区间二次双层规划的最好最优解

双层规划问题是一类具有递阶结构的优化问题.在不确定的双层规划优化问题中,目标函数系数或约束条件系数为区间数的双层规划模型在实际问题中有着广泛的应用.在二次-线性双层规划模型的基础上,提出了上、下层目标函数以及约束条件系数均具有区间系数的二次-线性双层规划模型,给出了求解其最好最优解的方法.首先,通过选取约束条件中不同的基矩阵,求得区间二次-线性双层规划的可能最优解.再比较求得的全部可能最优解,便可得到区间二次-线性双层规划模型的最好最优解.最后给出数值算例验证该方法的有效性.     

全系数模糊的广义模糊变量线性规划

构建并讨论了一类约束条件与目标函数均含模糊系数的广义模糊变量线性规划问题.首先,简单介绍了模糊结构元理论并基于模糊结构元理论定义了表征模糊数面积信息的散度指标.其次,兼顾距离与面积信息,给出了三角模糊数与拟三角模糊数比较排序的新方法,将全系数模糊的广义模糊变量线性规划转化为普通的多目标线性规划.最后,借鉴分层规划的思想,结合模糊数本身性质,给出了此类问题的一种简化求解方法.     

一个有经济意义的多目标规划模型的调整

本文在线性约束条件下 ,同时考虑三个目标函数的最优化 ,即线性函数、二次函数、分式函数 .对于已知的线性规划的最优基可行解 ,通过调整二次函数和分式函数中的系数向量和系数矩阵 ,使其成为这两个规划的最优解 .模型的改进有经济意义的解释     

具有带状系数矩阵的无限维线性规划的解法

研究一类每个约束条件有两个变量且每个变量出现在两个约束条件中的无限维线性规划.引入松弛变量后,得到约束方程组的系数矩阵为无限阶带状矩阵,用它的左逆以及属于零的特征向量可以表示这类问题的最优解.获得目标函数值收敛的一个充分条件.     

线性规划消耗系数矩阵灵敏度分析的某些探讨

讨论了线性规划模型中 ,消耗系数矩阵 A中某个基变量或某个约束方程的系数向量变化以及增减约束方程时 ,对最优基、最优解、目标函数值和影子价格的影响 .     

整数规划中割平面法的研究

割平面法是求解整数规划问题常用方法之一.用割平面法求解整数规划的基本思路是:先用单纯形表格方法去求解不考虑整数约束条件的松弛问题的最优解,如果获得的最优解的值都是整数,即为所求,运算停止.如果所得最优解不完全是整数,即松弛问题最优解中存在某个基变量为非整数值时,就从最优表中提取出关于这个基变量的约束等式,再从这个约束式出发构造一个割平面方程加入最优表中,再求出新的最优解,这样不断重复的构造割平面方程,直到找到整数解为止.主要研究以下四个关键点:一是研究从最优表中提取出的、关于基变量的约束等式出发,通过将式中的系数进行整数和非负真分数的分解,从而得到一个小于等于0的另外一个不等式的推导过程;二是总结出从小于等于0的那个约束不等式出发构造割平面方程的四种方法;三是分析构造割平面方程的这四种方法相互之间的区别和联系;四是探讨割平面法的几何意义.通过对这四个方面的分析和研究,对割平面法进行透彻的剖析,使读者能够全面把握割平面法.     

代入法求解条件极值的一点补遗

以二元函数为例,阐明把约束条件代入目标函数、从而将多元函数的条件极值转化为无条件极值这种常见求解方法的理论依据;并分析该方法本身的缺陷,得出采用方法容易遗漏极值点的结论;并利用隐函数存在定理得出一个附加要求．确定了该方法的适用范围．     

多维参数线性规划的解分割和方向导数

基于最优分割的概念,消费系数多维参数化的线性规划的最优值函数和临界域特征被研究.证明了最优值函数的有效域可分解为有限多个相对内部不交的临界域,这是不同于基于最优基概念所得结论.并给出一个求解任何点处任何方向导数的线性规划方法.     

