Validating Optimization with Uncertain Constraints

We consider optimization with uncertain or probabilistic constraints under the availability of limited data or Monte Carlo samples. In this situation, the obtained solutions are subject to statistical noises that affect both the feasibility and the objective performance. To guarantee feasibility, common approaches in data-driven optimization impose constraint reformulations that are "safe" enough to ensure solution feasibility with high confidence. Often times, selecting this safety margin relies on loose statistical estimates, in turn leading to overly conservative and suboptimal solutions. We propose a validation-based framework to balance the feasibility-optimality tradeoff more efficiently, by leveraging the typical low-dimensional structure of solution paths in these data-driven reformulations instead of estimates based on the whole decision space utilized by past approaches. We demonstrate how our approach can lead to a feasible solution with less conservative safety adjustment and confidence guarantees.