Scalarization and optimality conditions for vector equilibrium problems

In this paper, we investigated vector equilibrium problems and gave the scalarization results for weakly efficient solutions, Henig efficient solutions, and globally efficient solutions to the vector equilibrium problems without the convexity assumption. Using nonsmooth analysis and the scalarization results, we provided the necessary conditions for weakly efficient solutions, Henig efficient solutions, globally efficient solutions, and superefficient solutions to vector equilibrium problems. By the assumption of convexity, we gave sufficient conditions for those solutions. As applications, we gave the necessary and sufficient conditions for corresponding solutions to vector variational inequalities and vector optimization problems.     

多目标优化问题鲁棒有效解与真有效解之间的关系

在一定条件下研究了多目标优化问题鲁棒有效解与真有效解之间的关系及鲁棒有效解的最优性条件.首先,给出多目标优化问题鲁棒弱有效解的概念,研究它与鲁棒有效解和真有效解之间的关系,举例说明了相关结果的合理性.其次,在次类凸和伪凸性假设下研究了鲁棒有效解的必要性条件和充分性条件.     

Optimality conditions for various efficient solutions involving coderivatives: From set-valued optimization problems to set-valued equilibrium problems

We present a new approach to the study of a set-valued equilibrium problem (for short, SEP) through the study of a set-valued optimization problem with a geometric constraint (for short, SOP) based on an equivalence between solutions of these problems. As illustrations, we adapt to SEP enhanced notions of relative Pareto efficient solutions introduced in set optimization by Bao and Mordukhovich and derive from known or new optimality conditions for various efficient solutions of SOP similar results for solutions of SEP as well as for solutions of a vector equilibrium problem and a vector variational inequality.We also introduce the concept of quasi weakly efficient solutions for the above problems and divide all efficient solutions under consideration into the Pareto-type group containing Pareto efficient, primary relative efficient, intrinsic relative efficient, quasi relative efficient solutions and the weak Pareto-type group containing quasi weakly efficient, weakly efficient, strongly efficient, positive properly efficient, Henig global properly efficient, Henig properly efficient, super efficient and Benson properly efficient solutions. The necessary conditions for Pareto-type efficient solutions and necessary/sufficient conditions for weak Pareto-type efficient solutions formulated here are expressed in terms of the Ioffe approximate coderivative and normal cone in the Banach space setting and in terms of the Mordukhovich coderivative and normal cone in the Asplund space setting.     

E-Borwein真有效解的刻画

基于Borwein真有效解的思想,利用free disposal集提出了向量优化问题的一类近似Borwein真有效解概念,建立了其与E-Benson真有效解间的等价关系     

Boundedness and Nonemptiness of the Efficient Solution Sets in Multiobjective Optimization

For a given multiobjective optimization problem, we study recession properties of the sets of efficient solutions and properly efficient solutions. We work out various consequences based on the obtained recession properties, including a characterization for the boundedness and nonemptiness of the set of (properly) efficient solutions when the problem is a convex problem. We also show that the boundedness and nonemptiness of the set of efficient solutions is equivalent to that of the set of properly efficient solutions under an additional mild condition. Finally, we provide some new verifiable necessary conditions for the nonemptiness of the set of efficient solutions in terms of the associated recession functions and recession cones.     

群体多目标规划αβ-较多联合有效解类的几何特性

对于群体多目标规划问题，文[1]和[2]分别引进了它的联合有效解类和带参数α的α-较多联合有效解类，并且建立了这些解类的最优性条件．文[3]则研究了联合有效解类的几何特性．本文借助供选方案集的带两个参数α和β的αβ-较多有效数，定义了群体多目标规划问题的更一般的αβ-较多联合有效解类，并且研究了这些解的几何特性，得到了若干必要条件和充分条件．     

Optimality conditions in convex multiobjective SIP

Mathematical Programming - The purpose of this paper is to characterize the weak efficient solutions, the efficient solutions, and the isolated efficient solutions of a given vector optimization...     

多目标规划ak-较多有效解类的有效性充分条件

在文［３］引进多目标规划问题的带双参数ａ和ｋ的ａｋ－较多有效解, 并且给出了此类解的有效性必要条件。本文在一定凸性的假设下证明了ａｋ－较多 有效解和ａｋ－弱较多有效解的几个有效性充分条件．     

Minimum values over the efficient set in multiple objective decision making

In multiple objective decision making (MODM), it is often helpful to provide the decision maker (DM) with bounds on the values of each of the objectives. Ideal solutions are relatively easy to calculate and provide upper bounds on the value of each objective over the set of efficient solutions. Ideal solutions also provide lower bounds on the value of each objective over the ideal set. However, the lower bounds over the set of efficient solutions can be strictly less than the ideal lower bounds, but are, in general, more difficult to determine. Thus MODM procedures which utilize the ideal lower bound may overlook elements of the set of efficient solutions. This study explores the differences between the subset of the set of efficient solutions to a MODM problem bounded by its ideal solutions and the complete efficient set.     

