多目标决策中的ε—共轭对偶理论

本文首先讨论了ε—有效解的性质,证明了ε—有效解集的连通性。第二,在通常的Pareto有效解的意义下,利用ε—次微分和ε—共轭映射,讨论了Pareto有效解的共轭对偶定理、拉格朗日对偶定理和鞍点定理。还证明了ε—次微分的存在性定理。§1 ε—有效解和连通性近年来,对多目标最优化的共轭对偶理论已有了许多讨论。Tanino,T.[1]在Pareto有效解的意义下利用向量值函数的次微分给出了多目标最     

关于多值映照的几点注记

§1.引言多值映照在数学规划、对策论.不动点理论、非光滑分析等方面有广泛应用,它已成为非线性分析的一个重要内容.目前已有不少有关的文章和书籍出版.文献[1]对凸分析和凸多值映照作了十分简练的阐述,并以它作工具对极值理论作了统一处理. 本文将对多值映照的各种连续性定义的关系和对偶定理作了几点注记,在讨论这些注     

非线性规划的Ω共轭对偶理论

本文通过推广凸共轭函数和次梯度的概念,建立了非线性规划问题的一类对偶理论——Ω共轭对偶理论.研究结果表明,许多关于非线性最优化对偶性方面的结论都是本文的特殊情况.     

群体多目标最优化的Lagrange对偶性

本文建立了目标和约束为不对称的群体多目标最优化问题的Lagrange对偶规划，在问题的联合弱有效解意义下，得到群体多目标最优化Lagrange型的弱对偶定理、基本对偶定理、直接对偶定理和逆对偶定理。     

E-凸多目标规划的最优性及Wolfe型对偶

先引入了一类带不等式和等式约束的E-凸多目标优化问题(MP),给出了该类问题的一个最优性充分条件;其次,建立了该类规划问题(MP)的一类Wolfe型对偶模型(WD),并在E-凸条件下,获得了弱对偶定理,强对偶定理和逆对偶定理.     

一类锥约束多目标优化问题的高阶对偶研究

在一类锥约束单目标优化问题的一阶对偶模型基础之上,建立了锥约束多目标优化问题的二阶和高阶对偶模型.在广义凸性假设下,给出了弱对偶定理,在Kuhn-Tucker约束品性下,得到了强对偶定理.最后,在弱对偶定理的基础上,利用Fritz-John型必要条件建立了逆对偶定理.     

不确定多目标优化鲁棒真有效解的最优性与对偶

本文研究一类不确定性多目标优化问题鲁棒真有效解的最优性条件和对偶理论.首先,借助鲁棒真有效解的标量化定理,在鲁棒型闭凸锥约束品性下,建立了不确定多目标优化问题真有效解的最优性条件;其次,针对原不确定多目优化的Wolfe型对偶问题,得到关于鲁棒真有效解的强、弱对偶定理.     

鲁棒凸优化问题拟近似解的刻划

本文研究鲁棒凸优化问题拟近似解的最优性条件和对偶理论.首先利用鲁棒优化方法,在由约束函数的共轭函数的上图给出的闭凸锥约束规格条件下,建立了拟近似解的最优性充要条件.其次给出了鲁棒凸优化问题拟近似解在Wolf型和Mond-weir型对偶模型下的强(弱)对偶定理.最后给出具体实例验证了本文获得的结果.     

非光滑多目标半无限规划问题的混合型对偶

该文研究了非光滑多目标半无限规划问题的混合型对偶.首先,利用Lagrange函数介绍了非光滑多目标半无限规划混合型对偶的弱有效解和有效解的定义.其次,利用Dini?伪凸性建立了非光滑多目标半无限规划混合型对偶的弱对偶定理、强对偶定理和逆对偶定理.该文所得结果推广了已有文献中的主要结果.     

非线性规划的(H,Ω)对偶理论

对偶理论是非线性规划理论的一个重要组成部分,目前较成熟和完善的仅是凸规划的对偶理论.对于非凸规划对偶问题的研究仅有少量的工作完成,其结果也不令人满意.文献[1]就凸共轭函数进行了推广,建立了(H,(?))共轭函数理论,这一理论为凸对偶向非凸对偶迈进提供了基础.本文应用这一(H,(?))共轭函数理论,提出并建立了非线性规划的(H,(?))对偶理论.应用表明,在特殊簇 H 及(?)下,迄今为止几乎所有非线性规划的对偶理论都是这一对偶框架下的特殊形式,因此可以说,它是对偶理论的一个突破.     

Duality theory in multiobjective programming

In this paper, a multiobjective programming problem is considered as that of finding the set of all nondominated solutions with respect to the given domination cone. Two point-to-set maps, the primal map and the dual map, and the vector-valued Lagrangian function are defined, corresponding to the case of a scalar optimization problem. The Lagrange multiplier theorem, the saddle-point theorem, and the duality theorem are derived by using the properties of these maps under adequate convexity assumptions and regularity conditions.     

