参数规划中最优值函数的次线性及齐次拟凹凸性

本文主要对参数最优化问题P(u): max f(x,u) s.t.x∈C(u) 的最优值函数的次线性和齐次拟凹凸性进行了系统研究,同时还探讨了通过特殊化P(u)的目标函数或约束条件而得到的其它几个参数最优化问题。许多新结果对一般的抽象空间,如线性空间、线性拓扑空间、线性赋范空间或Banach空间等亦是有效的。有关结论可应用于许多最优控制问题和经济数学,也可应用到分式规划的研究中去。     

构建数学模型 培养解题能力

把实际问题转化为一个数学问题,通常称为数学模型.提高中学数学教学质量,最重要的是学生学到有用的数学,构建数学模型也是中学数学教学改革的方向.自实施新课标以来,以物理、化学、生物、医学等学科知识为背景的跨学科综合题颇受关注.有些问题用常规方法难以解决,往往需要构建一个与之有关的数学模型.建立数学模型的过程称为数学建模.数学建模就是要把现实     

数学规划的稳定性

<正> 我们知道,如果一个生产实际问题可以归纳成数学规划问题去解决吋,在形成数学模型的过程中,目标函数和约束条件中的已知系数由于观察、实验、或测量得不够精确,产生一些误差.这些误差对数学模型的真实性有没有影响呢?换言之,如一个具有最优解的数学规划问题,将其已知系数作微小变动后,是否还有最优解?回答是,不一定有.例如凸规划:     

一个平行的广义Kuhn—Tucker必要条件

本文引入能行锥的概念,得到一个新的约束品性,给出了最优化问题在一般约束条件下,目标函数f(x)在x 取得局部极小值的一个平行的广义Kuhn-Tucker 必要条件。     

调优操作法(EVOP)在制碱工业中的应用

一引言 随着生产力的高度发达,其生产设备与技术也愈复杂,即产品从投入原材料开始到最终产品形成的全过程中,影响产品质量与产量的因素就愈多.现代化工企业尤其如此.如何使一个复杂的、影响因素很多的生产线经常保持在最佳工作状态?该问题的数学本质就是多参数最优化问题. 多参数最优化的途径,一般可分为两种;(1).有数学模型的调优;(2)无数学模型的调优.前者就是可计算性的参数优化问题,它需要一个反映调优目标与各参数之间因果关系的数学模型,但对化工装置,十分符合机理的数学模型尚不多.后者不需数学模型,要在这种前提下完成参数优化工作,…     

多目标数学规划的稳定性

<正> §1.引言 当我们用数学规划去描述和求解某些实际问题的时候,特别是在最优设计问题中,评价最优性的目标往往不只一个,这就构成了所谓多目标数学规划问题(或称向量极值问题).近年来,在国内外已经引起了一些从事于数学规划研究的人越来越大的兴趣.尽管关于“最优性”的含意各不相同,定义也多,但都是在多目标数学规划问题的“有效解”     

现实生活中最优化问题的数学模型构造

在实际生产、现实生活和科学研究中,许多情形下往往要求操作、经营和决策者考虑怎样才能以最低的成本、最短的时间获取最大的效益,这类问题在数学中称为最优化问题.这类问题的求解,需要我们通过对问题中给出的有关信息和数据进行分析、加工,灵活的运用数学知识构造恰当的数学模     

交互作用和多因素最优化──澄清一个历史性误会

在实践中,对于多因素(经过数量化后,数学中指多个自变量)的现象,常提出最优化的要求.一般分成两个领域.一是多因素试验,寻求最佳的工艺参数、配方、配比、设计参数等。另一是计算机设计,通过数学模型或者编好的程序,寻求目标函数的最优解.数学中,一般归结为或称作解非线性规划.(有不少项目,不是追求多目标综合效果或者总目标的最优解,而是急于寻求满足当前需要的解,所谓优良解.这二者在主流上往往矛盾不大,虽然对于长久的项目来说应当倾向于前者).本文进行的理论讨论,限于解决多因素最优化的要求,而不是指解决其它性质的课题(此后不再重复指…     

