  Optimization of distortion riskmetrics with distributional uncertainty has
wide applications in finance and operations research. Distortion riskmetrics
include many commonly applied risk measures and deviation measures, which are
not necessarily monotone or convex. One of our central findings is a unifying
result that allows us to convert an optimization of a non-convex distortion
riskmetric with distributional uncertainty to a convex one, leading to great
tractability. A sufficient condition to the unifying equivalence result is the
novel notion of closedness under concentration, a variation of which is also
shown to be necessary for the equivalence. Our results include many special
cases that are well studied in the optimization literature, including but not
limited to optimizing probabilities, Value-at-Risk, Expected Shortfall, Yaari's
dual utility, and differences between distortion risk measures, under various
forms of distributional uncertainty. We illustrate our theoretical results via
applications to portfolio optimization, optimization under moment constraints,
and preference robust optimization.
