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In this work, we present algorithmically tractable safe approximations ofdistributionally robust optimization (DRO) problems. The considered ambiguitysets can exploit information on moments as well as confidence sets. Typically,reformulation approaches using duality theory need to make strong assumptionson the structure of the underlying constraints, such as convexity in thedecisions or concavity in the uncertainty. In contrast, here we present aduality-based reformulation approach for DRO problems, where the objective ofthe adverserial is allowed to depend on univariate indicator functions. Thisrenders the problem nonlinear and nonconvex. In order to be able to reformulatethe semiinfinite constraints nevertheless, an exact reformulation is presentedthat is approximated by a discretized counterpart. The approximation isrealized as a mixed-integer linear problem that yields sufficient conditionsfor distributional robustness of the original problem. Furthermore, it isproven that with increasingly fine discretizations, the discretizedreformulation converges to the original distributionally robust problem. Theapproach is made concrete for a challenging, fundamental task in particleseparation that appears in material design. Computational results for realisticsettings show that the safe approximation yields robust solutions ofhigh-quality and can be computed within short time.

A Safe Approximation Based on Mixed-Integer Optimization for Non-Convex Distributional Robustness Governed by Univariate Indicator Functions