随机广义纳什均衡问题带有条件风险价值约束的确定性模型及其求解方法

主要考虑随机广义纳什均衡问题(SGNEP),由于随机变量的存在,SGNEP通常无解.对此问题,文章首先给出一阶必要性条件并利用NCP函数得到优化模型的目标函数,为降低所得解的"风险",再利用条件风险价值(CVaR)给出约束条件,从而构造出求解SGNEP的一个低风险模型,并将此模型所得解视为SGNEP的解.然而,直接求解该低风险模型可能会遇到两个问题:一是该模型含有非光滑约束,二是目标函数和约束条件包含期望值.考虑到这两个问题,采用光滑化和罚样本均值近似方法提出该模型的近似问题,并进一步给出近似问题最优解的收敛性结果.最后,文章给出数值算例,以验证所提方法的可行性.     

基于遗传算法和满意度函数的武器系统维修性分配规划问题研究

武器系统维修性分配是一个复杂的多目标规划问题.常规求解方法得到的结果难以满足其约束条件,并且难以体现不同目标间重要性的差别.据此,通过引入遗传算法(GA)和满意度函数解决了上述问题.首先利用GA求解单目标的最优解,建立各目标的满意度函数并综合为一个总满意度函数,最后利用GA求总满意度的最大值,即可获得模型的满意解.仿真算例表明,GA算法可适用于解决武器系统维修性分配问题,同时多目标规划中采用满意度函数法比线性加权法更可行.     

“运价”为负的运输模型计算机求解问题的处理

为解决一些计算机软件求解"运价"既有正值又有负值运输模型时"不可求解"的问题,本文采用"运价同额增减法"决策模型转换的方法,将原模型的"运价"全部转换为正值后再用计算机软件求解,并分别编写了EXCEL求解模板和求解程序对该方法的计算加以印证。结果表明,采用该方法求解得出的最优解(最优决策方案)与原模型求得的最优解完全一样,而最优值(最优决策效果)减去虚增(或加上虚减)的部分就是原模型的最优值。采用这种方法能成功地解决一些计算机软件"不可求解"的问题。     

A single-step HSS method for non-Hermitian positive definite linear systems

In this paper, based on the Hermitian and skew-Hermitian splitting (HSS) iteration method, a single-step HSS (SHSS) iteration method is introduced to solve the non-Hermitian positive definite linear systems. Theoretical analysis shows that, under a loose restriction on the iteration parameter, the SHSS method is convergent to the unique solution of the linear system. Furthermore, we derive an upper bound for the spectral radius of the SHSS iteration matrix, and the quasi-optimal parameter is obtained to minimize the above upper bound. Numerical experiments are reported to the efficiency of the SHSS method; numerical comparisons show that the proposed SHSS method is superior to the HSS method under certain conditions.     

A TWO-GRID METHOD FOR THE C0 INTERIOR PENALTY DISCRETIZATION OF THE MONGE-AMP(E)RE EQUATION

The purpose of this paper is to analyze an efficient method for the solution of the nonlinear system resulting from the discretization of the elliptic Monge-Ampère equation by a $C^0$ interior penalty method with Lagrange finite elements. We consider the two-grid method for nonlinear equations which consists in solving the discrete nonlinear system on a coarse mesh and using that solution as initial guess for one iteration of Newton's method on a finer mesh. Thus both steps are inexpensive. We give quasi-optimal $W^{1,\infty}$ error estimates for the discretization and estimate the difference between the interior penalty solution and the two-grid numerical solution. Numerical experiments confirm the computational efficiency of the approach compared to Newton's method on the fine mesh.     

