集值优化问题在有效意义下的最优性条件

本文在赋范空间中,讨论集值优化问题的有效元导数型最优性条件.当目标映射和约束映射的下方向导数存在时,在近似锥次类凸假设下利用有效点的性质和凸集分离定理得到了集值优化问题有效元导数型Kuhn-Thcker必要条件,在可微Г-拟凸性的假设下得到了Kuhn-Tucker最优性充分条件;此外利用集值映射沿弱方向锥的导数的特性给出了有效解最优性的另一种刻画.     

一类集值映射向量优化问题的ε-有效解

集值映射向量优化问题是最优化理论中的一个重要方向.在集值映射为生成锥内部-锥一类凸(简记为ic-锥类凸)的假设条件下,利用择一定理,给出了集值映射向量优化问题ε-弱有效解和ε-有效解的最优性条件和ε-Lagrange乘子定理,是弱有效解和有效解相应结果的推广.     

非光滑向量均衡问题近似拟全局真有效解的最优性条件

本文研究一类非光滑向量均衡问题(Vector Equilibrium Problem)(VEP)关于近似拟全局真有效解的最优性条件.首先,利用凸集的拟相对内部型分离定理和Clarke次微分的性质,得到了问题(VEP)关于近似拟全局真有效解的最优性必要条件.其次,引入近似伪拟凸函数的概念,并给出具体实例验证其存在性,且在该凸性假设下建立了问题(VEP)关于近似拟全局真有效解的充分条件.最后,利用Tammer函数以及构建满足一定性质的非线性泛函,得到了问题(VEP)近似拟全局真有效解的标量化定理.     

弧连通锥-凸数学规划的最优性条件

在弧连通锥-凸假设下讨论Hausdorff局部凸空间中的一类数学规划的最优性条件问题.首先,利用择一定理得到了锥约束标量优化问题的一个必要最优性条件.其次,利用凸集分离定理证明了无约束向量优化问题关于弱极小元的标量化定理和一个一致的充分必要条件.所得结果深化和丰富了最优化理论及其应用的内容.     

近似次微分刻画的约束向量均衡问题的最优性条件北大核心CSCD

该文研究一类约束向量均衡问题(CVEP)近似拟弱有效解的最优性条件和对偶定理.首先,建立了问题(CVEP)近似拟弱有效解关于近似次微分形式的最优性必要条件.其次,引入了一种广义凸性的概念,称之为近似伪拟type-I函数,并在其假设下,获得了问题(CVEP)近似拟弱有效解的最优性充分条件.最后,引入了问题(CVEP)的广义近似Mond-Weir对偶模型,并建立其与原问题间关于近似拟弱有效解的对偶定理.     

向量优化中基于拟内部的弱C(\varepsilon)-有效解的标量化

首先获得了co-radiant集的一些拟内部性质. 进而在邻近C(\varepsilon)-次似凸性假设条件下, 建立了相应的择一性定理, 并给出了基于拟内部的集值向量优化问题弱C(\varepsilon)-有效解的线性标量化结果.     

向量优化中基于拟内部的弱C(ε)-有效解的标量化

首先获得了co-radiant集的一些拟内部性质.进而在邻近C(ε)-次似凸性假设条件下,建立了相应的择一性定理,并给出了基于拟内部的集值向量优化问题弱C(ε)-有效解的线性标量化结果.     

Benson真有效意义下向量集值优化的广义Fritz John条件

引入了一种有关集值映射的切导数和强、弱*伪凸的概念。借助凸集分离定理及锥分离定理建立了Benson真有效意义下向量集值优化导数型的FritzJohn最优性条件,并对条件的充分性进行了讨论。当特殊到单值映射时这些最优性条件与经典的结果完全吻合。     

Benson真有效意义下向量集值优化的广义Fritz-John条件

借助Clarke切锥并用上图引入了关于集值映射的Clarke切导数.借助于一种新的择一性定理建立了向量集值优化问题在弱Benson真有效意义下的广义Fritz-John最优性条件,而且证明在一种伪凸的假设下,这种最优性条件还为充分的.     

