove convergence, since we present a family of counterexamples. We finally give a demonstration that no jamming can occur for quadratic objective functions.This work was supported by the Natural Sciences and Engineering Research Council of Canada     

On the convergence of sequential number-theoretic method for optimization

1. IntroductionIn the papers [l] and [2] H. Niederreiter and K. McCurley gave a quasi-Monte Carlolnethod for the approxiInate computation of the extreme values of a mu1tivariab1e function.In l989 K.T. Fa11g and Y. Wang["'l proposed a sequential algorithm fOr optinlization bya number-theoretic method (abbr. SNTO), which is nlore effective than the Niederreiter'smethod in some cases, but it lacks a complete convergence result concerlling the a1gorithm.In the present note we will approach…     

On the superlinear local convergence of a filter-SQP method

Transition to superlinear local convergence is shown for a modified version of the trust-region filter-SQP method for nonlinear programming introduced by Fletcher, Leyffer, and Toint [8]. Hereby, the original trust-region SQP-steps can be used without an additional second order correction. The main modification consists in using the Lagrangian function value instead of the objective function value in the filter together with an appropriate infeasibility measure. Moreover, it is shown that the modified trust-region filter-SQP method has the same global convergence properties as the original algorithm in [8].Mathematics Subject Classification (2000): 90C55, 65K05, 90C30     

On the global convergence of Chebyshev's iterative method

In [A. Melman, Geometry and convergence of Euler's and Halley's methods, SIAM Rev. 39(4) (1997) 728–735] the geometry and global convergence of Euler's and Halley's methods was studied. Now we complete Melman's paper by considering other classical third-order method: Chebyshev's method. By using the geometric interpretation of this method a global convergence theorem is performed. A comparison of the different hypothesis of convergence is also presented.     

On the convergence properties of second-order multiplier methods

The purpose of this note is to provide some estimates relating to Newton-type methods of multipliers. These estimates can be used to infer that convergence in such methods can be achieved for an arbitrary choice of the initial multiplier vector by selecting the penalty parameter sufficiently large.This work was supported by Grant No. NSF ENG 74-19332.     

On the convergence of the LJ search method

The convergence of the Luus-Jaakola search method for unconstrained optimization problems is established.Notation E _n Euclideann-space - f Gradient off(x) - ² f Hessian matrix - (·) ^T Transpose of (·) - I Index set {1, 2, ...,n} - [x _i1 ^*(j) ] Point around which search is made in the (j + 1)th iteration, i.e., [x _1l ^*(j) ,x _2l ^*(j) ,...,x _n1 ^*(j) ] - r _i ⁽ⁱ⁾ Range ofx _il ^*(i) in the (j + 1)th iteration - l ₁ min_i {r _i ⁽⁰⁾ } - l ₂ min_i {r _i ⁽⁰⁾ } - A _j Region of search in thejth iteration, i.e., {x E _n:x _il ^*(j-1) –0.5r _i ^(j-1) x _ix _il ^*(j-1) +0.5r _i ^(j-1) ,i I} - S _j Closed sphere with center origin and radius _j - Reduction factor in each iteration - 1– - (·) Gamma function Many discussions with Dr. S. N. Iyer, Professor of Electrical Engineering, College of Engineering, Trivandrum, India, are gratefully acknowledged. The author has great pleasure to thank Dr. K. Surendran, Professor, Department of Electrical Engineering, P.S.G. College of Technology, Coimbatore, India, for suggesting this work.     

On the global and cubic convergence of a quasi-cyclic Jacobi method

Summary In this paper we consider the global and the cubic convergence of a quasi-cyclic Jacobi method for the symmetric eigenvalue, problem. The method belongs to a class of quasi-cyclic methods recently proposed by W. Mascarenhas. Mascarenhas showed that the methods from his class asymptotically converge cubically per quasi-sweep (one quasi-sweep is equivalent to 1.25 cyclic sweeps) provided the eigenvalues are simple. Here we prove the global convergence of our method and derive very sharp asymptotic convergence bounds in the general case of multiple eigenvalues. We discuss the ultimate cubic convergence of the method and present several numerical examples which all well comply with the theory.This work was supported in part by the University of Minnesota Army High Performance Computing Research Center and the U.S Army Contract DAAL03-89-C-0038. The paper was partly written while this author was a visiting faculty in the Department of Mathematics, University of Kansas, Lawrence, Kansas. The first version of this paper was made in July 1990 while this author was visiting AHPCRC.     

Rate of convergence of a generalization of Newton's method

The Newton's method for finding the root of the equation (t)=0 can be easily generalized to the case where is monotone, convex, but not differentiable. Then, the convergence is superlinear. The purpose of this note is to show that the convergence is only superlinear. Indeed, for all (1, 2), we exhibit an example where the convergence of the iterates is exactly .     

Quadratic convergence of a special quasi-cyclic Jacobi method

This paper considers the ultimate asymptotic convergence of a block- oriented, quasi-cyclic Jacobi method for symmetric matrices. The conclusion applies to the new one-sided Jacobi method for computing the singular value decomposition, recently proposed by Drmač and Veselić. Using a simple qualitative analysis, the discussion indicates that a quadratic off-norm reduction per quasi-sweep is to be expected in all perceivable cases.      

An adaptive trust region method and its convergence

In this paper, a new trust region subproblem is proposed. The trust radius in the new subproblem adjusts itself adaptively. As a result, an adaptive trust region method is constructed based on the new trust region subproblem. The local and global convergence results of the adaptive trust region method are proved. Numerical results indicate that the new method is very efficient.