Discontinuous Galerkin hp-BEM with quasi-uniform meshes

We present and analyze a discontinuous variant of the $hp$ -version of the boundary element Galerkin method with quasi-uniform meshes. The model problem is that of the hypersingular integral operator on an (open or closed) polyhedral surface. We prove a quasi-optimal error estimate and conclude convergence orders which are quasi-optimal for the $h$ -version with arbitrary degree and almost quasi-optimal for the $p$ -version. Numerical results underline the theory.     

Modified parameterized inexact Uzawa method for singular saddle-point problems

As well known, each of the consistent singular saddle-point (CSSP) problems has more than one solutions, and most of the iteration methods can only be proved to converge to one of the solutions of the CSSP problem. However, we do not know which solution it is and whether this solution depends on the initial iteration guesses. In this work, we introduce a new iteration method by slightly modifying the parameterized inexact Uzawa (PIU) iteration scheme. Theoretical analysis shows that, under suitable restrictions on the involved iteration parameters, the iteration sequence produced by the new method converges to the solution \(\mathcal {A}^{\dag }b\) for any initial guess, no matter the singular saddle-point system \(\mathcal {A}~x=b\) is consistent or inconsistent, where \(\mathcal {A}^{\dag }\) denotes the Moore-Penrose inverse of matrix \(\mathcal {A}\). In addition, the quasi-optimal iteration parameters and the corresponding quasi-optimal convergence factor are determined. Numerical examples are given to verify the correctness of the theoretical results and the effectiveness of our new method.     

奇异杂交边界点方法求解泊松方程

As a boundary-type meshless method,the singular hybrid boundary node method(SHBNM)is based on the modified variational principle and the moving least square(MLS)approximation,so it has the advantages of both boundary element method(BEM)and meshless method.In this paper,the dual reciprocity method(DRM)is combined with SHBNM to solve Poisson equation in which the solution is divided into particular solution and general solution.The general solution is achieved by means of SHBNM,and the particular solution is approximated by using the radial basis function(RBF).Only randomly distributed nodes on the bounding surface of the domain are required and it doesn't need extra equations to compute internal parameters in the domain.The postprocess is very simple.Numerical examples for the solution of Poisson equation show that high convergence rates and high accuracy with a small node number are achievable.     

Quadratic finite element approximation of the Signorini problem

Applying high order finite elements to unilateral contact variational inequalities may provide more accurate computed solutions, compared with linear finite elements. Up to now, there was no significant progress in the mathematical study of their performances. The main question is involved with the modeling of the nonpenetration Signorini condition on the discrete solution along the contact region. In this work we describe two nonconforming quadratic finite element approximations of the Poisson-Signorini problem, responding to the crucial practical concern of easy implementation, and we present the numerical analysis of their efficiency. By means of Falk's Lemma we prove optimal and quasi-optimal convergence rates according to the regularity of the exact solution.

     

三维弹塑性结构下限分析的边界元方法

基于极限分析的下限定理,建立了用常规边界元方法进行三维理想弹塑性结构极限分析的求解算法.下限分析所需的弹性应力场可直接由边界元方法求得.所需的自平衡应力场由一组带有待定系数的自平衡应力场基矢量的线性组合进行模拟,这些自平衡应力场基矢量由边界元弹塑性迭代计算得到.下限分析问题最终被归结为一系列未知变量较少的非线性数学规划子问题并通过复合形法进行求解.给出的计算结果表明该算法有较高的精度和计算效率.     

Crew rostering problem in a public transport company

In this paper, we present an applied study commissioned by Metro Bilbao on how to establish a more egalitarian annual allocation of work to drivers. Task allocation is mixed, with some tasks allocated on a rotating basis and others not. The model proposed is solved as a sequence of four types of integer programming problem. The solution obtained is quasi-optimal: all drivers carry out practically the same tasks over the full year. The main contribution of this paper is its method for combining semi-rotating allocation with a planning time frame divided into five periods of three different types with a workload distributed in a non uniform fashion over the days of the week, and with constraints agreed with employees to obtain an egalitarian solution. This method is being implemented at Metro Bilbao, and Eusko Tren has commissioned a study into a similar method by the authors.