On Data-Driven Robust Optimization With Multiple Uncertainty Subsets: Unified Uncertainty Set Representation and Mitigating Conservatism

TL;DR

The paper addresses conservatism and computational challenges in data-driven robust optimization when uncertainties are represented as unions of multiple subsets. It introduces a monolithic mixed-integer uncertainty set representation that unifies all subset choices into a single MILP, enabling a single subproblem within a column-and-constraint generation framework and avoiding exponential growth with horizon length $N$. To mitigate conservatism, the authors blend RO with a distributionally robust optimization concept by optimizing over a discrete probability vector $\mathbf{p}$ within a KL-divergence-based ambiguity set $\mathcal{P}$, leading to a less conservative objective while avoiding full high-dimensional distribution modeling. They provide algorithmic developments (CCG variants and reformulations) and demonstrate, across three case studies, that the proposed schemes achieve reduced conservatism with favorable computational performance relative to standard RO and Wasserstein-based DRO, highlighting practical applicability for robust predictive control and planning under uncertainty.

Abstract

Constructing uncertainty sets as unions of multiple subsets has emerged as an effective approach for creating compact and flexible uncertainty representations in data-driven robust optimization (RO). This paper focuses on two separate research questions. The first concerns the computational challenge in applying these uncertainty sets in RO-based predictive control. To address this, a monolithic mixed-integer representation of the uncertainty set is proposed to uniformly describe the union of multiple subsets, enabling the computation of the worst-case uncertainty scenario across all subsets within a single mixed-integer linear programming (MILP) problem. The second research question focuses on mitigating the conservatism of conventional RO formulations by leveraging the structure of the uncertainty set. To achieve this, a novel objective function is proposed to exploit the uncertainty set structure and integrate the existing RO and distributionally robust optimization (DRO) formulations, yielding less conservative solutions than conventional RO formulations while avoiding the high-dimensional continuous uncertainty distributions and incurring high computational burden typically associated with existing DRO formulations. Given the proposed formulations, numerically efficient computation methods based on column-and-constraint generation (CCG) are also developed. Extensive simulations across three case studies are performed to demonstrate the effectiveness of the proposed schemes.

Figures (3)

• Figure 1: Simulation results of Case Study 1 with the proposed approach and the conventional approach.
• Figure 2: The chemical process network for Case Study 3.
• Figure 3: Error bar of the expected NPV with different ambiguity set size $\rho$.