A Safe Approximation Based on Mixed-Integer Optimization for Non-Convex Distributional Robustness Governed by Univariate Indicator Functions
Jana Dienstbier, Frauke Liers, Florian Rösel, Jan Rolfes
TL;DR
This work addresses DRO problems that include non-convex univariate indicator functions by developing a safe, discretized approximation that reduces to a mixed-integer linear program (MILP). The approach relies on a semi-infinite reformulation, moment- and envelope-based ambiguity sets that admit linear dualization, and a discretization that yields a provably safe inner approximation. The authors prove that the inner problem of the safe approximation converges to the true inner value as the discretization becomes finer, and demonstrate MILP-representability of the safe model under mild conditions. Computational experiments in a particle-separation setting show that robust solutions maintain high quality and can be computed quickly, highlighting the practical applicability of the method to challenging non-convex DRO problems.
Abstract
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.