  Data-driven distributionally robust optimization is a recently emerging
paradigm aimed at finding a solution that is driven by sample data but is
protected against sampling errors. An increasingly popular approach, known as
Wasserstein distributionally robust optimization (DRO), achieves this by
applying the Wasserstein metric to construct a ball centred at the empirical
distribution and finding a solution that performs well against the most
adversarial distribution from the ball. In this paper, we present a general
framework for studying different choices of a Wasserstein metric and point out
the limitation of the existing choices. In particular, while choosing a
Wasserstein metric of a higher order is desirable from a data-driven
perspective, given its less conservative nature, such a choice comes with a
high price from a robustness perspective - it is no longer applicable to many
heavy-tailed distributions of practical concern. We show that this seemingly
inevitable trade-off can be resolved by our framework, where a new class of
Wasserstein metrics, called coherent Wasserstein metrics, is introduced. Like
Wasserstein DRO, distributionally robust optimization using the coherent
Wasserstein metrics, termed generalized Wasserstein distributionally robust
optimization (GW-DRO), has all the desirable performance guarantees:
finite-sample guarantee, asymptotic consistency, and computational
tractability. The worst-case expectation problem in GW-DRO is in general a
nonconvex optimization problem, yet we provide new analysis to prove its
tractability without relying on the common duality scheme. Our framework, as
shown in this paper, offers a fruitful opportunity to design novel Wasserstein
DRO models that can be applied in various contexts such as operations
management, finance, and machine learning.
