Interior-point methods for linear programming: a review

The paper reviews some recent advances in interior-point methods for linear programming and indicates directions in which future progress can be made. Most of the interior-point methods belong to any of three categories: affine-scaling methods, potential reduction methods and central path methods. These methods are discussed together with infeasible interior methods and homogeneous self-dual methods for linear programming. Also discussed are some theoretical issues in interior-point methods like dependence of complexity bounds on some non-traditional measures different from the input length L of the problem. Finally, the paper concludes with remarks on the comparison of interior-point methods with the simplex method based on their performance on NITLIB suite, a standard collection of test problems.     

QUADRATIC INVARIANTS AND SYMPLECTIC STRUCTURE OF GENERAL LINEAR METHODS

1. IntroductionInvestigating whether a numerical method inherits some dynamical properties possessed bythe differential equation systems being integrated is an important field of numerical analysisand has received much attention in recent years See the review articlesof Sanz-Serna[9] and Section 11.16 of Hairer et. al.[2] for more detail concerning the symplectic methods. Most of the work on canonical iotegrators has dealt with one-step formulaesuch as Runge-Kutta methods and Runge'methods …     

HIGH-ACCURACY EXPLICIT TWO-STEP METHODS WITH MINIMAL PHASE-LAG FOR y″= f（t,y）

In this paper, two families of high accuracy explicit two-step methods with minimal phase-lag are developed for the numerical integration of special second-order periodic initial-value problems. In comparison with some methods in [1-4,6], the advantage of these methods has a higher accuracy and minimal phase-lag. The methods proposed in this paper can be considered as a generalization of some methods in [1,3,4]. Numerical examples indicate that these new methods are generally more accurate than the methods used in [3,6]. second order periodic initial-value problems, phase-lag, local truncation error     

两类解非线性约束问题的超记忆可行方向法

本文将无约束超记忆梯度法推广到非线性不等式约束优化问题上来,给出了两类形式很一般的超记忆可行方向法,并在非退化及连续可微等较弱的假设下证明了其全局收敛性.适当选取算法中的参量及记忆方向,不仅可得到一些已知的方法及新方法,而且还可能加快算法的收敛速度.     

On interval predictor-corrector methods

One can approximate numerically the solution of the initial value problem using single or multistep methods. Linear multistep methods are used very often, especially combinations of explicit and implicit methods. In floating-point arithmetic from an explicit method (a predictor), we can get the first approximation to the solution obtained from an implicit method (a corrector). We can do the same with interval multistep methods. Realizing such interval methods in floating-point interval arithmetic, we compute solutions in the form of intervals which contain all possible errors. In this paper, we propose interval predictor-corrector methods based on conventional Adams-Bashforth-Moulton and Nyström-Milne-Simpson methods. In numerical examples, these methods are compared with interval methods of Runge-Kutta type and methods based on high-order Taylor series. It appears that the presented methods yield comparable approximations to the solutions.     

变分不等式的几类求解方法

本文转为系统地分析和概述了变分不等式问题中几类占有重要地位的求解方法,包括方法产生的背景,主要结果及应用等,这几类算法分别为连续算法,（拟）牛顿型算法,一般迭代模型,投影算法,投影收缩算法等。     

A REGULARIZED CONJUGATE GRADIENT METHOD FOR SYMMETRIC POSITIVE DEFINITE SYSTEM OF LINEAR EQUATIONS

AbstractA class of regularized conjugate gradient methods is presented for solving the large sparse system of linear equations of which the coefficient matrix is an ill-conditioned symmetric positive definite matrix. The convergence properties of these methods are discussed in depth, and the best possible choices of the parameters involved in the new methods are investigated in detail. Numerical computations show that the new methods are more efficient and robust than both classical relaxation methods and classical conjugate direction methods.     

互补问题算法的新进展

互补问题是一类重要的优化问题,在最近３０多年的时间里,人们为求解它而提出了许多算法,该文主要介绍１９９０－１９９７年之间出现的某些新算法,它们大致可归类为：（１）光滑方程法;（２）非光滑方程法;（３）可微无约束优化法;（４）ＧＬＰ投影法;（５）内点法;（６）磨光与非内点连续法,文中对每类算法及相应的收敛性结果做了描述与评论,并列出有关文献。     

Computation of Carlson's Multiple Hypergeometric Function R for Bayesian Applications

Abstract

Carlson's multiple hypergeometric functions arise in Bayesian inference, including methods for multinomial data with missing category distinctions and for local smoothing of histograms. To use these methods one needs to calculate Carlson functions and their ratios. We discuss properties of the functions and explore computational methods for them, including closed form methods, expansion methods, Laplace approximations, and Monte Carlo methods. Examples are given to illustrate and compare methods.     

高阶优化算法分析简介

高阶优化算法是利用目标函数的高阶导数信息进行优化的算法,是最优化领域中的一个新兴的研究方向.高阶算法具有更低的迭代复杂度,但是需要求解一个更难的子问题.主要介绍三种高阶算法,分别为求解凸问题的高阶加速张量算法和A-HPE框架下的最优张量算法,以及求解非凸问题的ARp算法.同时也介绍了怎样求解高阶算法的子问题.希望通过对高阶算法的介绍,引起更多学者的关注与重视.     

A class of one-parameter alternating direction implicit methods for two-dimensional wave equations with discrete and distributed time-variable delays

For solving the initial-boundary value problem of two-dimensional wave equations with discrete and distributed time-variable delays, in the present paper, we first construct a class of basic one-parameter methods. In order to raise the computational efficiency of this class methods, we remold the methods as one-parameter alternating direction implicit (ADI) methods. Under the suitable conditions, the remolded methods are proved to be stable and convergent of second order in both of time and space. With several numerical experiments, the computational effectiveness and theoretical exactness of the methods are confirmed. Moreover, it is illustrated that the proposed one-parameter ADI method has the better advantage in computational efficiency than the basic one-parameter methods.     

