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In 2010, Gordan \v{Z}itkovi\'{c} introduced the notion of convex compactnessfor a convex subset of a linear topological space and gave some importantapplications to both nonlinear analysis and mathematical economics in [ Gordan\v{Z}itkovi\'{c}, Convex compactness and its applications, Math. FinanceEconom. 3(1) (2010) 1--12 ]. Motivated by Gordan \v{Z}itkovi\'{c}'s idea, inthis 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 equivalenceclasses of random variables from a probability space $(\Omega,\mathcal{F},P)$to the scalar field $K$ of real numbers or complex numbers, endowed with thetopology of convergence in probability. This paper continues to develop thetheory of $L^0$--convex compactness by establishing various kinds ofcharacterization theorems for $L^0$--convex subsets of a class of importanttopological modules--complete random normed modules, in particular, we make useof the theory of random conjugate spaces to give a characterization theorem ofJames type for a closed $L^0$--convex subset of a complete random normedmodule. As applications, we successfully generalize some basic theorems ofclassical convex optimization and variational inequalities from a convexfunction on a reflexive Banach space to an $L^0$--convex function on a randomreflexive random normed module. Since the usual weak compactness method failsin the random setting of this paper and in particular, since the difficultiescaused by the partial order structure of the range of an $L^0$--valued functionalso frequently occurs in the study of problems involved in this paper, we areforced to discover a series of new skills to meet the needs of this paper.

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