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 both 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, the paper models ambiguity by decision-dependent collections of cumulative distribution functions viewed as subsets of a metric space of upper semicontinuous functions. We derive a series of results for this setting including estimates of the metric, the hypo-distance, and a new proof of the equivalence with weak convergence. Second, we utilize the theory of lopsided convergence to establish existence, convergence, and approximation of solutions of optimization problems with stochastic ambiguity. For the first time, we estimate the lop-distance between bifunctions and show that this leads to bounds on the solution quality for problems with stochastic ambiguity. Among other consequences, these results facilitate the study of the “price of robustness” and related quantities.