AN ADAPTIVE NONMONOTONIC TRUST REGION METHOD WITH CURVILINEAR SEARCHES

In this paper,an algorithm for unconstrained optimization that employs both trustregion techniques and curvilinear searches is proposed.At every iteration,we solve thetrust region subproblem whose radius is generated adaptively only once.Nonmonotonicbacktracking curvilinear searches are performed when the solution of the subproblem isunacceptable.The global convergence and fast local convergence rate of the proposedalgorithms are established under some reasonable conditions.The results of numericalexperiments are reported to show the effectiveness of the proposed algorithms.     

网络最短的最优解集结构

The shortest path problem in a network G is to find shortest paths between some specified source vertices and terminal vertices when the lengths of edges are given. The structure of the optimal solutions set on the shortest paths is studied in this paper. First,the conditions of having unique shortest path between two distinguished vertices s and t in a network G are discussed;Second,the structural properties of 2-transformation graph (?) on the shortest-paths for G are presented heavily.     

FRAMELET BASED DECONVOLUTION

In this paper, two framelet based deconvolution algorithms are proposed. The basic idea of framelet based approach is to convert the deconvolution problem to the problem of inpainting in a frame domain by constructing a framelet system with one of the masks being the given （discrete） convolution kernel via the unitary extension principle of [26], as introduced in [6-9] . The first algorithm unifies our previous works in high resolution image reconstruction and infra-red chopped and nodded image restoration, and the second one is a combination of our previous frame-based deconvolution algorithm and the iterative thresholding algorithm given by [14, 16]. The strong convergence of the algorithms in infinite dimensional settings is given by employing proximal forward-backward splitting （PFBS） method. Consequently, it unifies iterative algorithms of infinite and finite dimensional setting and simplifies the proof of the convergence of the aluorithms of [6].     

A Stochastic Approximation Frame Algorithm with Adaptive Directions

Stochastic approximation problem is to find some root or extremum of a non- linear function for which only noisy measurements of the function are available.The classical algorithm for stochastic approximation problem is the Robbins-Monro (RM) algorithm,which uses the noisy evaluation of the negative gradient direction as the iterative direction.In order to accelerate the RM algorithm,this paper gives a flame algorithm using adaptive iterative directions.At each iteration,the new algorithm goes towards either the noisy evaluation of the negative gradient direction or some other directions under some switch criterions.Two feasible choices of the criterions are pro- posed and two corresponding flame algorithms are formed.Different choices of the directions under the same given switch criterion in the flame can also form different algorithms.We also proposed the simultanous perturbation difference forms for the two flame algorithms.The almost surely convergence of the new algorithms are all established.The numerical experiments show that the new algorithms are promising.     

NONMONOTONE CURVED SEARCH METHODS FOR UNCONSTRAINED OPTIMIZATION

In this paper two nonmonolone curved search (NCS) algorithms fur unconstrained optimization are presented. The NCS algorithms possess both a global convergence properly and a quadratic rale of convergence. Some numerical results are also reported which show that the NCS algorithn is superior to the usual curved search (UCS)aIgorithm for typical lest problems.     

Generalized Jacobi and Gauss-Seidel Methods for Solving Linear System of Equations

The Jacobi and Gauss-Seidel algorithms are among the stationary iterative methods for solving linear system of equations. They are now mostly used as precondition-ers for the popular iterative solvers. In this paper a generalization of these methods are proposed and their convergence properties are studied. Some numerical experiments are given to show the efficiency of the new methods.     

Convergence and complexity of arbitrary order adaptive mixed element methods for the Poisson equation

This paper discusses convergence and complexity of arbitrary,but fixed,order adaptive mixed element methods for the Poisson equation in two and three dimensions.The two main ingredients in the analysis,namely the quasi-orthogonality and the discrete reliability,are achieved by use of a discrete Helmholtz decomposition and a discrete inf-sup condition.The adaptive algorithms are shown to be contractive for the sum of the error of flux in L2-norm and the scaled error estimator after each step of mesh refinement and to be quasi-optimal with respect to the number of elements of underlying partitions.The methods do not require a separate treatment for the data oscillation.     

求解$P_*(\kappa)$阵线性互补问题的宽邻域Dikin型高阶内点算法

Based on the idea of Dikin-type primal-dual affine scaling method for linear programming, we describe a high-order Dikin-type algorithm for P. （κ）-matrix linear complementarity problem in a wide neighborhood of the central path, and its polynomial-time complexity bound is given. Finally, two numerical experiments are provided to show the effectiveness of the proposed algorithms.     

Predictor-Corrector Smoothing Methods for Monotone LCP

In this paper, we analyze the global and local convergence properties of two predictor-corrector smoothing methods, which are based on the framework of the method in [1], for monotone linear complementarity problems (LCPs). The difference between the algorithm in [1] and our algorithms is that the neighborhood of smoothing central path in our paper is different to that in [1]. In addition, the difference between Algorithm 2.1 and the algorithm in [1] exists in the calculation of the predictor step. Comparing with the results in [1],the global and local convergence of the two methods can be obtained under very mild conditions. The global convergence of the two methods do not need the boundness of the inverse of the Jacobian. The superlinear convergence of Algorithm 2.1‘ is obtained under the assumption of nonsingularity of generalized Jacobian of Φ(x,y) at the limit point and Algorithm 2.1 obtains superlinear convergence under the assumption of strict complementarity at the solution. The efficiency of the two methods is tested by numerical experiments.     

