Stochastic linear optimization under partial uncertainty and incomplete information using the notion of probability multimeasure

Abstract

A scalar stochastic linear optimization problem under linear constraints is studied, introducing the concept of a probability multimeasure on the underlying probability space. This allows deriving a deterministic equivalent formulation, transforming the original stochastic problem into a set‑valued optimization problem. The authors develop methods to estimate expected values w.r.t. a probability multimeasure, and extend classic results like the strong law of large numbers, the Glivenko–Cantelli theorem, and the central limit theorem to this setting. They also define sampling procedures tied to probability multimeasures and introduce the notion of cumulative distribution multifunction, discussing its properties in the deterministic counterpart problem