Variational Theory for Optimization under Stochastic Ambiguity
本文提出一个统一框架，通过概率分布族描述随机模糊性，并利用度量空间和lopsided收敛理论，研究优化问题的解的存在性、收敛性和近似，适用于处理参数不确定性和描述不完整性的优化场景。
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 of lopsided convergence to establish existence, convergence, and approximation of solutions of optimization problems with stochastic ambiguity. For the first time, we estimate a 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.