一类非线性互补问题的信赖域算法

In this paper,an ODE-type trust region algorithm for solving a class of nonlinear complementarity problems is proposed.A feature of this algorithm is that only the solution of linear systems of equations is required at each iteration,thus avoiding the need for solving a quadratic subproblem with a trust region bound.Under some conditions,it is proven that this algorithm is globally and locally superlinear convergent.The limited numerical examples show its efficiency.     

NGLM:一类全局收敛的Newton-GMRES方法

本文提出了一类具有全局收敛性质的Newton-GMRES方法—NGLM方法．该方法是对经典Newton—GMRES方法的推广．NGLM方法的全局策略是当在非精确Newton方向上后退不能成功时，转而在一个子空间上运用信赖域方法确定迭代步长．理论分析与数值实验均表明，NGLM方法改善了Newton—GMRES方法的强健性．     

A new simple model trust-region method with generalized Barzilai-Borwein parameter for large-scale optimization

In this paper a new trust region method with simple model for solving large-scale unconstrained nonlinear optimization is proposed. By employing the generalized weak quasi-Newton equations, we derive several schemes to construct variants of scalar matrices as the Hessian approximation used in the trust region subproblem. Under some reasonable conditions, global convergence of the proposed algorithm is established in the trust region framework. The numerical experiments on solving the test problems with dimensions from 50 to 20,000 in the CUTEr library are reported to show efficiency of the algorithm.     

带非线性不等式约束优化问题的信赖域算法

借助于KKT条件和NCP函数,提出了求解带非线性不等式约束优化问题的信赖域算法.该算法在每一步迭代时,不必求解带信赖域界的二次规划子问题,仅需求一线性方程组系统.在适当的假设条件下,它还是整体收敛的和局部超线性收敛的.数值实验结果表明该方法是有效的.     

Truncated Trust Region Methods Based on Preconditioned Iterative Subalgorithms for Large Sparse Systems of Nonlinear Equations

This paper is devoted to globally convergent methods for solving large sparse systems of nonlinear equations with an inexact approximation of the Jacobian matrix. These methods include difference versions of the Newton method and various quasi-Newton methods. We propose a class of trust region methods together with a proof of their global convergence and describe an implementable globally convergent algorithm which can be used as a realization of these methods. Considerable attention is concentrated on the application of conjugate gradient-type iterative methods to the solution of linear subproblems. We prove that both the GMRES and the smoothed COS well-preconditioned methods can be used for the construction of globally convergent trust region methods. The efficiency of our algorithm is demonstrated computationally by using a large collection of sparse test problems.     

Convergence Rate of The Trust Region Method for Nonlinear Equations Under Local Error Bound Condition

In this paper, we present the new trust region method for nonlinear equations with the trust region converging to zero. The new method preserves the global convergence of the traditional trust region methods in which the trust region radius will be larger than a positive constant. We study the convergence rate of the new method under the local error bound condition which is weaker than the nonsingularity. An example given by Y.X. Yuan shows that the convergence rate can not be quadratic. Finally, some numerical results are given. This work is supported by Chinese NSFC grants 10401023 and 10371076, Research Grants for Young Teachers of Shanghai Jiao Tong University, and E-Institute of Computational Sciences of Shanghai Universities. An erratum to this article is available at .     

An improved trust region algorithm for nonlinear equations

In this paper, we present a new trust region algorithm for the system of singular nonlinear equations with the regularized trust region subproblem. The new algorithm preserves the global convergence of the traditional trust region algorithm, and has the quadratic convergence under some suitable conditions. Finally, some numerical results are given.     

