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

In this work, we present an algorithmically tractable safe approximation of distributionally robust optimization (DRO) problems that contain univariate indicator functions. The latter appear in different applications, but render the model nonlinear and nonconvex. The considered ambiguity sets can exploit moment information. Typically, reformulation approaches using duality theory need to make strong assumptions on the structure of the underlying constraints, such as convexity in the decisions or concavity in the uncertainty which cannot be assumed in our setting. We nevertheless present an equivalent semi-infinite reformulation that is subsequently approximated by a discretized counterpart. Under mild assumptions, the latter provides a safe approximation that is formulated as a tractable mixed-integer linear problem, which can be solved by available standard software. Obtained solutions are guaranteed to be feasible for the original distributionally robust problem. Although we show that in general convergence to the true DRO problem cannot be expected, we furthermore prove that the approximation of the adversarial problem indeed converges to its true value for increasingly fine discretization. On the practical side, the approach is made concrete for a challenging, fundamental task in material design, namely in particle separation. Computational results for a realistic setting show that the safe approximation yields robust solutions of high-quality and can be computed within short time.