A Max-Min-Max Algorithm for Large-Scale Robust Optimization

Abstract: Robust optimization (RO) is a powerful paradigm for decision making under uncertainty. Existing algorithms for solving RO, including the reformulation approach and the cutting-plane method, do not scale well, hindering the application of RO to large-scale decision problems. In this paper, we devise a first-order algorithm for solving RO based on a novel max-min-max perspective. Our algorithm operates directly on the model functions and sets through the subgradient and projection oracles, which enables the exploitation of problem structures and is especially suitable for large-scale RO. Theoretically, we prove that the oracle complexity of our algorithm for attaining an $\varepsilon$-approximate optimal solution is $\mathcal{O}(\varepsilon^{-3})$ or $\mathcal{O}(\varepsilon^{-2})$, depending on the smoothness of the model functions. The algorithm and its theoretical results are then extended to RO with projection-unfriendly uncertainty sets. We also show via extensive numerical experiments that the proposed algorithm outperforms the reformulation approach, the cutting-plane method and two other recent first-order algorithms.