An ODE-Based Trust Region Filter Algorithm for Unconstrained Optimization

In this article, an ODE-based trust region filter algorithm for unconstrained optimization is proposed. It can be regarded as a combination of trust region and filter techniques with ODE-based methods. Unlike the existing trust-region-filter methods and ODE-based methods, a distinct feature of this method is that at each iteration, a reduced linear system is solved to obtain a trial step, thus avoiding solving a trust region subproblem. Under some standard assumptions, it is proven that the algorithm is globally convergent. Preliminary numerical results show that the new algorithm is efficient for large scale problems.     

A hybrid trust region algorithm for unconstrained optimization

This paper presents a hybrid trust region algorithm for unconstrained optimization problems. It can be regarded as a combination of ODE-based methods, line search and trust region techniques. A feature of the proposed method is that at each iteration, a system of linear equations is solved only once to obtain a trial step. Further, when the trial step is not accepted, the method performs an inexact line search along it instead of resolving a new linear system. Under reasonable assumptions, the algorithm is proven to be globally and superlinearly convergent. Numerical results are also reported that show the efficiency of this proposed method.     

一类LC1规划问题的超线性收敛ODE信赖域算法

In this paper, a new trust region algorithm for unconstrained LC1 optimization problems is given. Compare with those existing trust regiion methods, this algorithm has a different feature: it obtains a stepsize at each iteration not by soloving a quadratic subproblem with a trust region bound, but by solving a system of linear equations. Thus it reduces computational complexity and improves computation efficiency. It is proven that this algorithm is globally convergent and locally superlinear under some conditions.     

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.     

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.     

LC1约束规划问题的超线性收敛ODE型信赖域算法

本文对带线性等式约束的LC^1优化问题提出了一个新的ODE型信赖域算法，它在每一次迭代时，不必求解带信赖域界的子问题，仅解一线性方程组而求得试验步。从而可以降低计算的复杂性，提高计算效率，在一定的条件下，文中还证明了该算法是超线性收敛的。     

An ODE-based trust region method for unconstrained optimization problems

In this paper, a new trust region algorithm is proposed for solving unconstrained optimization problems. This method can be regarded as a combination of trust region technique, fixed step-length and ODE-based methods. A feature of this proposed method is that at each iteration, only a system of linear equations is solved to obtain a trial step. Another is that when a trial step is not accepted, the method generates an iterative point whose step-length is defined by a formula. Under some standard assumptions, it is proven that the algorithm is globally convergent and locally superlinear convergent. Preliminary numerical results are reported.     

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.     

Inexact trust region method for large sparse systems of nonlinear equations

The main purpose of this paper is to prove the global convergence of the new trust region method based on the smoothed CGS algorithm. This method is surprisingly convenient for the numerical solution of large sparse systems of nonlinear equations, as is demonstrated by numerical experiments. A modification of the proposed trust region method does not use matrices, so it can be used for large dense systems of nonlinear equations.     

Convergence of a projected gradient method with trust region for nonlinear constrained optimization†

In this paper we describe a projected gradient algorithm with trust region, introducing a nondifferentiable merit function for solving nonlinear constrained optimization problems. We show that this method is globally convergent even if conditions are weak. It is also proved that, when the strict complementarity condition holds, the proposed algorithm can be solved by an equality constrained problem, allowing locally rate of superlinear convergence.     

A modified SLP algorithm and its global convergence

This paper concerns a filter technique and its application to the trust region method for nonlinear programming (NLP) problems. We used our filter trust region algorithm to solve NLP problems with equality and inequality constraints, instead of solving NLP problems with just inequality constraints, as was introduced by Fletcher et al. [R. Fletcher, S. Leyffer, Ph.L. Toint, On the global converge of an SLP-filter algorithm, Report NA/183, Department of Mathematics, Dundee University, Dundee, Scotland, 1999]. We incorporate this filter technique into the traditional trust region method such that the new algorithm possesses nonmonotonicity. Unlike the tradition trust region method, our algorithm performs a nonmonotone filter technique to find a new iteration point if a trial step is not accepted. Under mild conditions, we prove that the algorithm is globally convergent.     

Two accelerated nonmonotone adaptive trust region line search methods

Hybridizing monotone and nonmonotone approaches, we employ a modified trust region ratio in which more information is provided about the agreement between the exact and the approximate models. Also, we use an adaptive trust region radius as well as two accelerated Armijo-type line search strategies to avoid resolving the trust region subproblem whenever a trial step is rejected. We show that the proposed algorithm is globally and locally superlinearly convergent. Comparative numerical experiments show practical efficiency of the proposed accelerated adaptive trust region algorithm.     

