In 2010, Gordan \v{Z}itkovi\'{c} introduced the notion of convex compactness for a convex subset of a linear topological space and gave some important applications to both nonlinear analysis and mathematical economics in [ Gordan \v{Z}itkovi\'{c}, Convex compactness and its applications, Math. Finance Econom. 3(1) (2010) 1--12 ]. Motivated by Gordan \v{Z}itkovi\'{c}'s idea, in this paper we introduce the notion of $L^0$--convex compactness for an $L^0$--convex subset of a topological module over the topological algebra $L^0(\mathcal{F},K)$, where $L^0(\mathcal{F},K)$ is the algebra of equivalence classes of random variables from a probability space $(\Omega,\mathcal{F},P)$ to the scalar field $K$ of real numbers or complex numbers, endowed with the topology of convergence in probability. This paper continues to develop the theory of $L^0$--convex compactness by establishing various kinds of characterization theorems for $L^0$--convex subsets of a class of important topological modules--complete random normed modules, in particular, we make use of the theory of random conjugate spaces to give a characterization theorem of James type for a closed $L^0$--convex subset of a complete random normed module. As applications, we successfully generalize some basic theorems of classical convex optimization and variational inequalities from a convex function on a reflexive Banach space to an $L^0$--convex function on a random reflexive random normed module. Since the usual weak compactness method fails in the random setting of this paper and in particular, since the difficulties caused by the partial order structure of the range of an $L^0$--valued function also frequently occurs in the study of problems involved in this paper, we are forced to discover a series of new skills to meet the needs of this paper.

中文翻译：

---

L0-凸紧致性及其在随机凸优化和随机变分不等式中的应用

2010 年，Gordan \v{Z}itkovi\'{c} 为线性拓扑空间的凸子集引入了凸紧性的概念，并在 [ Gordan \v{Z} itkovi\'{c}，凸紧性及其应用，数学。金融经济学。3(1) (2010) 1--12 ]。受 Gordan \v{Z}itkovi\'{c} 思想的启发，在本文中，我们引入了 $L^0$--凸紧致性的概念，用于 $L^0$--拓扑模块的凸子集在拓扑代数 $L^0(\mathcal{F},K)$ 上，其中 $L^0(\mathcal{F},K)$ 是来自概率空间 $(\ Omega,\mathcal{F},P)$ 到实数或复数的标量场 $K$，具有概率收敛的拓扑结构。本文继续发展$L^0$--凸紧性理论，通过建立$L^0$--一类重要拓扑模的凸子集--完全随机赋范模的各种表征定理，特别是，我们利用随机共轭空间理论，给出了一个完全随机赋范模的闭$L^0$--凸子集的James型表征定理。作为应用，我们成功地将经典凸优化和变分不等式的一些基本定理从自反 Banach 空间上的凸函数推广到随机自反随机赋范模块上的 $L^0$--凸函数。由于通常的弱紧致方法在本文的随机设置中失败，特别是，