数学解题 中的模型化思考

数学模型是联系客观世界与数学的桥梁 .数学模型是用数学语言来模拟空间形式和数量关系的模型 .广义地看 ,一切数学概念、公式、理论体系、算法系统都可称为数学模型 ,如 :算术是计算盈亏的模型 ,几何是物体外形的模型等 .狭义地看 ,只有反映特定问题的数学结构才称为数学模型 ,如一次函数是匀速直线运动的模型 ,不定方程是鸡兔同笼问题的模型等 .数学模型方法是针对要解决的问题来构造相应的数学模型、再通过对数学模型的研究去解决实际问题的一种数学方法 .数学模型方法在解题中的基本步骤是 :( 1 )从要解决的问题中恰当构建相应的数学模…     

前言

数学物理问题是应用数学的一个重要分支.在这类问题中,人们基于基本的物理定律用数学语言(方程)描述实际的物理过程,并研究数学模型的适定性和解的形态等.数学物理反问题主要是指由已知可以测量到的信息,基于数学物理模型,重构未知信息的问题.由于数学物理反问题的研究更多地来源于一些重要的实际问题,从而引起了国内外数学工作者的不断重视.研究成果往往可以为一些重要技术、关键问题的解决提供想法和工具,因此,     

Nash fuzzy equilibrium: Theory and application to a spatial duopoly

Consider a non-cooperative n-persons game. Each gambler has a set of mixed strategies at his disposal. The payoffs are some physical or immaterial objects. The game is a fuzzy game because (1) gamblers have more or less precise preferences for the payoffs and (2) the outcoming of payoffs is uncertain. The uncertainty can be expressed either by a distribution of possibility or by a distribution of probability. The product set of a gambler's mixed strategies is convex and compact and the payoff functions are continuous. Then a closed and convex fuzzy point-to-set mapping is defined on the product set of strategies and, by using a Butnariu theorem, the existence of a fixed point for this fuzzy point-to-set mapping is proved. The issue allows us to generalize a famous Nash result: a n-persons non-cooperative fuzzy game with mixed strategies has at least one equilibrium point. In the second part of the paper an economic application is devoted to the statement of the equilibrium existence conditions in a spatial duopoly. The model is not only more general than the classical ones, but also more relevant because new results are obtained.     

Aubin cores and bargaining sets for convex cooperative fuzzy games

In this paper, we deal with Aubin cores and bargaining sets in convex cooperative fuzzy games. We first give a simple and direct proof to the well-known result (proved by Branzei et al. (Fuzzy Sets Syst 139:267–281, 2003)) that for a convex cooperative fuzzy game v, its Aubin core C(v) coincides with its crisp core C ^cr (v). We then introduce the concept of bargaining sets for cooperative fuzzy games and prove that for a continuous convex cooperative fuzzy game v, its bargaining set coincides with its Aubin core, which extends a well-known result by Maschler et al. for classical cooperative games to cooperative fuzzy games. We also show that some results proved by Shapley (Int J Game Theory 1:11–26, 1971) for classical decomposable convex cooperative games can be extended to convex cooperative fuzzy games.     

Fuzzy Mathematical Programming: A Unified Approach Based On Fuzzy Relations

Fuzzy mathematical programming problems (FMP) form a subclass of decision - making problems where preferences between alternatives are described by means of objective function(s) defined on the set of alternatives. The formulation a FMP problem associated with the classical MP problem is presented. Then the concept of a feasible solution and optimal solution of FMP problem are defined. These concepts are based on generalized equality and inequality fuzzy relations. Among others we show that the class of all MP problems with (crisp) parameters can be naturally embedded into the class of FMP problems with fuzzy parameters. We also show that the feasible and optimal solutions being fuzzy sets are convex under some mild assumptions.     

Fuzzy extensions of bargaining sets and their existence in cooperative fuzzy games

Recently, the concept of classical bargaining set given by Aumann and Maschler in 1964 has been extended to fuzzy bargaining set. In this paper, we give a modification to correct some weakness of this extension. We also extend the concept of the Mas-Colell's bargaining set (the other major type of bargaining sets) to its corresponding fuzzy bargaining set. Our main effort is to prove existence theorems for these two types of fuzzy bargaining sets. We will also give necessary and sufficient conditions for these bargaining sets to coincide with the Aubin Core in a continuous superadditive cooperative fuzzy game which has a crisp maximal coalition of maximum excess at each payoff vector. We show that both Aumann-Maschler and Mas-Colell fuzzy bargaining sets of a continuous convex cooperative fuzzy game coincide with its Aubin core.     

