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

TL;DR

This paper introduces a unified mixed-integer approach for data-driven robust optimization with multiple uncertainty subsets, reducing conservatism and computational complexity in predictive control applications.

Contribution

It proposes a monolithic MILP representation for unions of uncertainty subsets and a novel objective to mitigate conservatism, integrating RO and DRO methods.

Findings

Efficient MILP formulation for uncertainty unions

Reduced conservatism compared to traditional RO

Validated effectiveness through three case studies

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.