Mann-Type Steepest-Descent and Modified Hybrid Steepest-Descent Methods for Variational Inequalities in Banach Spaces

In this paper, we propose three different kinds of iteration schemes to compute the approximate solutions of variational inequalities in the setting of Banach spaces. First, we suggest Mann-type steepest-descent iterative algorithm, which is based on two well-known methods: Mann iterative method and steepest-descent method. Second, we introduce modified hybrid steepest-descent iterative algorithm. Third, we propose modified hybrid steepest-descent iterative algorithm by using the resolvent operator. For the first two cases, we prove the convergence of sequences generated by the proposed algorithms to a solution of a variational inequality in the setting of Banach spaces. For the third case, we prove the convergence of the iterative sequence generated by the proposed algorithm to a zero of an operator, which is also a solution of a variational inequality.     

A multi-step class of iterative methods for nonlinear systems

In this article, the numerical solution of nonlinear systems using iterative methods are dealt with. Toward this goal, a general class of multi-point iteration methods with various orders is constructed. The error analysis is presented to prove the convergence order. Also, a thorough discussion on the computational complexity of the new iterative methods will be given. The analytical discussion of the paper will finally be upheld through solving some application-oriented problems.     

解线性方程组的预条件迭代方法

In this paper,we study the preconditioning iterative methods for the solution of the linear system and provide a convergence theorem of this method,it improves some recent results,We prove that if all parameters are in [0,1],the convergence rates for the MGS (Modified Gauss-Seidel)type methods are better than those of the corresponding SOR type methods.     

An algorithm for a bilevel problem with equilibrium and fixed point constraints

We prove the solution existence and propose an iterative algorithm for solving equilibrium problems over the common solutions of a monotone equilibrium problem and the fixed points of a strictly pseudocontractive mapping in Hilbert spaces. The algorithm is a combination of the proximal point and Halpern methods. Strong convergence is established.     

广义H-矩阵的乘积的性质

1 引言 广义M-矩阵和广义H-矩阵的理论在许多实际问题的研究中有着非常重要的作用,如欧拉方程数值求解中出现的线性系统的块迭代法的收敛性问题,以及动力系统的研究等.     

Hybrid methods with regularization for minimization problems and asymptotically strict pseudocontractive mappings in the intermediate sense

In this paper we introduce an iterative algorithm for finding a common element of the fixed point set of an asymptotically strict pseudocontractive mapping S in the intermediate sense and the solution set of the minimization problem (MP) for a convex and continuously Frechet differentiable functional in Hilbert space. The iterative algorithm is based on several well-known methods including the extragradient method, CQ method, Mann-type iterative method and hybrid gradient projection algorithm with regularization. We obtain a strong convergence theorem for three sequences generated by our iterative algorithm. In addition, we also prove a new weak convergence theorem by a modified extragradient method with regularization for the MP and the mapping S.     

Asynchronous gradient algorithms for a class of convex separable network flow problems

We consider the single commodity strictly convex network flow problem. The dual of this problem is unconstrained, differentiable, and well suited for solution via distributed or parallel iterative methods. We present and prove convergence of gradient and asynchronous gradient algorithms for solving the dual problem. Computational results are given and analysed.     

一致凸Banach空间非扩张映象带误差的Ishikawa型的三重迭代序列的收敛性

在一致凸Banach空间上,研究了半紧的非扩张压缩映象‖Tx-Ty‖≤‖x-y‖的Ishikawa型的三重迭代序列的收敛性问题,建立并证明了带误差的Ishikawa三重迭代逼近收敛定理,从而独特的推广了Mann和Ishikawa迭代方法,改进和发展了文献[1]-[7]的主要结果.     

Analysis and Computational Methods of Dirichlet Boundary Optimal Control Problems for 2D Boussinesq Equations

We study Dirichlet boundary optimal control problems for 2D Boussinesq equations. The existence of the solution of the optimization problem is proved and an optimality system of partial differential equations is derived from which optimal controls and states may be determined. Then, we present some computational methods to get the solution of the optimality system. The iterative algorithms are given explicitly. We also prove the convergence of the gradient algorithm.     

A New Class of Monotone Mappings and a New Class of Variational Inclusions in Banach Spaces

In this paper, we introduce and study a new class of variational inclusions in Banach spaces. As it concerns the methods of solution, we introduce a new class of monotone mappings. We define a proximal mapping associated with this mappings and show its Lipschitz continuity. By using the technique of proximal mapping, we construct a new iterative algorithm. Under some suitable conditions, we prove the convergence of iterative sequences generated by the algorithm. Our results improve and generalize many known results.     

Une methode iterative de resolution d’une inequation variationnelle

We prove the convergence of an iterative method to a solution of a variational inequality.      

