Iterative Gradient Descent Methods for Solving Linear Equations

Computational Mathematics and Mathematical Physics - The paper presents the results on the use of gradient descent algorithms for constructing iterative methods for solving linear equations. A...     

On the Solution of Linear Programs by Jacobian Smoothing Methods

We introduce a class of algorithms for the solution of linear programs. This class is motivated by some recent methods suggested for the solution of complementarity problems. It reformulates the optimality conditions of a linear program as a nonlinear system of equations and applies a Newton-type method to this system of equations. We investigate the global and local convergence properties and present some numerical results. The algorithms introduced here are somewhat related to the class of primal–dual interior-point methods. Although, at this stage of our research, the theoretical results and the numerical performance of our method are not as good as for interior-point methods, our approach seems to have some advantages which will also be discussed in detail.     

On the Equivalence of the Simplex Methods and a Multiplier-Alike Method for Linear Programming

In linear programming, the simplex method has been viewed for a long time as an efficient tool. Interior methods have attracted a lot of attention since they were proposed recently. It seems plausible intuitively that there is no reason why a good linear programming algorithm should not be allowed to cross the boundary of the feasible region when necessary. However, such an algorithm is seldom studied. In this paper, we will develop first a framework of a multiplier-alike algorithm for linear programming which allows its trajectory to move across the boundary of the feasible region. Second, we illustrate that such a framework has the potential to perform as well as the simplex method by showing that these methods are equivalent in a well-defined sense, even though they look so different.     

Interior-Point Methods with Decomposition for Solving Large-Scale Linear Programs

This paper deals with an algorithm incorporating the interior-point method into the Dantzig–Wolfe decomposition technique for solving large-scale linear programming problems. The algorithm decomposes a linear program into a main problem and a subproblem. The subproblem is solved approximately. Hence, inexact Newton directions are used in solving the main problem. We show that the algorithm is globally linearly convergent and has polynomial-time complexity.     

Relaxation Methods for Strictly Convex Regularizations of Piecewise Linear Programs

We give an algorithm for minimizing the sum of a strictly convex function and a convex piecewise linear function. It extends several dual coordinate ascent methods for large-scale linearly constrained problems that occur in entropy maximization, quadratic programming, and network flows. In particular, it may solve exact penalty versions of such (possibly inconsistent) problems, and subproblems of bundle methods for nondifferentiable optimization. It is simple, can exploit sparsity, and in certain cases is highly parallelizable. Its global convergence is established in the recent framework of B -functions (generalized Bregman functions). Accepted 29 October 1996     

On Distributionally Robust Chance-Constrained Linear Programs

In this paper, we discuss linear programs in which the data that specify the constraints are subject to random uncertainty. A usual approach in this setting is to enforce the constraints up to a given level of probability. We show that, for a wide class of probability distributions (namely, radial distributions) on the data, the probability constraints can be converted explicitly into convex second-order cone constraints; hence, the probability-constrained linear program can be solved exactly with great efficiency. Next, we analyze the situation where the probability distribution of the data is not completely specified, but is only known to belong to a given class of distributions. In this case, we provide explicit convex conditions that guarantee the satisfaction of the probability constraints for any possible distribution belonging to the given class.Communicated by B. T. PolyakThis work was supported by FIRB funds from the Italian Ministry of University and Research.     

关于求解带线性互补约束的数学规划问题正则方法的一个注记

本文主要研究在某些较弱条件下求解带线性互补约束的数学规划问题(MPLCC)正则方法的收敛性.若衡约束规划线性独立约束规范条件(MPEC-LICQ)在由正则方法产生的点列的聚点处成立,且迭代点列满足二阶必要条件,同时,若比在文[7]中渐近弱非退化条件Ⅰ更弱的渐近弱非退化条件Ⅱ在聚点处也成立,那么所有这些聚点都是B-稳定点.此外,在弱MPEC-LICQ,二阶必要条件及渐近弱退化条件Ⅱ下,我们仍能证明通过正则方法所得的聚点都是B-稳定点.     

Primal-Dual Simplex Method for Multiobjective Linear Programming

We develop a primal-dual simplex algorithm for multicriteria linear programming. It is based on the scalarization theorem of Pareto optimal solutions of multicriteria linear programs and the single objective primal-dual simplex algorithm. We illustrate the algorithm by an example, present some numerical results, give some further details on special cases and point out future research. The paper was written during a visit of the first author to the University of Sevilla financed by a grant of the Andalusian Consejería de Educación. The research of the first author was partially supported by University of Auckland Grant 3602178/9275. The research of the second and third authors was partially financed by Spanish Grants BFM2001-2378, BFM2001-4028, MTM2004-0909 and HA2003-0121. We thank Anthony Przybylski for the implementation and making his results available. We thank the anonymous referees, whose comments have helped us to improve the presentation of the paper.     

Problem Specification for Linear Programs

A language for specifying linear programs is proposed. The specificationlanguage is designed so as to enable the user to define theinput for a particular linear program in terms of a given tabulardatabase consisting of multidimensional tables.     

