The Benefit of Uncertainty Coupling in Robust and Adaptive Robust Optimization

TL;DR

This paper tackles conservatism in robust and adaptive robust optimization by introducing a generic coupled-uncertainty framework that intersects a constraint-wise uncertainty set with a coupling set. It develops tight, computable bounds on how much the objective can improve under coupling for both RO and ARO, across right-hand-side and coefficient-uncertainty scenarios, and extends the analysis to nonlinear problems. The authors show that coupling can enhance adaptability relative to static solutions, quantify this advantage via shrinkage factors, and connect these bounds to practical performance through extensive computational experiments in supply chain, portfolio management, and network lot sizing. The results offer a principled way to anticipate when coupling will yield gains and guide the design of uncertainty sets for more realistic, less conservative optimization under uncertainty.

Abstract

Despite the modeling power for problems under uncertainty, robust optimization (RO) and adaptive robust optimization (ARO) can exhibit too conservative solutions in terms of objective value degradation compared to the nominal case. One of the main reasons behind this conservatism is that, in many practical applications, uncertain constraints are directly designed as constraint-wise without taking into account couplings over multiple constraints. In this paper, we define a coupled uncertainty set as the intersection between a constraint-wise uncertainty set and a coupling set. We study the benefit of coupling in alleviating conservatism in RO and ARO. We provide theoretical tight and computable upper and lower bounds on the objective value improvement of RO and ARO problems under coupled uncertainty over constraint-wise uncertainty. In addition, we relate the power of adaptability over static solutions with the coupling of uncertainty set. Computational results demonstrate the benefit of coupling in applications.

Figures (13)

• Figure 1: Example of a supply chain network.
• Figure 2: Two scenarios of uncertainty sets for the supply chain example.
• Figure 3: Illustration of projections of a coupled set $\overline{U}$. The set $\Pi(\overline{U})$ would be the cube $\Pi_1(\overline{U}) \times \Pi_2(\overline{U})$, which contains $\overline{U}$.
• Figure 4: Illustration of shrinkage factors for the static supply chain example.
• Figure 5: Illustration of shrinkage factors for the adaptive supply chain example.

Theorems & Definitions (81)

• Example 2.1
• Example 2.2
• Definition 3.1
• Definition 3.2
• Definition 3.3
• Definition 3.4
• Lemma 3.1
• proof
• Lemma 3.2
• proof