线性不等式组的简单对偶非线性方法

将线性不等式组问题转化为一个形式简单的对偶空间非线性极值问题,本提出了一类新的求解线性不等式组的方法-简单对偶非线性方法,它在理论上是多项式算法,并可以从任意点启动,可以应用共轭梯度方法有效地求解大规模线性不等式组问题。本给出了不同的算法实现,数值实验结果表明,简单对偶非线性方法是有效的。     

求解线性不等式组的方法

本提出了一个新的求解线性不等式组可行解的方法－－无约束极值方法。通过在线性不等式组的非空可行域的相对内域上建立一个非线性极值问题，根据对偶关系，得到了一个对偶空间的无约束极值及原始，对偶变量之间的简单线性映射关系，这样将原来线性不等式组问题的求解转化为一个无约束极值问题。中主要讨论了求解无约束极值问题的共轭梯度算法。同时，在寻找不等式组可行解的过程中，定义了穿越方向，这样大大减少计算量。中最后数值实验结果表明此算法是有效的。     

非线性目标函数问题的求解策略

线性规划问题是不等式的一项重要应用之一,其考查目的是利用不等式的几何意义求与不等式相关的最值问题.根据目标函数的不同可以分为线性目标函数及非线性目标函数,以下介绍常见的非线性目标函数问题的求解策略.     

一类非线性积分不等式组中未知函数的估计

研究了一类二维非线性积分不等式组,该不等式组积分号外有非常数因子,不能用向量形式的Gronwall-Bellman型积分不等式进行估计.先利用Bernoulli不等式把非线性问题转化成线性问题,利用变量替换技巧和放大技巧研究只含有一个未知函数的积分不等式,接着利用两个引理和变量替换技巧和放大技巧给出不等式组中两个未知函数的估计.结果可用于研究积分、微分动力系统解的性质.     

基于动力系统的线性不等式组的解法

本文提出了一种新的求解线性不等式组可行解的方法-动力系统方法.假设线性不等式组的可行域为非空,在可行域的相对内域上建立一个非线性极值问题,根据对偶关系,得到一个对偶空间的无约束极值问题以及原始、对偶变量之间的简单线性映射关系,进而得到了一个结构简单的动力系统模型.文中主要讨论了动力系统的隐式格式,通过证明模型具有较好的计算稳定性.同时,在寻找不等式组可行解的过程中,定义了穿越方向,这样可以减少计算量.数值实验结果表明此算法是有效的.     

不等式

1.本单元重、难点分析本单元的重点:不等关系与不等式的基本性质;一元二次不等式、二元一次不等式(组)的解法及应用;简单的线性规划问题;基本不等式a+b2≥槡ab(a≥0,b≥0).本单元的难点:不等式的基本性质的理解及应用;简单的线性规划问题的求解;基本不等式的灵活应用.     

关于非线性不等式组Levenberg-Marquardt算法的收敛性(英文)

本文研究了一类非线性不等式组的求解问题.利用一列目标函数两次可微的参数优化问题来逼近非线性不等式组的解,光滑Levenberg-Marquardt方法来求解参数优化问题,在一些较弱的条件下证明了文中算法的全局收敛性,数值实例显示文中算法效果较好.     

一个求线性代数方程组非负解的算法及其在线性规划中的应用

1.引言关于线性规划的多项式算法,哈奇扬于1979年首先把一个线性规划问题化成一个线性不等式组的求解问题,然后用椭球方法求解线性不等式组,并证明是多项式时间可解的。Karmarkar于1984年也给出了一个求解线性规划的多项式时间解法,他     

不等式

1．本单元重、难点分析 本单元的重点：不等关系,不等式的基本性质（不等式变形的重要依据）;一元二次不等式、二元一次不等式（组）的解法及应用;简单的线性规划问题。     

从2012年高考题看非线性规划问题的解法

现行教科书中的线性规划问题,其中的不等式、方程都是一次的,图象都是直线,故线性规划问题即是线性方程、不等式问题,可用直线的几何性质解答,即数形结合法中的"以形代数"法(教科书都用"以形代数")解答,也可不用图象而用纯代数方法解答(教科书却不讲).当方程、不等式不全是一次时,严格说不应称作线性规划问题,但解法原理与线性     

Convergence and polynomiality of primal-dual interior-point algorithms for linear programming with selective addition of inequalities

This paper presents the convergence proof and complexity analysis of an interior-point framework that solves linear programming problems by dynamically selecting and adding relevant inequalities. First, we formulate a new primal–dual interior-point algorithm for solving linear programmes in non-standard form with equality and inequality constraints. The algorithm uses a primal–dual path-following predictor–corrector short-step interior-point method that starts with a reduced problem without any inequalities and selectively adds a given inequality only if it becomes active on the way to optimality. Second, we prove convergence of this algorithm to an optimal solution at which all inequalities are satisfied regardless of whether they have been added by the algorithm or not. We thus provide a theoretical foundation for similar schemes already used in practice. We also establish conditions under which the complexity of such algorithm is polynomial in the problem dimension and address remaining limitations without these conditions for possible further research.     