一类约束非光滑优化的非单调信赖域算法

提出了求解一类带一般凸约束的复合非光滑优化的信赖域算法 .和通常的信赖域方法不同的是 :该方法在每一步迭代时不是迫使目标函数严格单调递减 ,而是采用非单调策略 .由于光滑函数、逐段光滑函数、凸函数以及它们的复合都是局部Lipschitz函数 ,故本文所提方法是已有的处理同类型问题 ,包括带界约束的非线性最优化问题的方法的一般化 ,从而使得信赖域方法的适用范围扩大了 .同时 ,在一定条件下 ,该算法还是整体收敛的 .数值实验结果表明 :从计算的角度来看 ,非单调策略对高度非线性优化问题的求解非常有效     

Implicitly restarted projection algorithm for solving optimization problems

An algorithm for solving the problem of minimizing a quadratic function subject to ellipsoidal constraints is introduced. This algorithm is based on the impHcitly restarted Lanczos method to construct a basis for the Krylov subspace in conjunction with a model trust region strategy to choose the step. The trial step is computed on the small dimensional subspace that lies inside the trust region.

One of the main advantages of this algorithm is the way that the Krylov subspace is terminated. We introduce a terminationcondition that allows the gradient to be decreased on that subspace.

A convergence theory for this algorithm is presented. It is shown that this algorithm is globally convergent and it shouldcope quite well with large scale minimization problems. This theory is sufficiently general that it holds for any algorithm that projects the problem on a lower dimensional subspace.     

一类变分不等式问题的信赖域算法

基于J.M.Peng研究一类变分不等式问题（简记为VIP）时所提出的价值函数，本文提出了求解强单调的VIP的一个新的信赖域算法。和已有的处理VIP的信赖域方法不同的是：它在每步迭代时，不必求解带信赖域界的子问题，仅解一线性方程组而求得试验步。这样，计算的复杂性一般来说可降低。在通常的假设条件下，文中还证明了算法的整体收敛性。最后，在梯度是半光滑和约束是矩形域的假设下，该算法还是超线性收敛的。     

A trust region algorithm for minimization of locally Lipschitzian functions

The classical trust region algorithm for smooth nonlinear programs is extended to the nonsmooth case where the objective function is only locally Lipschitzian. At each iteration, an objective function that carries both first and second order information is minimized over a trust region. The term that carries the first order information is an iteration function that may not explicitly depend on subgradients or directional derivatives. We prove that the algorithm is globally convergent. This convergence result extends the result of Powell for minimization of smooth functions, the result of Yuan for minimization of composite convex functions, and the result of Dennis, Li and Tapia for minimization of regular functions. In addition, compared with the recent model of Pang, Han and Rangaraj for minimization of locally Lipschitzian functions using a line search, this algorithm has the same convergence property without assuming positive definiteness and uniform boundedness of the second order term. Applications of the algorithm to various nonsmooth optimization problems are discussed.This author's work was supported in part by the Australian Research Council.This author's work was carried out while he was visiting the Department of Applied Mathematics at the University of New South Wales.     

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.     

非单调技术预条件弧线路径信赖域解无约束优化

本文提供修正近似信赖域类型路经三类预条件弧线路径方法解无约束最优化问题.使用对称矩阵的稳定Bunch-Parlett易于形成信赖域子问题的弧线路径,使用单位下三角矩阵作为最优路径和修正梯度路径的预条件因子.运用预条件因子改进Hessian矩阵特征值分布加速预条件共轭梯度路径收敛速度.基于沿着三类路径信赖域子问题产生试探步,将信赖域策略与非单调线搜索技术相结合作为新的回代步.理论分析证明在合理条件下所提供的算法是整体收敛性,并且具有局部超线性收敛速率,数值结果表明算法的有效性.     

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．     

一个求解一类投影问题的总体线性收敛的迭代法

所求的解就是c在p上的投影。 对于问题(1.1),He基于求解线性互补问题的投影收缩(PC)法,把投影问题转化为等价的广义线性互补问题,提出了一个求解这类问题的迭代方法。 原始的PC方法只能证明迭代是全局收敛的,而无法估计其收敛速度。为此,[4]和[5]对原始的PC方法作了改进,提出了固定步长的PC法并证明了其收敛速度是线性的。但在实际应用中,固定步长的PC法比原始的PC法慢的多,而且在求步长时,还要估计约束矩阵范数的大小。 本文基于[5]的思想,对于(1.1)提出了一个新的PC方法,该方法是全局线性收敛的。 本文中用到的符号说明如下:x_i表示x的第i个分量。如果u∈(?)且Ω(?)(?)为凸闭集,则P_Ω[u]定义为u到Ω上的投影。特别地,u_+定义为u到非负卦限(?)上的投影,对于一个正定矩阵G,范数||u++G表示(u~TGu)(?)。