Painlevé-Kuratowski Convergences of the Solution Sets for Perturbed Vector Equilibrium Problems without Monotonicity

In this paper, we obtain some stability results for perturbed vector equilibrium problems. Under new assumptions, which are weaker than the assumption of C-strict monotonicity, we provide sufficient conditions for the Painlev^-Kuratowski Convergence of the weak efficient solution sets and efficient solution sets for the perturbed vector equilibrium problems with a sequence of mappings converging in real linear metric spaces. These results extend and improve some known results in the literature.

Painlev-Kuratowski Convergences of the Solution Sets to Perturbed Generalized Systems

In this paper, we obtain the Painleve-Kuratowski Convergence of the efficient solution sets, the weak efficient solution sets and various proper efficient solution sets for the perturbed generalized system with a sequence of mappings converging in a real locally convex Hausdorff topological vector spaces.

一类带映射差的非凸向量优化问题解的稳定性

在实际生活中,问题数据常常受到干扰,求原问题解时,常常利用近似问题解去逼近原问题解。使用这种方法进行求解时,原问题解集的稳定性是一个重要的前提条件。本文考虑一类带映射差的非凸向量优化问题,研究了近似问题数据收敛于原问题数据时,通过对映射差的两个映射凸性和收敛性的限制,获得了带映射差的非凸向量优化问题在Painlevé-Kuratowski收敛性意义下有效解的稳定性结果。

集值优化问题E-Henig有效解的稳定性

改善集下的Henig有效解统一了Henig有效解和近似Henig有效解,其稳定性分析在数值计算中不可或缺,同时集值优化问题是当前优化领域研究的热点问题,研究基于改善集下的集值优化问题E-Henig有效解的稳定性具有重要的理论意义和实用价值。首先,针对集值优化问题,基于改善集的概念,引入集值优化问题的E-Henig有效解,统一了集值优化问题近似Henig有效解和Henig有效解;其次,在集值优化问题目标映射和约束条件均扰动的情况下,借助Painlevé-Kuratowski收敛性,建立集值映射水平集的闭凸性、有界性及回收锥的相关性质;然后,借助所获得的集值映射水平集的闭凸性、有界性及回收锥的性质,在集值优化问题目标映射和约束条件均扰动的情况下,分别建立严格真拟C-凸集值优化问题E-弱有效点集、E-Henig有效点集和E-Henig有效解的稳定性结果。所得结果首次聚焦于集值优化问题基于改善集概念下的弱有效点集、Henig有效点集及Henig有效解集的稳定性结果,相较于以往文献大都只关注集值优化问题Henig有效解的存在性、最优性条件、对偶性性质,大大完善了集值优化问题Henig有效解的理论结果,并为数值计算的稳定性分析提供了方法和技巧。

向量优化问题有效点集的稳定性

在不需要紧性假设下,利用拟C-凸函数及回收锥的性质,建立了向量优化问题有效点集的稳定性,获得了一列目标函数和可行集均扰动情形下的向量优化问题与对应的向量优化问题有效点集的Painlevé-Kuratowski内收敛性结果。所得结果推广和改进了相关文献(Attouch H,RiahiH.Stability results for Ekeland’s-variational principle and cone extremal solution;Huang X X.Stabilityin vector-valued and set-valued optimization)中的相应结果,并给出例子说明了所得结果的正确性。

有效点和弱有效点的稳定性

利用集合序列的P-K收敛的概念,在控制锥是正则锥但未必具有有界基的情况下,讨论了向量值优化和集值优化问题的有效点的稳定性,改进了[2]中相应的结论。另外,还给出了向量值优化和集值优化问题的弱有效点的稳定性,得到了线性算子下的稳定性结果。

向量优化问题Benson真有效解的稳定性

首先利用非线性标量化技术,建立了向量优化问题Benson真有效解与一类标量优化问题解之间的等价关系.然后借助建立的等价性结果,在向量优化问题目标映射和约束条件均扰动的情况下,得到了向量优化问题Benson真有效点集和解集的抗干扰稳定性结果.该结果首次借助标量化技术,在扰动问题序列Painleve-Kuratowski收敛于目标优化问题的情况下,研究了向量优化问题Benson真有效解的抗干扰性,该结果对数值计算分析有重要的理论价值.

向量优化问题的几个收敛性结果(英文)

在本文中,获得了集合的弱有效元与真有效元的几个收敛性结果.然后,讨 论了集值映射向量优化问题(VP)和它的近似问题(VP)_n,在较强的假设条件下,获得了 (VP)_n的真有效解的几个收敛性结果.