Existence results for proper efficient solutions of vector equilibrium problems and applications

In this paper, we present sufficient conditions for the existence of Henig efficient solutions, superefficient solutions and Henig globally efficient solutions of a vector equilibrium problem in topological vector spaces, using a well-known separation theorem in infinite dimensional spaces. As an application, using a scalarization technique, existence results for proper efficient solutions of generalized vector variational inequalities are given.     

Efficiency and proper efficiency in vector maximization with respect to cones

Vector maximization problems arise when more than one objective function is to be maximized over a given feasibility region. The concepts of efficiency and proper efficiency have played a useful role in the analysis of such problems. Recently these concepts have been extended to vector maximization problems in which the underlying domination cone is a convex cone. In this paper, efficient and properly efficient solutions for the vector maximization problem in which the underlying domination cone is any nontrivial, closed convex cone are examined. Differences between properly and improperly efficient solutions are established. Characterizations of efficient and properly efficient solutions are presented, and conditions under which efficient solutions exist and fail to exist are derived.     

Asplund空间中非凸向量均衡问题近似解的最优性条件

在Asplund空间中,研究了非凸向量均衡问题近似解的最优性条件.借助Mordukhovich次可微概念,在没有任何凸性条件下获得了向量均衡问题εe-拟弱有效解,εe-拟Henig有效解,εe-拟全局有效解以及εe-拟有效解的必要最优性条件.作为它的应用,还给出了非凸向量优化问题近似解的最优性条件.     

Weakly and properly nonessential objectives in multiobjective optimization problems

In this work we characterize objective functions which do not change the set of efficient solutions (weakly efficient solutions, properly efficient solutions). Necessary and sufficient conditions for an objective function to be weakly nonessential (properly nonessential) are presented. We establish relations between weakly nonessential, properly nonessential and nonessential functions.     

多目标优化问题的完全标量化

本文首先利用松弛变量和广义Tchebycheff范数的推广形式提出一类新的标量化优化问题.进一步,通过调整几种参数范围获得一般多目标优化问题弱有效解、有效解和真有效解的一些完全标量化刻画.此外,本文提出例子对主要结果进行说明,利用相应的标量化方法判定给定的多目标优化问题的可行解是否是弱有效解、有效解和真有效解.     

上半连续集值优化解在图像逼近意义下的稳定性

给出上半连续集值映射优化问题在图像拓扑逼近意义下的本质弱有效解和本质有效解的概念.利用通有稳定性研究的usco方法, 证明了上半连续集值映射优化问题.在图像拓扑逼近意义下,弱有效解映射在定义域和映射同时扰动下是紧值上半连续的,从而是通有下半连续的,即在Baire纲意义下, 绝大多数上半连续集值映射优化问题, 在图像逼近意义下其弱有效解是稳定的,或者说是本质的. 证明了上半连续集值映射优化问题在图像逼近意义下有效解映射上半连续的一个充要条件,也即是有效解通有稳定的一个重要条件.     

Connectedness of the Set of Efficient Solutions for Generalized Systems

We introduce the concept of positive proper efficient solutions to the generalized system in this paper. We show that, under some suitable conditions, the set of positive proper efficient solutions is dense in the set of efficient solutions to the generalized system. We discuss also the connectedness of the set of efficient solutions for the generalized system with monotone bifunctions in real locally convex Hausdorff topological vector spaces. This research was partially supported by the National Natural Science Foundation of China (10561007), the Natural Science Foundation of Jiangxi Province, China, and a grant from the National Science Council of ROC.     

Efficient Solutions and Bounds on Tradeoffs

The modified Tchebycheff method, widely used to generate efficient solutions to a vector optimization problem, provides means to identify properly efficient solutions with a preimposed common bound on all tradeoffs. In this paper, we show how to generate weakly efficient solutions when different bounds are preimposed on the subsets of tradeoffs.     

On optimality conditions for vector variational inequalities

Optimality conditions for weak efficient, global efficient and efficient solutions of vector variational inequalities with constraints defined by equality, cone and set constraints are derived. Under various constraint qualifications, necessary optimality conditions for weak efficient, global efficient and efficient solutions in terms of the Clarke and Michel–Penot subdifferentials are established. With assumptions on quasiconvexity of constraint functions sufficient optimality conditions are also given.     

锥意义下有效解的连续性

对于有效解的连续性和稳定性的研究,大多数结果局限于有限维空间中的讨论.从而对无限维空间中向量优化问题有效解的连续性,采取构造函数的方法、反证法和USCO的方法进行探讨,得到了锥意义下有效解本质的一个充分必要条件以及相应的其他结果.     

Connectedness of the sets of weak efficient solutions for generalized vector equilibrium problems

In this paper, a generalized vector equilibrium problem is introduced and studied. A scalar characterization of weak efficient solutions for the generalized vector equilibrium problem is obtained. By using the scalarization result, the existence of the weak efficient solutions and the connectedness of the set of weak efficient solutions for the generalized vector equilibrium problem are proved in locally convex spaces.