  In this paper, we consider a distributionally robust optimization (DRO) model
in which the ambiguity set is defined as the set of distributions whose
Kullback-Leibler (KL) divergence to an empirical distribution is bounded.
Utilizing the fact that KL divergence is an exponential cone representable
function, we obtain the robust counterpart of the KL divergence constrained DRO
problem as a dual exponential cone constrained program under mild assumptions
on the underlying optimization problem. The resulting conic reformulation of
the original optimization problem can be directly solved by a commercial conic
programming solver. We specialize our generic formulation to two classical
optimization problems, namely, the Newsvendor Problem and the Uncapacitated
Facility Location Problem. Our computational study in an out-of-sample analysis
shows that the solutions obtained via the DRO approach yield significantly
better performance in terms of the dispersion of the cost realizations while
the central tendency deteriorates only slightly compared to the solutions
obtained by stochastic programming.
