$L^0$--convex compactness and its applications to random convex optimization and random variational inequalities

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.