  This paper focuses on the contextual optimization problem where a decision is
subject to some uncertain parameters and covariates that have some predictive
power on those parameters are available before the decision is made. More
specifically, we focus on solving the Wasserstein-distance-based
distributionally robust optimization (DRO) model for the problem, which
maximizes the worst-case expected objective over an uncertainty set including
all distributions closed enough to a nominal distribution with respect to the
Wasserstein distance. We develop a stochastic gradient descent algorithm based
on the idea of data augmentation to solve the model efficiently. The algorithm
iteratively a) does a bootstrapping sample from the nominal distribution; b)
perturbs the adversarially and c) updates decisions. Accordingly, the
computational time of the algorithm is only determined by the number of
iterations and the complexity of computing the gradient of a single sample.
Except for efficiently solving the model, the algorithm provide additional
advantages that the proposed algorithm can cope with any nominal distributions
and therefore is extendable to solve the problem in an online setting. We also
prove that the algorithm converges to the optimal solution of the DRO model at
a rate of a $O(1/\sqrt{T})$, where $T$ is the number of iterations of
bootstrapping. Consequently, the performance guarantee of the algorithm is that
of the DRO model plus $O(1/\sqrt{T})$. Through extensive numerical experiments,
we demonstrate the superior performance of the proposed algorithm to several
benchmarks.
