向量优化问题的对偶性(英文)

Duality framework on vector optimization problems in a locally convex topological vector space are established by using scalarization with a cone-strongly increasing function.The dualities for the scalar convex composed optimization problems and for general vector optimization problems are studied.A general approach for studying duality in vector optimization problems is presented.     

有效解适合鞍点准则的条件

建立了单目标优化问题的用次微分表达的对偶问题,并用其对偶性给出多目标优化问题的有效解适合鞍点准则的条件.     

具有广义B-凸函数的非光滑多目标规划的最优性与向量Lagrange鞍点理论

利用广义B-凸函数等概念，讨论了一类非光滑多目标规划，给出了广义最优性充分条件和Mond-Weir型对偶结果，讨论了向量Lagrange乘子性质并证明了向量值鞍点定理。     

向量最优化问题的Lagrange对偶与择一定理

本文讨论无限维向量最优化问题的Lagrange对偶与弱对偶，建立了若干鞍点定理与弱鞍点定理．作为研究对偶问题的工具，建立了一个新的择一定理．     

Two Types of Approximate Saddle Points

In this paper, we study two types of approximate solutions for a vector optimization problem in Banach space setting. Our main concern is to define two new concepts of approximate saddle points and relate them to the above solution concepts. As a result, a dual is formulated, and duality results are established.     

Saddle Point Optimality Conditions in Fuzzy Optimization Problems

The fuzzy-valued Lagrangian function of fuzzy optimization problem via the concept of fuzzy scalar (inner) product is proposed. A solution concept of fuzzy optimization problem, which is essentially similar to the notion of Pareto solution in multiobjective optimization problems, is introduced by imposing a partial ordering on the set of all fuzzy numbers. Under these settings, the saddle point optimality conditions along with necessary and sufficient conditions for the absence of a duality gap are elicited.     

Strong Duality for Proper Efficiency in Vector Optimization

In this paper, we give counterexamples showing that the strong duality results obtained in Refs. 1–5 for several dual problems of multiobjective mathematical programs are false. We provide also the conditions under which correct results can be established.This research was supported by the Brain Korea 21 Project in 2003. The authors thank the referees for valuable remarks.     

集值优化问题的强有效解与Kuhn—Tucker鞍点

首先在局部凸Hausdorff拓扑向量空间中定义了集值优化问题的Kuhn—Tucker鞍点，在近似锥一次类凸集值映射下，讨论了集值优化问题的强有效解与Kuhn—Tucker鞍点之间的关系．     

集值优化问题的强有效解与Kuhn-Tucker鞍点

首先在局部凸Hausdorff拓扑向量空间中定义了集值优化问题的Kuhn-Tucker鞍点,在近似锥-次类凸集值映射下,讨论了集值优化问题的强有效解与Kuhn-Tucker鞍点之间的关系.     

向量值最优化问题的最优性条件与对偶性

本文我们首先给出一类向量值优化问题（VP）的正切锥真有效解的定义，在锥方向导数的假设下，讨论了一类单目标问题 的最优性必要条件；然后利用正切锥方向导数定义一类正切锥F－凸函数类，并给出了（VP）正切锥真有效解的充分性条件，最后我们亦讨论了（VP）在正切锥真有效解意义下的对偶性质。     

Optimality Conditions and Lagrange Duality for Vector Extremum Problems with Set Constraint

The necessary and sufficient optimality conditions for vector extremum problems with set constraint in an ordered linear topological space are given. Finally, Lagrange duality is established.     

拓扑向量空间中非光滑向量极值问题的最优性条件与对偶

本文提出了向量值函数的锥D-s凸，锥D-s拟凸，s右导数及锥D-s伪凸等新概念，探讨了锥D-s凸函数的有关性质，建立了带约束非光滑向量极值问题(VP)的最优性必要条件与涉及锥D-s凸(拟凸，伪凸)函数的约束极值问题(VP)的最优性充分条件，给出了原问题(VP)与其Mond-Weir型对偶问题的弱对偶与强对偶结论，揭示了(VP)的局部锥D-(弱)有效解与整体锥D-(弱)有效解，(VP)的锥D-弱有效解与锥D-有效解的关系，所得结果拓广了凸规划及部分广义凸规划的有关结论．     

约束向量优化问题的Fenchel-Lagrange对偶及鞍点

The aim of this paper is to apply a perturbation approach to deal with Fenchel- Lagrange duality based on weak efficiency to a constrained vector optimization problem. Under the stability criterion, some relationships between the solutions of primal problem and the Fenchel-Lagrange duality are discussed. Moreover, under the same condition, two saddle-points theorems are proved.     

不完全拉格朗日函数和具有(F α,ρ,d)凸性的数学规划的鞍点判别准则

用一个不完全拉格朗日函数研究一类具有(F,α,ρ,d)凸性假设的非线性规划问题的鞍点最优性判别准则,为了得到最优解与鞍点之间的关系,给出了(F,α,ρ,d)凸性中参数所要满足的条件．     

超有效元意义下集值优化的最优性条件

该文在Hausdorff局部凸拓扑向量空间考虑约束集值优化问题(SOP)在超有效意义下的Fritz John条件和Kuhn-Tucker条件.首先借助集值映射的下半可微的概念给出这种空间中集值映射导数的定义, 据此讨论了超有效元的Fritz John最优性条件.最后, 给出约束集值优化问题(SOP)取得超有效元的充分条件.     

Optimality Conditions for Vector Optimization Problems

In this paper, some necessary and sufficient optimality conditions for the weakly efficient solutions of vector optimization problems (VOP) with finite equality and inequality constraints are shown by using two kinds of constraints qualifications in terms of the MP subdifferential due to Ye. A partial calmness and a penalized problem for the (VOP) are introduced and then the equivalence between the weakly efficient solution of the (VOP) and the local minimum solution of its penalized problem is proved under the assumption of partial calmness. This work was supported by the National Natural Science Foundation of China (10671135), the Specialized Research Fund for the Doctoral Program of Higher Education (20060610005) and the National Natural Science Foundation of Sichuan Province (07ZA123). The authors thank Professor P.M. Pardalos and the referees for comments and suggestions.     

Optimality Conditions for Vector Optimization Problems of a Difference of Convex Mappings

In this paper, we study optimality conditions for vector optimization problems of a difference of convex mappings

On First Order Optimality Conditions for Vector Optimization

We develop first order optimality conditions for constrained vector optimization. The partial orders for the objective and the constraints are induced by closed and convex cones with nonempty interior. After presenting some well known existence results for these problems, based on a scalarization approach, we establish necessity of the optimality conditions under a Slater-like constraint qualification, and then sufficiency for the K-convex case. We present two alternative sets of optimality conditions, with the same properties in connection with necessity and sufficiency, but which are different with respect to the dimension of the spaces to which the dual multipliers belong. We introduce a duality scheme, with a point-to-set dual objective, for which strong duality holds. Some examples and open problems for future research are also presented,     

生成锥内部凸-锥-类凸集值优化问题的超有效性

本文讨论生成锥内部凸-锥-类凸集值向量优化问题的超有效解.在生成锥内部凸-锥类凸假设下,建立了集值向量优化问题在超有效意义下的标量化、Lagrangian乘子和鞍点定理