线性规划最优解集的表现

文献［１］讨论了有无穷多最优解的线性规划问题，并利用最优单纯形表格的检验数给出线性规划有无穷多最优解的判别法，本文利用最优基可行解的凸组合及最优极向的非负线性组合给出线性规划最优解集的表现，从而把线性规划最优解集的几何特征阐释清楚．     

模糊线性规划的最优解

本文说明模糊线性规划的模糊优越集C_f,在一般情况下是去掉端点x~((0))的线段。在线段上有且只有一个模糊线性规划问题的最优解。最后提出了解模糊线性规划的一个比较简便的算法。根据本文的结论,可以证明Zimmermann算法的最优值因此,Zimmermann算法的最后一步可简单地用代替,从而节省了大量的计算工作量。[3]指出,当模糊判决用乘法或凸组合运算时,导出的规划往往是非线性的,求解比较困难。然而,用本文的结论,问题能容易地得到解决。     

互补基解性质的应用研究

对线性规划互补基解性质进行了研究,得到了由线性规划问题最优基对应的单纯形表直接获得对偶线性规划问题最优基对应的单纯形表的一个有效方法,给出了应用实例.     

带权值的模糊多目标线性规划

本文提出了求解一般多目标性规划问题 (MOL P)的带权值的模糊多目标线性规划方法 .证明了在权值都大于零的条件下 ,与 (MOLP)原问题对应的带权值的模糊多目标线性规划问题的最优解为模糊有效解 ,从而为原问题的有效解 ,并作了实例验证 .     

一类模糊线性规划模型的模糊最优区间值

讨论一类既有模糊不等式约束又有模糊等式约束的全模糊系数线性规划问题。在给定的模糊隶属度水平下 ,将模型转化为区间数线性规划模型 ,通过确定区间模型的最佳目标函数和最大可行域以及最劣目标函数和最小可行域 ,求出目标函数的模糊最优区间值 ,从而为决策者提供更多的决策信息。最后给出一个数值例子。     

线性规划的最钝角松弛算法

本文提出一个基于最钝角原理的松弛算法求解线性规划问题。该算法依据最钝角原理略去部分约束得到一个规模较小的子问题,用原始单纯形算法解之;再添加所略去的约束恢复原问题,若此时全部约束条件均满足则已获得一个基本最优解,否则用对偶单纯形算法继续求解。初步的数值试验表明,新算法比传统两阶段单纯形算法快得多。     

基于单纯形方法的双层线性规划全局优化算法

通过分析双层线性规划可行域的结构特征和全局最优解在约束域的极点上达到这一特性,对单纯形方法中进基变量的选取法则进行适当修改后,给出了一个求解双层线性规划局部最优解方法,然后引进上层目标函数对应的一种割平面约束来修正当前局部最优解,直到求得双层线性规划的全局最优解.提出的算法具有全局收敛性,并通过算例说明了算法的求解过程.     

求解线性规划的一种单纯形矩阵法

线性规划的单纯形法一直是运筹学教学中的难点,是求解线性规划的一种重要方法.通过实例从代数角度探讨了单纯形法的迭代思想,提出了用单纯形矩阵求解线性规划的方法.同传统的单纯形表计算比较而言,此方法操作简单,不易出错,为线性规划的求解提供了一种行之有效的方法。     

线性规划中一个避免人工变元的方法的改进

有许多文献讨论了线性规划问题中单纯形方法的改进(如文献[1～5]等)。我们在文献[1]的基础上,突破了传统方法中要求单纯形表中的基变量始终非负的想法,给出了求解线性规划问题中一个新的避免人工变量的方法,使其计算量得到减少。     

方向选择约束单纯形算法

一、引言人们一直致力于求解线性规划的单纯形算法的改进工作.1976年,Powell 发表过降低基维数的改进单纯形算法,这个算法是将基矩阵的一个块用基矩阵的其它块的乘积来表示,虽然实现了降低基维数,节省了存贮空间,却增加了计算次数,减慢了计算速度.Sethi and Thompson 针对线性规划问题也提出过竞争和非竞争约束(candidate andnoncandidate constraints)的概念.他们发现,随机生成的实验问题,其总约束中大约只有15%—25%是竞争约束,并提出了一个仅对竞争约束进行旋转运算的单纯形算法.他们的算法,对某些特殊的线性规划提高了求解速度,但并不减少基的维数,并不节省内存空间,增加了程序复杂性.1984年,Sethi and Thompson 又提出 PAPA 算法,再次利用线性规划问题通常只有少量竞争约束这个事实来提高求解速度.但 PAPA 算法往往要在原问题的可行域外运行.况且,上面提到的各种算法,均不能从理论上表明,它们较标准改进单纯形算法到底节省了多少存贮单元和节省了多少计算次数.     

关于FLP问题一种新的单纯形算法的推广

指出FLP问题的一种新的单纯形算法[1]中主要结论成立的适用条件,并给出了该适用条件不成立时,一般条件下的推广.     