Estimates on the amplitude of the first Dirichlet eigenvector in discrete frameworks

Consider a finite absorbing Markov generator, irreducible on the non-absorbing states. PerronFrobenius theory ensures the existence of a corresponding positive eigenvector ψ. The goal of the paper is to give bounds on the amplitude max ψ/ min ψ. Two approaches are proposed: One using a path method and the other one, restricted to the reversible situation, based on spectral estimates. The latter approach is extended to denumerable birth and death processes absorbing at 0 for which infinity is an entrance boundary. The interest of estimating the ratio is the reduction of the quantitative study of convergence to quasi-stationarity to the convergence to equilibrium of related ergodic processes, as seen by Diaconis and Miclo(2014).     

CURVILINEAR PATHS AND TRUST REGION METHODS WITH NONMONOTONIC BACK TRACKING TECHNIQUE FOR UNCONSTRAINED OPTIMIZATION

1. Illtroductioncrust region method is a well-accepted technique in nonlinear optindzation to assure globalconvergence. One of the adVantages of the model is that it does not require the objectivefunction to be convex. Many differellt versions have been suggested in using trust regiontechnique. For each iteration, suppose a current iterate point, a local quadratic model of thefunction and a trust region with center at the point and a certain radius are given. A point thatminimizes the model f…     

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.     

Differential gradient methods

A class of recently developed differential descent methods for function minimization is presented and discussed, and a number of algorithms are derived which minimize a quadratic function in a finite number of steps and rapidly minimize general functions. The main characteristics of our algorithms are that a more general curvilinear search path is used instead of a ray and that the eigensystem of the Hessian matrix is associated with the function minimization problem. The curvilinear search paths are obtained by solving certain initial-value systems of differential equations, which also suggest the development of modifications of known numerical integration techniques for use in function minimization. Results obtained on testing the algorithms on a number of test functions are also given and possible areas for future research indicated.     

On the convergence of curvilinear search algorithms in unconstrained optimization

The purpose of this paper is to unify the conditions under which curvilinear algorithms for unconstrained optimization converge. Particularly, two gradient path approximation algorithms and a trust region curvilinear algorithm are examined in this context.     

A new type of condensation curvilinear path algorithm for unconstrained generalized geometric programming

A new type of condensation curvilinear path algorithm is proposed for unconstrained generalized geometric programming (GGP). First, a new type of condensation problem is presented based on the special structure of GGP. Then a particular curvilinear path for the problem is constructed, along which we get the approximate solution of the problem within a trust region. It is proved that the method is globally convergent and that the convergence rate is quadratic. Numerical experiments are given to show the effectiveness and feasibility of the algorithm.     

The Gauss-Newton Methods via Conjugate Gradient Path without Line Search Technique for Solving Nonlinear Systems

In this article, we propose the Gauss-Newton methods via conjugate gradient path for solving nonlinear systems. By constructing and solving a linearized model of the nonlinear systems, we obtain the iterative direction by employing the conjugate gradient path. In successive iterations, the approximate Jacobian of the nonlinear systems is updated by a Broyden formula to construct the conjugate path. The global convergence and local superlinear convergence rate of the proposed algorithms are established under some reasonable conditions. Finally, the numerical results are reported to show the effectiveness of the proposed algorithms.     

线性不等式约束优化的弧线路径信赖域算法

提供了弧线路径结合仿射内点信赖域策略的非单调回代算法解线性不等式约束的优化问题.基于仿射投影的信赖域子问题获得新的搜索方向,采用弧线路径的近似信赖域和线搜索结合技术得到回代步,获得新的步长.通过证明所提供的弧线路径具有一系列良好性质,从而在合理的条件下,证明所提供的算法不仅具有整体收敛性,而且保持算法的局部超线性收敛速率.数值测试表明了算法的有效性与可靠性.     

A quadratically-convergent fixed-point algorithm for economic equilibria and linearly constrained optimization

Fixed-point algorithms for computing equilibria in economies with production or stationary points in constrained optimization generally use point-to-set mappings and therefore converge slowly. An alternative implementation uses continuous functions with a higher dimensionality corresponding to the inclusion of activity levels or dual variables. Here we develop algorithms that only increase the dimensionality implicitly. The solution path is piecewise-linear as in other algorithms. However, when viewed in the low-dimensional space, the path within each simplex can be piecewise-linear rather than linear. Asymptotically, these paths are linear and quadratic convergence is attained.This research was partially supported by NSF Grant ENG76-08749.     

Simplicial zero-point algorithms: A unifying description

In this paper, we consider the limiting paths of simplicial algorithms for finding a zero point. By rewriting the zero-point problem as a problem of finding a stationary point, the problem can be solved by generating a path of stationary points of the function restricted to an expanding convex, compact set. The limiting path of a simplicial algorithm to find a zero point is obtained by choosing this set in an appropriate way. Almost all simplicial algorithms fit in this framework. Using this framework, it can be shown very easily that Merrill's condition is sufficient for convergence of the algorithms.