Data-driven Stochastic Programming with Distributionally Robust Constraints Under Wasserstein Distance: Asymptotic Properties

Worst-Case Higher Moment Coherent Risk Based on Optimal Transport with Application to Distributionally Robust Portfolio Optimization

“…The popular methods of constructing an ambiguity set basically focus on the moment-based ambiguity set, which is constructed by certain moment information (see, for example, [16][17][18][19][20][21][22]), and the metric-based one, which is defined as a "ball" in the sense of a certain probability metric such as the Prohorov metric (see [23]), the goodness-of-fit (see [24]), likelihood function (see [25]), φ-divergence (see, for example, [26][27][28][29][30]), Kullback-Leibler (KL) divergence (see, for example, [31,32]) and so forth. In particular, due to the outstanding properties of Wasserstein distance, defined as follows, Wasserstein distance is attracting a growing interest in DRO recently (see, for example, [33][34][35][36][37]).…”

“…where the ambiguity set P is defined as a ball in the space of probability distribution by using Wasserstein distance, which is centered at the discrete empirical probability distribution. It is worth noting that, in contrast to the expected (risk-neutral) performance of certain loss function in the objective of the general DRO model (see, for example, [33][34][35][36][37]), the objective of ( 4) involve the risk-aversion performance-mean-HMCR, which is a nonlinear functional with respect to the probability distribution P. The nonliearity of the objective of the inner infinite-dimensional maximization optimization problem gives rise to a challenge, that is, it is difficult to solve the resulting semi-infinite optimization problem obtained by duality theory. Hence, to the best of our knowledge, it is difficult to derive the tractable reformulation of (4).…”

Distributionally robust second-order stochastic dominance constrained optimization with Wasserstein ball

“…There are mainly two types of ambiguity sets in the existing literature. Moment-based ambiguity sets, whose resulting distributionally robust optimization problems have been widely studied [9,39], and distance-based ambiguity sets, which contain all the probability distributions close to some nominal distribution measured by some probability metrics, such as Kullback-Leibler divergence [24], φ-divergence [22], and Wasserstein distance [1,21,24,28,35]. Mohajerin Esfahani and Kuhn estimated the priori probability that the true distribution belongs to the Wasserstein ambiguity set and established finite sample and asymptotic guarantees for the distributionally robust solutions in [29].…”