  Distributionally robust optimization (DRO) is a widely used framework for
optimizing objective functionals in the presence of both randomness and
model-form uncertainty. A key step in the practical solution of many DRO
problems is a tractable reformulation of the optimization over the chosen model
ambiguity set, which is generally infinite dimensional. Previous works have
solved this problem in the case where the objective functional is an expected
value. In this paper we study objective functionals that are the sum of an
expected value and a variance penalty term. We prove that the corresponding
variance-penalized DRO problem over an $f$-divergence neighborhood can be
reformulated as a finite-dimensional convex optimization problem. This result
also provides tight uncertainty quantification bounds on the variance.
