On Random Convex Analysis
On Random Convex Analysis
复制标题
关于随机凸分析
DOI：
--
复制
发表时间：
2016-03
期刊：
影响因子：
--
通讯作者：
Xiaolin Zeng
中科院分区：
文献类型：
--
作者：
Mingzhi Wu;Bixuan Yang;George Yuan;Xiaolin Zeng
近年来，基于随机空间理论的思想，随机凸分析得到了发展，以处理随机环境中的相应问题，如条件凸风险测度分析以及相关的变分问题和优化问题。随机凸分析是随机局部凸模上的凸分析。由于随机局部凸模具有比一般的局部凸空间更复杂的拓扑和代数结构，建立随机凸分析将面临比经典凸分析更困难的数学挑战，因此随机凸分析仍有许多重要的基本问题未解决.本文致力于解决一些重要的理论问题。首先，我们在具有局部L^0 $-凸拓扑的随机局部凸模上建立了真下连续L^0 $-凸函数的下极限性质，完善了这类函数的Fenchel-Moreau对偶定理.然后，我们研究了真L^0 $-凸函数的连续性、局部L^0 $-Lipschitz连续性和几乎处处L^0 $-Lipschitz连续性之间的关系。然后，我们建立了定义在随机赋范模上的真L^0 $-凸函数的次可微性、Gateaux-可微性和Frechet-可微性之间的优美关系.最后，基于适当的下半连续$\bar{L}^0$-值函数的Ekeland变分原理，证明了$\vareps $-次微分可以用次微分来逼近.我们要强调的是，本文的成功在于同时考虑了随机局部凸模的$（\vareps，\lambda）$--拓扑和局部L^0 $--凸拓扑。
Recently, based on the idea of randomizing space theory, random convex analysis has been being developed in order to deal with the corresponding problems in random environments such as analysis of conditional convex risk measures and the related variational problems and optimization problems. Random convex analysis is convex analysis over random locally convex modules. Since random locally convex modules have the more complicated topological and algebraic structures than ordinary locally convex spaces, establishing random convex analysis will encounter harder mathematical challenges than classical convex analysis so that there are still a lot of fundamentally important unsolved problems in random convex analysis. This paper is devoted to solving some important theoretic problems. First, we establish the inferior limit behavior of a proper lower semicontinuous $L^0$--convex function on a random locally convex module endowed with the locally $L^0$--convex topology, which makes perfect the Fenchel--Moreau duality theorem for such functions. Then, we investigate the relations among continuity, locally $L^0$--Lipschitzian continuity and almost surely sequent continuity of a proper $L^0$--convex function. And then, we establish the elegant relationships among subdifferentiability, G\^ateaux--differentiability and Fr\'ech\'et--differentiability for a proper $L^0$--convex function defined on random normed modules. At last, based on the Ekeland's variational principle for a proper lower semicontinuous $\bar{L}^0$--valued function, we show that $\varepsilon$--subdifferentials can be approximated by subdifferentials. We would like to emphasize that the success of this paper lies in simultaneously considering the $(\varepsilon, \lambda)$--topology and the locally $L^0$--convex topology for a random locally convex module.
登录
查看更多内容
随机局部凸模块的两种拓扑得出的一些基本结果之间的关系
影响因子：
1.7
作者：
郭铁信
通讯作者：
郭铁信
DOI：
10.1090/memo/0625
发表时间：
1998-03
期刊：
--
影响因子：
--
作者：
G. Yuan
通讯作者：
G. Yuan
DOI：
10.1137/1.9781611971088
发表时间：
1976
期刊：
--
影响因子：
--
作者：
I. Ekeland;R. Téman
通讯作者：
I. Ekeland;R. Téman
DOI：
10.1007/978-3-662-48670-2_6
发表时间：
2012-11
期刊：
arXiv: Functional Analysis
影响因子：
--
作者：
Patrick Cheridito;Michael Kupper;Nicolas Vogelpoth
通讯作者：
Patrick Cheridito;Michael Kupper;Nicolas Vogelpoth
DOI：
--
发表时间：
2014-11
期刊：
arXiv: Functional Analysis
影响因子：
--
作者：
J. M. Zapata
通讯作者：
J. M. Zapata