  Distributionally robust optimization (DRO) has been introduced for solving
stochastic programs where the distribution of the random parameters is unknown
and must be estimated by samples from that distribution. A key element of DRO
is the construction of the ambiguity set, which is a set of distributions that
covers the true distribution with a high probability. Assuming that the true
distribution has a probability density function, we propose a class of
ambiguity sets based on confidence bands of the true density function. The use
of the confidence band enables us to take the prior knowledge of the shape of
the underlying density function into consideration (e.g., unimodality or
monotonicity). Using the confidence band constructed by density estimation
techniques as the ambiguity set, we establish the convergence of the optimal
value of DRO to that of the stochastic program as the sample size increases.
However, the resulting DRO problem is computationally intractable, as it
involves functional decision variables as well as infinitely many constraints.
To address this challenge, using the duality theory, we reformulate it into a
finite-dimensional stochastic program, which is amenable to a stochastic
subgradient scheme as a solution method. We compare our approach with existing
state-of-the-art DRO methods on the newsvendor problem and the portfolio
management problem, and the numerical results showcase the advantage of our
approach.