Fenchel duality theorem in multiobjective programming problems with set functions

In this paper, we characterize a vector-valued convex set function by its epigraph. The concepts of a vector-valued set function and a vector-valued concave set function are given respectively. The definitions of the conjugate functions for a vector-valued convex set function and a vector-valued concave set function are introduced. Then a Fenchel duality theorem in multiobjective programming problem with set functions is derived.     

Conjugate maps and duality in multiobjective optimization

This paper considers duality in convex vector optimization. A vector optimization problem requires one to find all the efficient points of the attainable value set for given multiple objective functions. Embedding the primal problem into a family of perturbed problems enables one to define a dual problem in terms of the conjugate map of the perturbed objective function. Every solution of the stable primal problem is associated with a certain solution of the dual problem, which is characterized as a subgradient of the perturbed efficient value map. This pair of solutions also provides a saddle point of the Lagrangian map.     

Vectorization of set-valued maps with respect to total ordering cones and its applications to set-valued optimization problems

As a result of our previous studies on finding the minimal element of a set in n-dimensional Euclidean space with respect to a total ordering cone, we introduced a method which we call “The Successive Weighted Sum Method” (Küçük et al., 2011 [1], [2]). In this study, we compare the Weighted Sum Method to the Successive Weighted Sum Method. A vector-valued function is derived from the special type of set-valued function by using a total ordering cone, which is a process we called vectorization, and some properties of the given vector-valued function are presented. We also prove that this vector-valued function can be used instead of the set-valued map as an objective function of a set-valued optimization problem. Moreover, by giving two examples we show that there is no relationship between the continuity of set-valued map and the continuity of the vector-valued function derived from this set-valued map.     

Conjugate Duality for Constrained Vector Optimization in Abstract Convex Frame

This article focuses on a conjugate duality for a constrained vector optimization in the framework of abstract convexity. With the aid of the extension for the notion of infimum to the vector space, a set-valued topical function and the corresponding conjugate map, subdifferentials are presented. Following this, a conjugate dual problem is proposed via this conjugate map. Then, inspired by some ideas in the image space analysis, some equivalent characterizations of the zero duality gap are established by virtue of the subdifferentials.     

On The Existence of a Countable Set of Positive Solutions for a Nonlocal Boundary-Value Problem with Vector-Valued Response

ABSTRACT

The existence of a countable set of positive solutions for a nonlocal boundary-value problem with vector-valued response is investigated by some variational methods based on the idea of the Fenchel conjugate. As a consequence of a duality developed here, we obtain the existence of a countable set of solutions for our problem that are minimizers to a certain integral functional. We derive (also in the superlinear case) a measure of a duality gap between primal and dual functional for approximate solutions.     

Lagrangian function and duality theory in multiobjective programming with set functions

Using the concept of vector-valued Lagrangian functions, we characterize a special class of solutions,D-solutions, of a multiobjective programming problem with set functions in which the domination structure is described by a closed convex coneD. Properties of two perturbation functions, primal map and dual map, are also studied. Results lead to a general duality theorem.The authors greatly appreciate helpful and valuable comments and suggestions received from the referee.     

Set-valued duality theory for multiple objective linear programs and application to mathematical finance

We develop a duality theory for weakly minimal points of multiple objective linear programs which has several advantages in contrast to other theories. For instance, the dual variables are vectors rather than matrices and the dual feasible set is a polyhedron. We use a set-valued dual objective map the values of which have a very simple structure, in fact they are hyperplanes. As in other set-valued (but not in vector-valued) approaches, there is no duality gap in the case that the right-hand side of the linear constraints is zero. Moreover, we show that the whole theory can be developed by working in a complete lattice. Thus the duality theory has a high degree of analogy to its classical counterpart. Another important feature of our theory is that the infimum of the set-valued dual problem is attained in a finite set of vertices of the dual feasible domain. These advantages open the possibility of various applications such as a dual simplex algorithm. Exemplarily, we discuss an application to a Markowitz-type bicriterial portfolio optimization problem where the risk is measured by the Conditional Value at Risk.     

向量优化中广义增广拉格朗日对偶理论及应用

作者介绍了一种基于向量值延拓函数的广义增广拉格朗日函数,建立了基于广义增广拉格朗日函数的集值广义增广拉格朗日对偶映射和相应的对偶问题,得到了相应的强对偶和弱对偶结果,将所获结果应用到约束向量优化问题.该文的结果推广了一些已有的结论.     

Duality and Exact Penalization for Vector Optimization via Augmented Lagrangian

In this paper, we introduce an augmented Lagrangian function for a multiobjective optimization problem with an extended vector-valued function. On the basis of this augmented Lagrangian, set-valued dual maps and dual optimization problems are constructed. Weak and strong duality results are obtained. Necessary and sufficient conditions for uniformly exact penalization and exact penalization are established. Finally, comparisons of saddle-point properties are made between a class of augmented Lagrangian functions and nonlinear Lagrangian functions for a constrained multiobjective optimization problem.