电力系统中地网腐蚀诊断的一种新的数学模型及其仿真计算

对电力系统中具有重大应用价值的地网腐蚀诊断问题抽象出仿真求解的一种新的数学模型:即求解带约束的非线性隐式方程组模型.但由于问题本身的物理特性决定了所建立的数学模型具有以下特点:一是非线性方程组为欠定方程组,而且非线性程度非常高;二是方程组的所有函数均为隐函数;三是方程组附加若干箱约束条件.这种特性给模型分析与算法设计带来巨大困难.对于欠定方程组的求解,文中根据工程实际背景,尽可能地扩充方程的个数,使之成为超定方程组,然后对欠定方程组和超定方程组分别求解并进行比较.将带约束的非线性隐函数方程组求解问题,转化为无约束非线性最小二乘问题,并采用矩阵求导等技术和各种算法设计技巧克服隐函数的计算困难,最后使用拟牛顿信赖域方法进行计算.大量的计算实例表明,文中所提出的数学模型及求解方法是可行的.与目前广泛采用的工程简化模型相比较,在模型和算法上具有很大优势.     

A superlinearly convergent ODE-type trust region algorithm for nonsmooth nonlinear equations

This paper presents a new trust region algorithm for solving nonsmooth nonlinear equation problems which posses the smooth plus non-smooth decomposition. At each iteration, this method obtains a trial step by solving a system of linear equations, hence avoiding the need for solving a quadratic programming subproblem with a trust region bound. From a computational point of view, this approach may reduce computational effort and hence improve computational efficiency. Furthermore, it is proved under appropriate assumptions that this algorithm is globally and locally super-linearly convergent. Some numerical examples are reported.     

信赖域方法在Hölderian局部误差界下的收敛性质

信赖域方法是求解非线性方程组的一种重要方法.本文研究了求解非线性方程组的信赖域半径趋于零的信赖域算法在Jacobi矩阵Hölderian连续条件下的全局收敛性质,以及其在Hölderian局部误差界和Jacobi矩阵Hölderian连续条件下的收敛速度.     

A NONMONOTONE TRUST REGION METHOD FOR NONLINEAR LEAST SQUARES PROBLEMS

In this paper we present a nonmonotone trust region method for nonlinear least squares problems with zero-residual and prove its convergence properties. The extensive numerical results are reported which show that the nonmonotone trust region method is generally superior to the usual trust region method.     

A new trust region method for nonlinear equations

In this paper, a new trust region method for the system of nonlinear equations is presented in which the determining of the trust region radius incorporates the information of its natural residual. The global convergence is obtained under mild conditions. Unlike traditional trust region method, the superlinear convergence of the method is proven under the local error bound condition. This condition is weaker than the nondegeneracy assumption which is necessary for superlinear convergence of traditional trust region method. We also propose an approximate algorithm for the trust region subproblem. Preliminary numerical experiments are reported. Acknowledgements.The authors are indebted to our supervisor, Professor Y.-X. Yuan, for his excellent guidance and Jorge J. Moré for his subroutine. And we would like to thank the referees for their valuable suggestions and comments.     

A SUPERLINEARLY CONVERGENT TRUST REGION ALGORITHM FOR LC^1 CONSTRAINED OPTIMIZATION PROBLEMS

In this paper, a new trust region algorithm for nonlinear equality constrained LC^1 optimization problems is given. It obtains a search direction at each iteration not by solving a quadratic programming subproblem with a trust region bound, but by solving a system of linear equations. Since the computational complexity of a QP-Problem is in general much larger than that of a system of linear equations, this method proposed in this paper may reduce the computational complexity and hence improve computational efficiency. Furthermore, it is proved under appropriate assumptions that this algorithm is globally and super-linearly convergent to a solution of the original problem. Some numerical examples are reported, showing the proposed algorithm can be beneficial from a computational point of view.     

Constrained Dogleg methods for nonlinear systems with simple bounds

We focus on the numerical solution of medium scale bound-constrained systems of nonlinear equations. In this context, we consider an affine-scaling trust region approach that allows a great flexibility in choosing the scaling matrix used to handle the bounds. The method is based on a dogleg procedure tailored for constrained problems and so, it is named Constrained Dogleg method. It generates only strictly feasible iterates. Global and locally fast convergence is ensured under standard assumptions. The method has been implemented in the Matlab solver CoDoSol that supports several diagonal scalings in both spherical and elliptical trust region frameworks. We give a brief account of CoDoSol and report on the computational experience performed on a number of representative test problems.     