线性规划单纯形法的三种实现形式探析

按照一般寻优算法原则,在定义可行方向和步长的基础上,从线性规划问题系数矩阵的列向量子空间出发,说明了单纯形法的顶点寻优过程是一个在约束条件的仿射空间和系数矩阵的零子空间交错前进的过程,并在此基础上归纳和总结了数据字典式单纯形表、经典单纯形表和简化单纯形表的实现形式及其迭代计算的特点和优势,并建议未来在《运筹学》教学中广泛推广这三种类型的单纯形表.     

On the Minimum Norm Solution of Linear Programs

This paper describes a new technique to find the minimum norm solution of a linear program. The main idea is to reformulate this problem as an unconstrained minimization problem with a convex and smooth objective function. The minimization of this objective function can be carried out by a Newton-type method which is shown to be globally convergent. Furthermore, under certain assumptions, this Newton-type method converges in a finite number of iterations to the minimum norm solution of the underlying linear program.     

Solving Stochastic Linear Programs with Restricted Recourse Using Interior Point Methods

In this paper we present a specialized matrix factorization procedure for computing the dual step in a primal-dual path-following interior point algorithm for solving two-stage stochastic linear programs with restricted recourse. The algorithm, based on the Birge-Qi factorization technique, takes advantage of both the dual block-angular structure of the constraint matrix and of the special structure of the second-stage matrices involved in the model. Extensive computational experiments on a set of test problems have been conducted in order to evaluate the performance of the developed code. The results are very promising, showing that the code is competitive with state-of-the-art optimizers.     

A Modified Bisection Simplex Method for Linear Programming

1.IntroductionThebisectionmethod,proposedinanearlierpaperbytheauth..[7]texploitsinformationoftheoptimalvaluetospeedupthesolutionprocess-However,themethodrequiresabracketontheoptimalvalueaspartofitsinput,anditspromisinglygoodperformancedependsonwhetherasuitablebracketisavailable;itmayevenfaltosolveaproblemiftheinitialintervalprovideddoesnotcontaintheoptimalvalueactuallyInthispaper,themethodismodifiedformoregeneralpurposeuse-Organizedinanalternativesimplerform,thenewversionnolongerneed8anybrathe…     

模糊线性规划问题的一种新的单纯形算法

提出求解模糊线性规划问题的一种新的思路 ,就是应用单纯形法先求解与 (FLP)相应的普通线性规划问题 ,通过模糊约束集与模糊目标集的隶属度的比较 ,获得两个集合交集的最优隶属度 ,将此最优隶属度代入最优单纯形表中 ,即可求得 (FLP)的解。本算法只需在一张适当的迭代表台上执行单纯形迭代过程 ,简捷方便适用     

Predictor-Corrector Smoothing Methods for Linear Programs with a More Flexible Update of the Smoothing Parameter

We consider a smoothing-type method for the solution of linear programs. Its main idea is to reformulate the corresponding central path conditions as a nonlinear system of equations, to which a variant of Newton's method is applied. The method is shown to be globally and locally quadratically convergent under suitable assumptions. In contrast to a number of recently proposed smoothing-type methods, the current work allows a more flexible updating of the smoothing parameter. Furthermore, compared with previous smoothing-type methods, the current implementation of the new method gives significantly better numerical results on the netlib test suite.     

Interval Methods for Semi-Infinite Programs

A new approach for the numerical solution of smooth, nonlinear semi-infinite programs whose feasible set contains a nonempty interior is presented. Interval analysis methods are used to construct finite nonlinear, or mixed-integer nonlinear, reformulations of the original semi-infinite program under relatively mild assumptions on the problem structure. In certain cases the finite reformulation is exact and can be solved directly for the global minimum of the semi-infinite program (SIP). In the general case, this reformulation is over-constrained relative to the SIP, such that solving it yields a guaranteed feasible upper bound to the SIP solution. This upper bound can then be refined using a subdivision procedure which is shown to converge to the true SIP solution with finite -optimality. In particular, the method is shown to converge for SIPs which do not satisfy regularity assumptions required by reduction-based methods, and for which certain points in the feasible set are subject to an infinite number of active constraints. Numerical results are presented for a number of problems in the SIP literature. The solutions obtained are compared to those identified by reduction-based methods, the relative performances of the nonlinear and mixed-integer nonlinear formulations are studied, and the use of different inclusion functions in the finite reformulation is investigated.     

Language Constructs for Modeling Stochastic Linear Programs

This paper introduces two tokens of syntax for algebraic modeling languages that are sufficient to model a wide variety of stochastic optimization problems. The tokens are used to declare random parameters and to define a precedence relationship between different random events. It is necessary to make a number of assumptions in order to use this syntax, and it is through these assumptions, which cover the areas of model structure, time, replication, and stages, that we can attain a deeper understanding of this class of problem.     

一类全局收敛的记忆梯度法及其线性收敛性

本文研究一类新的解无约束最优化问题的记忆梯度法,在强Wolfe线性搜索下证明了其全局收敛性．当目标函数为一致凸函数时,对其线性收敛速率进行了分析．数值试验表明算法是很有效的．     

线性规划对偶单纯形法的常识性经济解释

线性规划对偶单纯形法的数学描述是抽象且不易理解的 ,本文找到了一种常识性经济解释的描述 .既通俗易懂、又能抓住方法的实质 .这将从通俗易懂的思想方法上促进线性规划问题解法的普及推广应用