On random convex analysis
Tiexin Guo, Erxin Zhang, Mingzhi Wu, Bixuan Yang, George Yuan and, Xiaolin Zeng

TL;DR
This paper advances the theoretical foundation of random convex analysis by establishing duality, continuity, and differentiability properties of convex functions over random locally convex modules, addressing complex topological challenges.
Contribution
It proves the Fenchel--Moreau duality theorem, explores various forms of continuity and differentiability, and links subdifferentials with variational principles in the context of random convex analysis.
Findings
Established the inferior limit behavior of $L^0$--convex functions.
Proved relationships among different notions of continuity.
Linked subdifferentiability with variational principles.
Abstract
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 --convex function on a…
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Taxonomy
TopicsOptimization and Variational Analysis · Risk and Portfolio Optimization · Fuzzy Systems and Optimization