Fuzzy subobjects in a category and the theory of C-sets

In this paper we shall define the concept of a fuzzy subobject of an object in arbitrary categories. This concept is generated by the representation theorem of fuzzy sets. By using fuzzy subobjects one can include most of the fuzzy concepts defined in the literature, such as: fuzzy groups, fuzzy relations and fuzzy convex sets. In the second part of the paper we shall define a new concept; that of a $\text{C}$-set. This concept will generalize that of a fuzzy set and we shall also prove that $\text{C}$-sets can be represented by some sets of functors. More precisely, $\text{C}$-sets form a category which can be represented by a category of functors. The utility of $\text{C}$-sets resides in the fact that one can replace “ordering” by the more general concept of a morphism in category. The new representation of $\text{C}$-sets is weaker than that of fuzzy sets.     

A set-valued Lagrange theorem based on a process for convex vector programming

ABSTRACT

In this paper, we present a new set-valued Lagrange multiplier theorem for constrained convex set-valued optimization problems. We introduce the novel concept of Lagrange process. This concept is a natural extension of the classical concept of Lagrange multiplier where the conventional notion of linear continuous operator is replaced by the concept of closed convex process, its set-valued analogue. The behaviour of this new Lagrange multiplier based on a process is shown to be particularly appropriate for some types of proper minimal points and, in general, when it has a bounded base.     

On some characterizations of preinvex fuzzy mappings

Jeyakumar (Methods Oper. Res. 55:109–125, 1985) and Weir and Mond (J. Math. Anal. Appl. 136:29–38, 1988) introduced the concept of preinvex function. The preinvex functions have some interesting properties. For example, every local minimum of a preinvex function is a global minimum and nonnegative linear combinations of preinvex functions are preinvex. Invex functions were introduced by Hanson (J. Math. Anal. Appl. 80:545–550, 1981) as a generalization of differentiable convex functions. These functions are more general than the convex and pseudo convex ones. The type of invex function is equivalent to the type of function whose stationary points are global minima. Under some conditions, an invex function is also a preinvex function. Syau (Fuzzy Sets Syst. 115:455–461, 2000) introduced the concepts of pseudoconvexity, invexity, and pseudoinvexity for fuzzy mappings of one variable by using the notion of differentiability and the results proposed by Goestschel and Voxman (Fuzzy Sets Syst. 18:31–43, 1986). Wu and Xu (Fuzzy Sets Syst 159:2090–2103, 2008) introduced the concepts of fuzzy pseudoconvex, fuzzy invex, fuzzy pseudoinvex, and fuzzy preinvex mapping from \(\mathbb{R}^{n}\) to the set of fuzzy numbers based on the concept of differentiability of fuzzy mapping defined by Wang and Wu (Fuzzy Sets Syst. 138:559–591, 2003). In this paper, we present some characterizations of preinvex fuzzy mappings. The necessary and sufficient conditions for differentiable and twice differentiable preinvex fuzzy mapping are provided. These characterizations correct and improve previous results given by other authors. This fact is shown with examples. Moreover, we introduce additional conditions under which these results are valid.     

模糊Choquet-可积函数空间的凸锥结构

在一般有限模糊测度空间上研究了非负模糊Choquet-可积函数的模糊测度表示, 进而获得了非负模糊Choquet-可积函数空间和一般模糊Choquet-可积函数空间构成凸锥的充要条件.     

基于三角模糊数的凸合成模糊对策解的结构

将凸合成模糊对策的特征函数用三角模糊数的形式表示出来,并以三角模糊数表示局中人的参与度,从而建立了一个新的凸合成模糊合作对策的模型.在此模型的基础上,给出了凸合成模糊对策的三角核心和三角稳定集,并证明了上述解可由子对策的核心和稳定集表达出来.     

闭模糊集构成凸模糊集的充要条件

本文通过引入弱拟凸模糊集的概念，针对欧几里空间上的闭模糊集，给出了它构成凸模糊集的一个充要条件，从而丰富了凸模糊集的理论及其应用。