Kantorovich's Majorization and Functional Equations

Abstract

We consider systems of nonlinear difference equations arising when convergence analysis of an iterative method for solving operator equations in Banach spaces is carried out via Kantorovich's technique of majorization. The main challenge in this context is to determine the convergence domain of the corresponding majorant generator. As it turns out, dealing with this task leads to solution of functional equations of a certain kind. After considering several examples, we formulate two generic models and develop an approach to their solution.     

Analysis and approximation of a reliable model

In this paper we consider the problem of approximating the dynamical system that models reliability of a system consisting of two machines separated by a finite storage buffer. The system is described as a distributed parameter system defined by a coupled partial and ordinary differential equations and formulated as an abstract Cauchy problem. To derive the dynamical solution and some instantaneous indexes of the model, we present a simple finite difference scheme and establish the convergence of this scheme by employing Trotter–Kato Theorem. Numerical results are given to illustrate the effectiveness of the scheme.     

A penalty-free method with line search for nonlinear equality constrained optimization

A new line search method is introduced for solving nonlinear equality constrained optimization problems. It does not use any penalty function or a filter. At each iteration, the trial step is determined such that either the value of the objective function or the measure of the constraint violation is sufficiently reduced. Under usual assumptions, it is shown that every limit point of the sequence of iterates generated by the algorithm is feasible, and there exists at least one limit point that is a stationary point for the problem. A simple modification of the algorithm by introducing second order correction steps is presented. It is shown that the modified method does not suffer from the Maratos’ effect, so that it converges superlinearly. The preliminary numerical results are reported.     

非线性约束下的梯度投影法的收敛性

1960年Rosen提出的梯度投影法虽然已广泛应用得到成功,但其收敛问题20多年来一直得不到证明,同时也举不出一个反例。算法非闭是困难的原因。1986年何光中十分巧妙地证明了梯度投影法的收敛性:在n维欧氏空间中任何迭代序列的极限点均为Kuhn—Tucker点,本文将Rosen梯度投影法自然地推广到非线性约束情况,算法仍然非闭,证明了收敛性。证明的实质是从局部点态性质出发,得到一介全局收敛性的结论。     

Convergence to equilibria in the Lorenz system via monotone methods

In this paper, we provide some parameter values of the Lorenz system for which its flow is monotone with respect to the order induced by quadratic cones. This implies that its attractor is contained in a two-dimensional invariant manifold. An application of Dulac's criterion to the induced flow over that manifold leads to conditions under which every positive semiorbit converges to an equilibrium.     

I-fuzzy拓扑空间中的Moore-Smith收敛

用方进明提出的I-fuzzy拟重邻域系研究I-fuzzy拓扑空间中的Moore-Smith收敛性，给出它的应用，并获得了很多重要结论。     

求解大规模非Hermite线性方程组的Krylov子空间型方法的收敛性分析

本文用统一的方式研究了当系数矩阵Ａ亏损且其谱位于右（左）半开平面时很多求解大规模非Ｈｅｒｍｉｔｅ线性方程组的Ｋｒｙｌｏｖ子空间型方法的收敛性，建立了有关的理论收敛界，揭示了收敛速度和Ａ的谱之间的内在联系．结果证明，当如下三种情形之一出现时，这些方法的收敛速度将会减慢：Ａ亏损，其谱的分布不理想，或Ａ的Ｊｏｒｄａｎ基病态．在证明中，我们给出了Ｃｈｅｂｙｓｈｅｖ多项式的高阶导数在复平面中某椭圆域上的若干新性质，其中之一修正了文献中广泛使用的一个结果．     

解非凸规划问题的一种连续化方法

针对一类非线性规划问题的解存在的新等价性条件，给出了大范围收敛的连续化方法及证明了收敛性的结论．     

可交换随机变量序列的完全收敛性

本文探讨了可交换随机变量的完全收敛性问题，从充性和必要性两方面都作了研究，并把ｉ．ｉ．ｄ．情形下的部分结果作了为特例推出。     

Convergence acceleration of orthogonal series

The aim of this paper is to verify efficiency of two acceleration methods for orthogonal series (more strictly, for series defined at the beginning of Section 1). These methods are quite different although they use the same transform of such a series given there. The first method (Section 3) has some features common with Levin’s and Weniger’s methods. It may be profitably used in numerical calculations for a vast class of series. The second one (Sections 4 and 5) is somewhat similar to the Euler–Knopp transform of power series. Also this method is numerically realizable but more important is that for a narrower class of series, including some ones having applications in physics, it gives explicit analytic formulae of their transform.