Combating Conservativeness in Data-Driven Optimization under Uncertainty: A Solution Path Approach

TL;DR

This paper introduces a validation-based approach to reduce conservativeness in data-driven optimization under uncertainty by leveraging low-dimensional solution structures, ensuring statistical feasibility with less over-conservativeness.

Contribution

The paper proposes a novel validation strategy that avoids full feasible set estimation, achieving less conservative solutions with statistical guarantees in data-driven optimization.

Findings

Solutions satisfy statistical feasibility with light dimension dependence

Approach is asymptotically optimal and minimally conservative

Numerical experiments outperform established benchmarks

Abstract

In data-driven optimization, solution feasibility is often ensured through a "safe" reformulation of the uncertain constraints, such that an obtained data-driven solution is guaranteed to be feasible for the oracle formulation with high statistical confidence. Such approaches generally involve an implicit estimation of the whole feasible set that can scale rapidly with the problem dimension, in turn leading to over-conservative solutions. In this paper, we investigate a validation-based strategy to avoid set estimation by exploiting the intrinsic low dimensionality among all possible solutions output from a given reformulation. We demonstrate how our obtained solutions satisfy statistical feasibility guarantees with light dimension dependence, and how they are asymptotically optimal and thus regarded as the least conservative with respect to the considered reformulation classes. We apply this strategy to several data-driven optimization paradigms including (distributionally) robust optimization, sample average approximation and scenario optimization. Numerical experiments show encouraging performances of our strategy compared to established benchmarks.