Nonlinear Distributionally Robust Optimization
Mohammed Rayyan Sheriff Peyman Mohajerin Esfahani
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 non-linear 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 non-linear 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
non-linear DRO problem. Finally, for the minimum variance portfolio selection
problem we analyze the regularity conditions and compute the FW-oracle in
various settings, and validate the theoretical results numerically.