近似平衡约束向量优化问题解集的上Painlevé-Kuratowski收敛性

主要研究近似平衡约束向量优化问题解集的稳定性．考虑了一类近似平衡约束向量优化问题，在目标函数列Gamma-收敛，而不必是连续收敛情形下，分别获得近似平衡约束向量优化问题有效解集、弱有效解集和强有效解集的上Painleve-Kuratowski收敛性，所得结果推广和改进了最近文献的结果．

局部有限超空间的弱紧性和第一可数性

研究了拓扑空间X上的非空闭子集超空间CL(X)的Kuratowski-Painlevé-收敛与τlocfin-收敛的等价性,给出了CL(X)赋予局部有限拓扑τlocfin的三类弱紧性:ω-有界性,D-紧性和D-伪紧性,利用空间X的分解方法得到了(CL(X),τlocfin)满足第一可数公理的等价证明.

集值优化问题近似Henig真有效点的稳定性

针对集值优化问题近似Henig真有效点,提出在目标集值优化问题的映射及可行域均扰动的情形下,建立C凸集值优化问题近似Henig真有效点的稳定性结果,将近似Henig真有效点的稳定性研究从向量值优化问题推广到集值优化问题中。首先给出集值映射序列Γ_(C)收敛的概念,比较了集值映射序列Painlevé-Kuratowski收敛与Γ_(C)收敛性之间收敛强弱的关系,发现Painlevé-Kuratowski收敛弱于Γ_(C)收敛;其次将Painlevé-Kuratowski收敛应用于建立近似Henig真有效点的稳定性结果中,在扰动集值优化问题的问题数据Painlevé-Kuratowski收敛到目标集值优化问题的问题数据情况下,获得了集值优化问题近似Henig真有效点的抗干扰稳定性结果,该结果对数值计算分析中集值优化问题近似Henig真有效点的稳定性研究有着重要的理论分析价值。

自由支配集下近似平衡约束向量优化问题的稳定性研究

在自由支配集下,对一类近似平衡约束向量优化问题(AOPVF)的稳定性进行研究.首先,在较弱的凸性假设下获得了约束集映射的Berge-半连续性和约束集的闭性、凸性和紧性结果.然后,在目标函数列Gamma-收敛的假设下,分别得到了AOPVF弱有效解映射Berge-半连续和弱有效解集下Painlevé-Kuratowski收敛的充分条件,并给出例子说明结论是新颖和有意义的.

More Compactification for Differential Systems

This article is a review and promotion of the study of solutions of differential equations in the “neighborhood of infinity” via a non traditional compactification. We define and compute critical points at infinity of polynomial autonomuos differential systems and develop an explicit formula for the leading asymptotic term of diverging solutions to critical points at infinity. Applications to problems of completeness and incompleteness (the existence and nonexistence respectively of global solutions) of dynamical systems are provided. In particular a quadratic competing species model and the Lorentz equations are being used as arenas where our technique is applied. The study is also relevant to the Painlevé property and to questions of integrability of dynamical systems.

Painlevé-Kuratowski Stability of the Solution Sets to Perturbed Vector Generalized Systems

In this paper, stability results of solution mappings to perturbed vector generalized system are studied. Firstly, without the assumption of monotonicity, the Painleve-Kuratowski convergence of global efficient solution sets of a family of perturbed problems to the corresponding global efficient solution set of the generalized system is obtained, where the perturbations are performed on both the objective function and the feasible set. Then, the density and Painleve-Kuratowski convergence results of efficient solution sets are established by using gamma convergence, which is weaker than the assumption of continuous convergence. These results extend and improve the recent ones in the literature.

Convergence,Scalarization and Continuity of Minimal Solutions in Set Optimization

The paper deals with the study of two different aspects of stability in the given space as well as the image space,where the solution concepts are based on a partial order relation on the family of bounded subsets of a real normed linear space.The first aspect of stability deals with the topological set convergence of families of solution sets of perturbed problems in the image space and Painlevé–Kuratowski set convergence of solution sets of the perturbed problems in the given space.The convergence in the given space is also established in terms of solution sets of scalarized perturbed problems.The second aspect of stability deals with semicontinuity of the solution set maps of parametric perturbed problems in both the spaces.

集合优化问题解集的稳定性和扩展适定性

研究了集合优化问题近似解集的稳定性和扩展适定性。当目标函数具连续性和广义锥-拟凸性时,讨论了集合优化问题近似解集的Painlevé-Kuratowski上收敛和下收敛。结合适当假设条件,分析了集合优化问题的扩展适定性和弱扩展适定性。

关于半无限优化问题稳定性的注记

【目的】对半无限向量优化问题约束集映射的稳定性进行研究。【方法】首先,举例说明Peng等人在2018年的一篇文献中的相关结论是有缺陷的。然后借助函数的自然拟C-凸性,分别获得了约束集映射的Berge-下半连续性和约束集的Painlevé-Kuratowski收敛性。【结果】基于函数的自然拟C-凸性,获得了约束集映射的稳定性结果。【结论】克服了原文献中相关定理的缺陷,并举例说明所得的新结论改进了相应结果。