Analyzing the potential of a firm: an operations research approach

An approach to analyzing the potential of a firm, which is understood as the firm's ability to provide goods or (and) services to be supplied to a marketplace under restrictions imposed by a business environment in which the firm functions, is proposed. The approach is based on using linear inequalities and, generally, mixed variables in modelling this ability for a broad spectrum of industrial, transportation, agricultural, and other types of firms and allows one to formulate problems of analyzing the potential of a firm as linear programming problems or mixed programming problems with linear constraints. This approach generalizes the one proposed by the author earlier for a more narrow class of models and allows one to effectively employ a widely available software for solving practical problems of the considered kind, especially for firms described by large scale models of mathematical programming.     

General criteria for redundant and nonredundant linear inequalities

A system of linear inequalities subject to nonnegativity restrictions is considered. General criteria which are necessary and sufficient for a linear inequality to be redundant are derived. This general characterization provides a basis for unifying some of the existing techniques. After taking into consideration the existence of redundant linear inequalities, general necessary and sufficient criteria for a linear inequality to be nonredundant are also obtained. An example is given to illustrate the application of these new criteria.The author wishes to thank the referee for his comments.     

Finite convergence of algorithms for nonlinear programs and variational inequalities

Algorithms for nonlinear programming and variational inequality problems are, in general, only guaranteed to converge in the limit to a Karush-Kuhn-Tucker point, in the case of nonlinear programs, or to a solution in the case of variational inequalities. In this paper, we derive sufficient conditions for nonlinear programs with convex feasible sets such that any convergent algorithm can be modified, by adding a convex subproblem with a linear objective function, to guarantee finite convergence in a generalized sense. When the feasible set is polyhedral, the subproblem is a linear program and finite convergence is obtained. Similar results are also developed for variational inequalities.The research of the first author was supported in part by the Office of Naval Research under Contract No. N00014-86-K-0173.The authors are indebted to Professors Olvi Mangasarian, Garth McCormick, Jong-Shi Pang, Hanif Sherali, and Hoang Tuy for helpful comments and suggestions and to two anonymous referees for constructive remarks and for bringing to their attention the results in Refs. 13 and 14.     

Robust Stability and Stabilization of a Class of Nonlinear Itô-Type Stochastic Systems via Linear Matrix Inequalities

This article presents a new approach to robust quadratic stabilization of nonlinear stochastic systems. The linear rate vector of a stochastic system is perturbed by a nonlinear function, and this nonlinear function satisfies a quadratic constraint. Our objective is to show how linear constant feedback laws can be formulated to stabilize this type of stochastic systems and, at the same time maximize the bounds on this nonlinear perturbing function which the system can tolerate without becoming unstable. The new formulation provides a suitable setting for robust stabilization of nonlinear stochastic systems where the underlying deterministic systems satisfy the generalized matching conditions. Our sufficient conditions are written in matrix forms, which are determined by solving linear matrix inequalities (LMIs), which have significant computational advantage over any other existing techniques. Examples are given to demonstrate the results.     

Multiprocessor scheduling under precedence constraints: Polyhedral results

We consider the problem of scheduling a set of tasks related by precedence constraints to a set of processors, so as to minimize their makespan. Each task has to be assigned to a unique processor and no preemption is allowed. A new integer programming formulation of the problem is given and strong valid inequalities are derived. A subset of the inequalities in this formulation has a strong combinatorial structure, which we use to define the polytope of partitions into linear orders. The facial structure of this polytope is investigated and facet defining inequalities are presented which may be helpful to tighten the integer programming formulation of other variants of multiprocessor scheduling problems. Numerical results on real-life problems are presented.     

Proximal-Point Algorithm Using a Linear Proximal Term

Proximal-point algorithms (PPAs) are classical solvers for convex optimization problems and monotone variational inequalities (VIs). The proximal term in existing PPAs usually is the gradient of a certain function. This paper presents a class of PPA-based methods for monotone VIs. For a given current point, a proximal point is obtained via solving a PPA-like subproblem whose proximal term is linear but may not be the gradient of any functions. The new iterate is updated via an additional slight calculation. Global convergence of the method is proved under the same mild assumptions as the original PPA. Finally, profiting from the less restrictions on the linear proximal terms, we propose some parallel splitting augmented Lagrangian methods for structured variational inequalities with separable operators. B.S. He was supported by NSFC Grant 10571083 and Jiangsu NSF Grant BK2008255.     

Solution differentiability for variational inequalities

In this paper we study solution differentiability properties for variational inequalities. We characterize Fréchet differentiability of perturbed solutions to parametric variational inequality problems defined on polyhedral sets. Our result extends the recent result of Pang and it directly specializes to nonlinear complementarity problems, variational inequality problems defined on perturbed sets and to nonlinear programming problems.     

解线性不等式的神经网络

本文提出两个解线性不等式的Hopfiedl-Tank型的神经网络。第一个网络模拟同时松弛投影方法，第二个网络是二次规划方法。当线性不等式的解集非空时，这两个方法都给出该线性不等式的解。同时我们还给出了这两个网络的数值模拟。