半凸多目标优化ε-拟弱有效解的最优性条件及对偶定理

本文在相关文献考虑MP问题的基础上,增加了等式约束条件,即本文考虑了VP问题,并将已有文献中的凸性假设改为半凸性假设,得到VP问题的ε-拟弱有效解的相应最优性条件.接着,本文定义了VP问题的拉格朗日函数及其ε-拟弱鞍点,得到VP问题的ε-拟弱鞍点相应定理.最后,本文考虑了VP问题的对偶问题,获得了VP问题的弱对偶和强对偶定理.     

Optimality Conditions of Generalized Subconvexlike Set-Valued Optimization Problems Based on the Quasi-Relative Interior

In this paper, firstly, a new generalized subconvexlike set-valued map based on the quasi-relative interior is introduced. Secondly, by a separation theorem involving the quasi-relative interior, some separation properties are obtained. Finally, some optimality conditions are established. Our results improve some results in the literature.     

Higher-order Karush–Kuhn–Tucker optimality conditions for set-valued optimization with nonsolid ordering cones

In this paper, we consider higher-order Karush–Kuhn–Tucker optimality conditions in terms of radial derivatives for set-valued optimization with nonsolid ordering cones. First, we develop sum rules and chain rules in the form of equality for radial derivatives. Then, we investigate set-valued optimization including mixed constraints with both ordering cones in the objective and constraint spaces having possibly empty interior. We obtain necessary conditions for quasi-relative efficient solutions and sufficient conditions for Pareto efficient solutions. For the special case of weak efficient solutions, we receive even necessary and sufficient conditions. Our results are new or improve recent existing ones in the literature.     

On optimality conditions for quasi-relative efficient solutions in set-valued optimization

In the paper, we establish necessary and sufficient optimality conditions for quasi-relative efficient solutions of a constrained set-valued optimization problem using the Lagrange multipliers. Many examples are given to show that our results and their applications are more advantageous than some existing ones in the literature.     

Set-Valued Systems with Infinite-Dimensional Image and Applications

In infinite-dimensional spaces, we investigate a set-valued system from the image perspective. By exploiting the quasi-interior and the quasi-relative interior, we give some new equivalent characterizations of (proper, regular) linear separation and establish some new sufficient and necessary conditions for the impossibility of the system under new assumptions, which do not require the set to have nonempty interior. We also present under mild assumptions the equivalence between (proper, regular) linear separation and saddle points of Lagrangian functions for the system. These results are applied to obtain some new saddle point sufficient and necessary optimality conditions of vector optimization problems.     

非凸向量集值优化Benson真有效解的最优性条件与对偶

在无需偏序锥内部非空的情况下给出了非凸约束向量集值优化Benaon真有效解一种加细的最优性条件，并建立了向量集值优化Benson真有效解一种改进的Lagrange乘子型对偶，它比已有的Lagrange乘子型对偶具有较好的对偶性。     

较多有效解类的最优性条件和Lagrange对偶定理

利用较多锥的内部和闭包,引进多目标规划问题的严格强较多有效解等概念.根据它们的表示定理,建立各类较多有效解的最优性条件,并由此得到严格弱较多有效解的Lagrange直接对偶和严格强较多有效解的Lagrange逆对偶定理.     

Quasi-relative interior-type constraint qualifications ensuring strong Lagrange duality for optimization problems with cone and affine constraints

In this article we provide weak sufficient strong duality conditions for a convex optimization problem with cone and affine constraints, stated in infinite dimensional spaces, and its Lagrange dual problem. Our results are given by using the notions of quasi-relative interior and quasi-interior for convex sets. The main strong duality theorem is accompanied by several stronger, yet easier to verify in practice, versions of it. As exemplification we treat a problem which is inspired from network equilibrium. Our results come as corrections and improvements to Daniele and Giuffré (2007) [9].     

Efficient solutions and optimality conditions for vector equilibrium problems

Necessary optimality conditions for efficient solutions of unconstrained and vector equilibrium problems with equality and inequality constraints are derived. Under assumptions on generalized convexity, necessary optimality conditions for efficient solutions become sufficient optimality conditions. Note that it is not required here that the ordering cone in the objective space has a nonempty interior.