Inexact trust region PGC method for large sparse unconstrained optimization

We present an algorithm, partitioning group correction (PGC) algorithm based on trust region and conjugate gradient method, for large-scale sparse unconstrained optimization. In large sparse optimization, computing the whole Hessian matrix and solving the Newton-like equations at each iteration can be considerably expensive when a trust region method is adopted. The method depends on a symmetric consistent partition of the columns of the Hessian matrix and an inaccurate solution to the Newton-like equations by conjugate gradient method. And we allow that the current direction exceeds the trust region bound if it is a good descent direction. Besides, we studies a method dealing with some sparse matrices having a dense structure part. Some good convergence properties are kept and we contrast the computational behavior of our method with that of other algorithms. Our numerical tests show that the algorithm is promising and quite effective, and that its performance is comparable to or better than that of other algorithms available.     

A fourth order ADI method for semidiscrete parabolic equations

A fourth order fourstep ADI method is described for solving the systems of ordinary differential equations which are obtained when a (nonlinear) parabolic initial-boundary value problem in two dimensions is semi-discretized. The local time-discretization error and the stability conditions are derived. By numerical experiments it is demonstrated that the (asymptotic) fourth order behaviour does not degenerate if the time step increases to relatively large values. Also a comparison is made with the classical ADI method of Peaceman and Rachford showing the superiority of the fourth order method in the higher accuracy region, particularly in nonlinear problems.     

LINEAR SYSTEMS ASSOCIATED WITH NUMERICAL METHODS FOR CONSTRAINED OPITMIZATION

Linear systems associated with numerical methods for constrained optimization are discussed in this paper. It is shown that the corresponding subproblems arise in most well-known methods, no matter line search methods or trust region methods for constrained optimization can be expressed as similar systems of linear equations. All these linear systems can be viewed as some kinds of approximation to the linear system derived by the Lagrange-Newton method. Some properties of these linear systems are analyzed.     

Linear Systems Associated with Numerical Methods for Constrained Optimization

Linear systems associated with numerical methods for constrained optimization are discussed in thia paper ,It is shown that the corresponding subproblems arise in most well-known methods,no matter line search methods or trust region methods for constrained optimization can be expressed as similar systems of linear equations.All these linear systems can be viewed as some kinds of approximation to the linear system derived by the Lagrange-Newton method .Some properties of these linear systems are analyzed.     

A unified approach to global convergence of trust region methods for nonsmooth optimization

This paper investigates the global convergence of trust region (TR) methods for solving nonsmooth minimization problems. For a class of nonsmooth objective functions called regular functions, conditions are found on the TR local models that imply three fundamental convergence properties. These conditions are shown to be satisfied by appropriate forms of Fletcher's TR method for solving constrained optimization problems, Powell and Yuan's TR method for solving nonlinear fitting problems, Zhang, Kim and Lasdon's successive linear programming method for solving constrained problems, Duff, Nocedal and Reid's TR method for solving systems of nonlinear equations, and El Hallabi and Tapia's TR method for solving systems of nonlinear equations. Thus our results can be viewed as a unified convergence theory for TR methods for nonsmooth problems.Research supported by AFOSR 89-0363, DOE DEFG05-86ER25017 and ARO 9DAAL03-90-G-0093.Corresponding author.     

Finding all Solutions of Systems of Nonlinear Equations Using the Dual Simplex Method

An efficient algorithm is proposed for finding all solutions of nonlinear equations using linear programming (LP). This algorithm is based on a simple test (termed the LP test) for nonexistence of a solution to a system of nonlinear equations in a given region. In the conventional LP test, the system of nonlinear equations is transformed into an LP problem, to which the simplex method is applied. However, although the LP test is very powerful, it requires many pivotings for each region. In this paper, we use the dual simplex method in the LP test, which makes the average number of pivotings per region much smaller (less than one, for example) and makes the algorithm very efficient. By numerical examples, it is shown that the proposed algorithm can find all solutions of systems of 200 nonlinear equations in practical computation time.