Variance reduction in sample approximations of stochastic programs

This paper studies the use of randomized Quasi-Monte Carlo methods (RQMC) in sample approximations of stochastic programs. In numerical integration, RQMC methods often substantially reduce the variance of sample approximations compared to Monte Carlo (MC). It seems thus natural to use RQMC methods in sample approximations of stochastic programs. It is shown, that RQMC methods produce epi-convergent approximations of the original problem. RQMC and MC methods are compared numerically in five different portfolio management models. In the tests, RQMC methods outperform MC sampling substantially reducing the sample variance and bias of optimal values in all the considered problems.     

Feasible Interior Methods Using Slacks for Nonlinear Optimization

A slack-based feasible interior point method is described which can be derived as a modification of infeasible methods. The modification is minor for most line search methods, but trust region methods require special attention. It is shown how the Cauchy point, which is often computed in trust region methods, must be modified so that the feasible method is effective for problems containing both equality and inequality constraints. The relationship between slack-based methods and traditional feasible methods is discussed. Numerical results using the KNITRO package show the relative performance of feasible versus infeasible interior point methods.     

New higher-order methods for the simultaneous inclusion of polynomial zeros

Higher-order methods for the simultaneous inclusion of complex zeros of algebraic polynomials are presented in parallel (total-step) and serial (single-step) versions. If the multiplicities of each zeros are given in advance, the proposed methods can be extended for multiple zeros using appropriate corrections. These methods are constructed on the basis of the zero-relation of Gargantini’s type, the inclusion isotonicity property and suitable corrections that appear in two-point methods of the fourth order for solving nonlinear equations. It is proved that the order of convergence of the proposed methods is at least six. The computational efficiency of the new methods is very high since the acceleration of convergence order from 3 (basic methods) to 6 (new methods) is attained using only n polynomial evaluations per iteration. Computational efficiency of the considered methods is studied in detail and two numerical examples are given to demonstrate the convergence behavior of the proposed methods.     

Two-Stage Stochastic Runge-Kutta Methods for Stochastic Differential Equations

In this paper we discuss two-stage diagonally implicit stochastic Runge-Kutta methods with strong order 1.0 for strong solutions of Stratonovich stochastic differential equations. Five stochastic Runge-Kutta methods are presented in this paper. They are an explicit method with a large MS-stability region, a semi-implicit method with minimum principal error coefficients, a semi-implicit method with a large MS-stability region, an implicit method with minimum principal error coefficients and another implicit method. We also consider composite stochastic Runge-Kutta methods which are the combination of semi-implicit Runge-Kutta methods and implicit Runge-Kutta methods. Two composite methods are presented in this paper. Numerical results are reported to compare the convergence properties and stability properties of these stochastic Runge-Kutta methods.     

Quantitative parametric connections between methods for generating noninferior solutions in multiobjective optimization

In multiobjective optimization, the trade-off information between different objective functions is probably the most important piece of information in a solution process to reach the most preferred solution. Generating methods are the methods in generating noninferior solutions. Different generating methods provide decision makers trade-off information in different forms. Various generating methods are characterized in this paper and the quantitative parametric connections between these generating methods are established. The result in this paper is then used to consolidate trade-off information with different formats associated with different generating methods.     

Comparison of Two Kinds of Prediction-Correction Methods for Monotone Variational Inequalities

In this paper, we study the relationship between the forward-backward splitting method and the extra-gradient method for monotone variational inequalities. Both of the methods can be viewed as prediction-correction methods. The only difference is that they use different search directions in the correction-step. Our analysis explains theoretically why the extra-gradient methods usually outperform the forward-backward splitting methods. We suggest some modifications for the two methods and numerical results are given to verify the superiority of the modified methods.     

MULTISTEP DISCRETIZATION OF INDEX 3 DAES

1. IntroductionIn this paper, we will consider the multistep discrezations of the differential--algebraicequations (DAEs) in Hessenberg formwhere F e AN M M R", K E AN M L - AM, G E AN - RL, the initial value(yo, ic, no) at xo are assumed to be consistent, i.e., they satisfyWe supposes, F, G and K are sufficiently differentiable, and thatin a neighbourhood of the solution. Such problems often appear in the simulation ofmechanical systems with constraints and the singularly perturbed…     

Fully discrete dynamic mesh discontinuous Galerkin methods for the Cahn-Hilliard equation of phase transition

Fully discrete discontinuous Galerkin methods with variable mesh- es in time are developed for the fourth order Cahn-Hilliard equation arising from phase transition in materials science. The methods are formulated and analyzed in both two and three dimensions, and are proved to give optimal order error bounds. This coupled with the flexibility of the methods demonstrates that the proposed discontinuous Galerkin methods indeed provide an efficient and viable alternative to the mixed finite element methods and nonconforming (plate) finite element methods for solving fourth order partial differential equations.

     

A class of three-point root-solvers of optimal order of convergence

The construction of a class of three-point methods for solving nonlinear equations of the eighth order is presented. These methods are developed by combining fourth order methods from the class of optimal two-point methods and a modified Newton’s method in the third step, obtained by a suitable approximation of the first derivative based on interpolation by a nonlinear fraction. It is proved that the new three-step methods reach the eighth order of convergence using only four function evaluations, which supports the Kung-Traub conjecture on the optimal order of convergence. Numerical examples for the selected special cases of two-step methods are given to demonstrate very fast convergence and a high computational efficiency of the proposed multipoint methods. Some computational aspects and the comparison with existing methods are also included.