Convergence and certain control conditions for hybrid viscosity approximation methods

Very recently, Yao, Chen and Yao [20] proposed a hybrid viscosity approximation method, which combines the viscosity approximation method and the Mann iteration method. Under the convergence of one parameter sequence to zero, they derived a strong convergence theorem in a uniformly smooth Banach space. In this paper, under the convergence of no parameter sequence to zero, we prove the strong convergence of the sequence generated by their method to a fixed point of a nonexpansive mapping, which solves a variational inequality. An appropriate example such that all conditions of this result are satisfied and their condition β_n→0 is not satisfied is provided. Furthermore, we also give a weak convergence theorem for their method involving a nonexpansive mapping in a Hilbert space.     

Convergence conditions for restarted conjugate gradient methods with inaccurate line searches

Convergence properties of restarted conjugate gradient methods are investigated for the case where the usual requirement that an exact line search be performed at each iteration is relaxed.The objective function is assumed to have continuous second derivatives and the eigenvalues of the Hessian are assumed to be bounded above and below by positive constants. It is further assumed that a Lipschitz condition on the second derivatives is satisfied at the location of the minimum.A class of descent methods is described which exhibitn-step quadratic convergence when restarted even though errors are permitted in the line search. It is then shown that two conjugate gradient methods belong to this class.Sponsored by the United States Army under Contract No. DA-31-124-ARO-D-462.     

Convergence conditions for Gummel's method

Four data-smallness conditions that guarantee existence and uniqueness for solutions of stationary systems of equations in the physics of semiconductors are derived. Gummel's method converges under these conditions.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova Akademii Nauk SSSR, Vol. 185, pp. 146–159, 1990.     

Convergence conditions for the Seidel process

The convergence condition of the successive approximation process based on the Seidel method is derived for a system of two transcendental equations with allowance for specific functional dependences.Ukrainian External Polytechnical Institute, Khar'kov. Translated from Vychislitel'naya i Prikladnaya Matematika, No. 74, pp. 19–20, 1992;     

Convergence theorems for continuous descent methods

We examine continuous descent methods for the minimization of Lipschitzian functions defined on a general Banach space. We establish several convergence theorems for those methods which are generated by regular vector fields. Since the complement of the set of regular vector fields is -porous, we conclude that our results apply to most vector fields in the sense of Baires categories.     

Convergence results for multistep Runge-Kutta methods

Summary. Recently Ch. Lubich proved convergence results for Runge-Kutta methods applied to stiff mechanical systems. The present paper discusses the new ideas necessary to extend these results to general linear methods, in particular BDF and multistep Runge-Kutta methods. Received August 9, 1993 / Revised version received May 3, 1994     

Convergence of Newton-like-iterative methods

Summary Newton-like methods in which the intermediate systems of linear equations are solved by iterative techniques are examined. By applying the theory of inexact Newton methods radius of convergence and rate of convergence results are easily obtained. The analysis is carried out in affine invariant terms. The results are applicable to cases where the underlying Newton-like method is, for example, a difference Newton-like or update-Newton method.     

Convergence of inexact Newton methods for generalized equations

For solving the generalized equation $f(x)+F(x) \ni 0$ , where $f$ is a smooth function and $F$ is a set-valued mapping acting between Banach spaces, we study the inexact Newton method described by $$\begin{aligned} \left( f(x_k)+ D f(x_k)(x_{k+1}-x_k) + F(x_{k+1})\right) \cap R_k(x_k, x_{k+1}) \ne \emptyset , \end{aligned}$$ where $Df$ is the derivative of $f$ and the sequence of mappings $R_k$ represents the inexactness. We show how regularity properties of the mappings $f+F$ and $R_k$ are able to guarantee that every sequence generated by the method is convergent either q-linearly, q-superlinearly, or q-quadratically, according to the particular assumptions. We also show there are circumstances in which at least one convergence sequence is sure to be generated. As a byproduct, we obtain convergence results about inexact Newton methods for solving equations, variational inequalities and nonlinear programming problems.     

Convergence of SSOR methods for linear complementarity problems

In this paper, by applying the SSOR splitting, we propose two new iterative methods for solving the linear complementarity problem LCP (M,q). Convergence results for these two methods are presented when M is an H-matrix (and also an M-matrix). Finally, two numerical examples are given to show the efficiency of the presented methods.     

Convergence of gradient methods for optimal control problems

A general convergence theorem for gradient algorithms in normed spaces is given and is applied to the unconstrained optimal control problem. A further application is given to time-lag systems of neutral type.This work was completed while the author held a Science Research Council Postdoctoral Fellowship at Loughborough University of Technology, Loughborough, Leicestershire, England.     

Convergence of direct methods for paramonotone variational inequalities

We analyze one-step direct methods for variational inequality problems, establishing convergence under paramonotonicity of the operator. Previous results on the method required much more demanding assumptions, like strong or uniform monotonicity, implying uniqueness of solution, which is not the case for our approach.     

Convergence analysis for column-action methods in image reconstruction

Column-oriented versions of algebraic iterative methods are interesting alternatives to their row-version counterparts: they converge to a least squares solution, and they provide a basis for saving computational work by skipping small updates. In this paper we consider the case of noise-free data. We present a convergence analysis of the column algorithms, we discuss two techniques (loping and flagging) for reducing the work, and we establish some convergence results for methods that utilize these techniques. The performance of the algorithms is illustrated with numerical examples from computed tomography.     

Convergence spaces and diagonal conditions

Certain diagonal axioms due to Kowalsky, Cook and Fischer are studied and compared.     

Convergence conditions for the Brown-Robinson iterative method for bimatrix games

The article considers the convergence of the Brown-Robinson iterative method to find a mixed-strategy equilibrium in a bimatrix game. The known result on convergence to an equilibrium for a zero-sum game is generalized to a wider class of games that are reducible to zero-sum games by a composition of various transformations: addition of a constant to any column of the first-player payoff matrix; addition of a constant to any row in the second-player payoff matrix; multiplication of the payoff matrix by a positive constant α>0. Translated from Prikladnaya Matematika i Informatika, No. 2, pp. 69–83, 1999.     

Convergence of skew-symmetric iterative methods

We propose a new technique for studying the convergence of triangular skew-symmetric and product triangular skew-symmetric iterative methods (introduced earlier by the first author) based on the notion of a field of values of a matrix. We obtain formulas connecting the field of values of the initial matrix, that of the matrix which determines the iterative method, and eigenvalues of the iterative matrix. We prove that the mentioned methods can converge even if the initial matrix is not dissipative.