A simplex algorithm for piecewise-linear programming I: Derivation and proof

The simplex method for linear programming can be extended to permit the minimization of any convex separable piecewise-linear objective, subject to linear constraints. This three-part paper develops and analyzes a general, computationally practical simplex algorithm for piecewiselinear programming.Part I derives and justifies the essential steps of the algorithm, by extension from the simplex method for linear programming in bounded variables. The proof employs familiar finite-termination arguments and established piecewise-linear duality theory.Part II considers the relaxation of technical assumptions pertaining to finiteness, feasibility and nondegeneracy of piecewise-linear programs. Degeneracy is found to have broader consequences than in the linear case, and the standard techniques for prevention of cycling are extended accordingly.Part III analyzes the computational requirements of piecewise-linear programming. The direct approach embodied in the piecewise-linear simplex algorithm is shown to be inherently more efficient than indirect approaches that rely on transformation of piecewise-linear programs to equivalent linear programs. A concluding section surveys the many applications of piecewise-linear programming in linear programming,l ₁ estimation, goal programming, interval programming, and nonlinear optimization.This research has been supported in part by the National Science Foundation under grant MCS-8217261.     

Enumeration of All Possibly Optimal Vertices with Possible Optimality Degrees in Linear Programming Problems with a Possibilistic Objective Function

In this paper, we treat linear programming problems with fuzzy objective function coefficients. To such a problem, the possibly optimal solution set is defined as a fuzzy set. It is shown that any possibly optimal solution can be represented by a convex combination of possibly optimal vertices. A method to enumerate all possibly optimal vertices with their membership degrees is developed. It is shown that, given a possibly optimal extreme point with a higher membership degree, the membership degree of an adjacent extreme point is calculated by solving a linear programming problem and that all possibly optimal vertices are enumerated sequentially by tracing adjacent possibly optimal extreme points from a possibly optimal extreme point with the highest membership degree.     

多分类器融合系统中模糊测度的确定

有限事例集上,choquet模糊积分的计算可以转化成模糊测度的线性组合,故可以使用标准的优化技术来确定模糊测度。本文使用线性规划来确定模糊测度,数据库上的仿真实验表明,线性规划确定的模糊测度的系统融合精度比多数投票法和加权平均法的分类精度要高。     

模糊多目标半定规划的最优性条件

将模糊集理论应用到多目标半定规划中来,提出了有约束的模糊多目标半定规划模型,并首次给出了其最优有效解的定义.通过构造确定的隶属度函数,将以矩阵为决策变量的模糊多目标半定规划转化为一种目标函数的某些分量由约束函数决定的确定性多目标半定规划,并证明了前者最优有效解与后者有效解的一致性.在此基础之上,讨论了二者的最优性条件.     

Fuzzy linear programming under interval uncertainty based on IFS representation

The equivalence between the interval-valued fuzzy set (IVFS) and the intuitionistic fuzzy set (IFS) is exploited to study linear programming problems involving interval uncertainty modeled using IFS. The non-membership of IFS is constructed with three different viewpoints viz., optimistic, pessimistic, and mixed. These constructions along with their indeterminacy factors result in S-shaped membership functions in the fuzzy counterparts of the intuitionistic fuzzy linear programming models. The solution methodology of Yang et al. [45], and its subsequent generalization by Lin and Chen [33] are used to compute the optimal solutions of the three fuzzy linear programming models.     

An interactive fuzzy satisficing method for structured multiobjective linear fractional programs with fuzzy numbers

In this paper, by considering the experts' vague or fuzzy understanding of the nature of the parameters in the problem formulation process, multiobjective linear fractional programming problems with block angular structure involving fuzzy numbers are formulated. Using the a-level sets of fuzzy numbers, the corresponding nonfuzzy a-multiobjective linear fractional programming problem is introduced. The fuzzy goals of the decision maker for the objective functions are quantified by eliciting the corresponding membership functions including nonlinear ones. Through the introduction of extended Pareto optimality concepts, if the decision maker specifies the degree a and the reference membership values, the corresponding extended Pareto optimal solution can be obtained by solving the minimax problems for which the Dantzig-Wolfe decomposition method and Ritter's partitioning procedure are applicable. Then a linear programming-based interactive fuzzy satisficing method with decomposition procedures for deriving a satisficing solution for the decision maker efficiently from an extended Pareto optimal solution set is presented. An illustrative numerical example is provided to demonstrate the feasibility of the proposed method.     

基于多目标规划法的模糊线性回归分析

讨论输入、输出均为模糊数,回归系数为实数时的模糊线性回归分析。由于模糊最小二乘线性回归容易受异常值的影响,而最小一乘法能有效地降低回归模型的误差。为此,基于最小一乘法,建立多目标规划模型并将其转化为非线性规划问题进行求解,从而实现模糊线性回归模型的参数估计。最后,结合一个数值实例,验证和比较该方法的合理性和优越性。     

Linear programming in measure spaces

This paper studies a linear programming problem in measure spaces (LPM). Several results are obtained. First, the optimal value of LPM can be equal to the optimal value of the dual problem (DLPM), but the solution of DLPM may be not exist in its feasible region. Sccond, :he relations between the optimal solution of LPM and the extreme point of the feasible region of LPM are discussed. In order to investigate the conditions under which a feasible solution becomes an extremal point, the inequality constraint of LPM is transformed to an equality constraint. Third, the LPM can be reformulated to be a general capacity problem (GCAP) or a linear semi-infinite programming problem (LSIP = SIP), and under appropriate restrictioiis, the algorithm developed by the authors in [7] and [8] are applicable for developing an approximation scheme for the optimal solution of LPM