L-0-convex compactness and its applications to random convex optimization and random variational inequalities
L-0-convex compactness and its applications to random convex optimization and random variational inequalities
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L-0-凸紧致性及其在随机凸优化和随机变分不等式中的应用
DOI：
10.1080/02331934.2020.1727901
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发表时间：
2020
期刊：
影响因子：
2.2
通讯作者：
Wu Mingzhi
中科院分区：
文献类型：
--
作者：
Guo Tiexin;Zhang Erxin;Wang Yachao;Wu Mingzhi
本文首先引入了拓扑代数上一类特殊的闭凸子集--Hausdorff拓扑模的闭凸子集的-凸紧性的概念，这里是从概率空间到实数或复数的标量域K的随机变量等价类的代数，具有概率收敛的拓扑。然后，本文通过建立一类重要拓扑模-完备随机赋范模的闭凸子集的各种关于凸紧性的刻画定理，继续发展了-凸紧性理论，特别地，我们充分利用随机共轭空间的理论建立了完备随机赋范模的闭凸子集的James型关于凸紧性的刻画定理，这也令人惊讶地暗示我们的-凸紧性的概念在完备随机赋范模的闭凸子集的背景下与GordanŽitković的凸紧性概念一致。作为我们结果的第一个应用，我们给出了随机凸优化(或称-凸优化)的一个基本定理，它包含了Hansen和Richard的著名结果作为特例。作为第二个应用，我们给出了一个随机变分不等式解的存在定理，它将H.Brezis的经典结果从自反Banach空间推广到随机自反完备随机赋范模。应该强调的是，由于通常的弱紧性方法在这种情况下不再适用，因此为第二种应用提出了一种新的方法，即-凸紧性方法。此外，我们关于随机凸优化的基本定理也可以应用于条件凸风险度量的优化问题的研究，这将在以后的论文中给出。
First, this paper introduces the notion of-convex compactness for a special class of closed convex subsets–closed-convex subsets of a Hausdorff topological module over the topological algebra, whereis the algebra of equivalence classes of random variables from a probability spaceto the scalar fieldKof real numbers or complex numbers, endowed with the topology of convergence in probability. Then, this paper continues to develop the theory of-convex compactness by establishing various kinds of characterization theorems on-convex compactness for closed-convex subsets of a class of important topological modules – complete random normed modules, in particular, we make full use of the theory of random conjugate spaces to establish the characterization theorem of James type on-convex compactness for a closed-convex subset of a complete random normed module, which also surprisingly implies that our notion of-convex compactness coincides with Gordan Žitković's notion of convex compactness in the context of a closed-convex subset of a complete random normed module. As the first application of our results, we give a fundamental theorem on random convex optimization (or,-convex optimization), which includes Hansen and Richard's famous result as a special case. As the second application, we give an existence theorem of solutions of random variational inequalities, which generalizes H. Brezis' classical result from a reflexive Banach space to a random reflexive complete random normed module. It should be emphasized that a new method, namely the-convex compactness method, is presented for the second application since the usual weak compactness method is no longer applicable in the present case. Besides, our fundamental theorem on random convex optimization can be also applied in the study of optimization problems of conditional convex risk measures, which will be given in our future papers.