Strong convergence theorems for countable families of multivalued nonexpansive mappings and systems of equilibrium and variational inequality problems

In this paper, we introduce a new iterative scheme by hybrid methods and prove strong convergence of the scheme for approximation of a common fixed point of two countably infinite families of multi valued nonexpansive mappings which is also a solution to system equilibrium problems and system of variational inequality problems in a real Hilbert space. Our results extend important recent results.     

Iterative methods for solving proximal split minimization problems

In this paper, we propose two iterative algorithms for finding the minimum-norm solution of a split minimization problem. We prove strong convergence of the sequences generated by the proposed algorithms. The iterative schemes are proposed in such a way that the selection of the step-sizes does not need any prior information about the operator norm. We further give some examples to numerically verify the efficiency and implementation of our new methods and compare the two algorithms presented. Our results act as supplements to several recent important results in this area.     

增生算子方程的具误差的Ishikawa迭代序列的强收敛定理

本文研究了Banach空间中Lipschitz的增生算子T的方程的解的迭代逼近问题.利用Ishikawa迭代法,证明了具误差的Ishikawa迭代序列强收敛到方程的唯一解,得到了一般的收敛率估计式.     

Extended Projection Methods for Monotone Variational Inequalities

In this paper, we prove that each monotone variational inequality is equivalent to a two-mapping variational inequality problem. On the basis of this fact, a new class of iterative methods for the solution of nonlinear monotone variational inequality problems is presented. The global convergence of the proposed methods is established under the monotonicity assumption. The conditions concerning the implementability of the algorithms are also discussed. The proposed methods have a close relationship to the Douglas–Rachford operator splitting method for monotone variational inequalities.     

具有参数的不带有导数的平方收敛的迭代法

1.引 言 考虑数值求解非线性方程 f(x)=0, (1)其中实值函数f(x)在实零点x*的某邻域U(x*)内连续可微且f'(x)≠0. 牛顿法是科学与工程计算中数值求解(1)的常用数值方法.虽然它一般至少是二阶收敛的,但它需要调用导数值,这使其应用受到限制.我们修改牛顿法,用割线代替切线可得不带     

Convergence and spectral analysis of the Frontini-Sormani family of multipoint third order methods from quadrature rule

Point of attraction theory is an important tool to analyze the local convergence of iterative methods for solving systems of nonlinear equations. In this work, we prove a generalized form of Ortega-Rheinbolt result based on point of attraction theory. The new result guarantees that the solution of the nonlinear system is a point of attraction of iterative scheme, especially multipoint iterations. We then apply it to study the attraction theorem of the Frontini-Sormani family of multipoint third order methods from Quadrature Rule. Error estimates are given and compared with existing ones. We also obtain the radius of convergence of the special members of the family. Two numerical examples are provided to illustrate the theory. Further, a spectral analysis of the Discrete Fourier Transform of the numerical errors is conducted in order to find the best method of the family. The convergence and the spectral analysis of a multistep version of one of the special member of the family are studied.     

Iterative Methods for a Linearly Perturbed Algebraic Matrix Riccati Equation Arising in Stochastic Control

We start with a discussion of coupled algebraic Riccati equations arising in the study of linear-quadratic optimal control problems for Markov jump linear systems. Under suitable assumptions, this system of equations has a unique positive semidefinite solution, which is the solution of practical interest. The coupled equations can be rewritten as a single linearly perturbed matrix Riccati equation with special structures. We study the linearly perturbed Riccati equation in a more general setting and obtain a class of iterative methods from different splittings of a positive operator involved in the Riccati equation. We prove some special properties of the sequences generated by these methods and determine and compare the convergence rates of these methods. Our results are then applied to the coupled Riccati equations of jump linear systems. We obtain linear convergence of the Lyapunov iteration and the modified Lyapunov iteration, and confirm that the modified Lyapunov iteration indeed has faster convergence than the original Lyapunov iteration.     

Numerical solution of quasi-variational inequalities arising in stochastic game theory

We study the finite-difference approximation for the quasi-variational inequalities for a stochastic game involving discrete actions of the players and continuous and discrete payoff. We prove convergence of iterative schemes for the solution of the discretized quasi-variational inequalities, with estimates of the rate of convergence (via contraction mappings) in two particular cases. Further, we prove stability of the finite-difference schemes, and convergence of the solution of the discrete problems to the solution of the continuous problem as the discretization mesh goes to zero. We provide a direct interpretation of the discrete problems in terms of finite-state, continuous-time Markov processes.     

Necessary and sufficient conditions for the convergence of two- and three-point Newton-type iterations

Necessary and sufficient conditions under which two- and three-point iterative methods have the order of convergence р (2 ≤ р ≤ 8) are formulated for the first time. These conditions can be effectively used to prove the convergence of iterative methods. In particular, the order of convergence of some known optimal methods is verified using the proposed sufficient convergence tests. The optimal set of parameters making it possible to increase the order of convergence is found. It is shown that the parameters of the known iterative methods with the optimal order of convergence have the same asymptotic behavior. The simplicity of choosing the parameters of the proposed methods is an advantage over the other known methods.