  This article focuses on a class of distributionally robust optimization (DRO)
problems where, unlike the growing body of the literature, the objective
function is potentially nonlinear in the distribution. Existing methods to
optimize nonlinear functions in probability space use the Frechet derivatives,
which present both theoretical and computational challenges. Motivated by this,
we propose an alternative notion for the derivative and corresponding
smoothness based on Gateaux (G)-derivative for generic risk measures. These
concepts are explained via three running risk measure examples of variance,
entropic risk, and risk on finite support sets. We then propose a G-derivative
based Frank-Wolfe (FW) algorithm for generic nonlinear optimization problems in
probability spaces and establish its convergence under the proposed notion of
smoothness in a completely norm-independent manner. We use the set-up of the FW
algorithm to devise a methodology to compute a saddle point of the nonlinear
DRO problem. Finally, we validate our theoretical results on two cases of the
entropic and variance risk measures in the context of portfolio selection
problems. In particular, we analyze their regularity conditions and "sufficient
statistic", compute the respective FW-oracle in various settings, and confirm
the theoretical outcomes through numerical validation.
