Variational Theory for Optimization under Stochastic Ambiguity

Stochastic ambiguity provides a rich class of uncertainty models that includes those in stochastic, robust, risk-based, and semi-infinite optimization and that accounts for uncertainty about parameter values as well as incompleteness of the description of uncertainty. We provide a novel, unifying perspective on optimization under stochastic ambiguity that rests on two pillars. First, ambiguity is formulated in terms of the (cumulative) probability distribution associated with the random elements; more specifically, ambiguity is expressed by letting this distribution belong to a subfamily of distributions that might, or might not, depend on the decision variable. We derive a series of estimates by introducing a metric for the space of distribution functions based on the hypo-distance between upper semicontinuous functions. In the process, we show that this metric is consistent with convergence in distribution (= weak$^\star$ convergence) of the associated probability measures. Second, we rely on the theory...