Distributionally Robust Optimization with Multimodal Decision-Dependent Ambiguity Sets

TL;DR

This work develops a generic two-stage distributionally robust optimization framework, called multimodal D^3RO, to handle uncertainty that is both multimodal and decision-dependent. It introduces a φ-divergence based ambiguity set for mode probabilities and complementary moment-based or Wasserstein-based ambiguity sets for the distributions within each mode, deriving tractable reformulations in special cases such as Variation Distance and χ^2-distance. Theoretical results show multimodality can improve in-sample and out-of-sample performance over single-modal DRO, and a separation-based decomposition algorithm with finite convergence guarantees enables scalable solution of large instances. Computational studies on facility location and shipment planning demonstrate the approach's superior performance and speed-ups against traditional single-modal or decision-independent frameworks, highlighting practical benefits for robust decision-making under complex uncertainty.

Abstract

We consider a two-stage distributionally robust optimization (DRO) model with multimodal uncertainty, where both the mode probabilities and uncertainty distributions could be affected by the first-stage decisions. To address this setting, we propose a generic framework by introducing a $φ$-divergence based ambiguity set to characterize the decision-dependent mode probabilities and further consider both moment-based and Wasserstein distance-based ambiguity sets to characterize the uncertainty distribution under each mode. We identify two special $φ$-divergence examples (variation distance and $χ^2$-distance) and provide specific forms of decision dependence relationships under which we can derive tractable reformulations. Furthermore, we investigate the benefits of considering multimodality in a DRO model compared to a single-modal counterpart through an analytical analysis. Additionally, we develop a separation-based decomposition algorithm to solve the resulting multimodal decision-dependent DRO models with finite convergence and optimality guarantee under certain settings. We provide a detailed computational study over two example problem settings, the facility location problem and shipment planning problem with pricing, to illustrate our results, which demonstrate that omission of multimodality or decision-dependent uncertainties within DRO frameworks result in inadequately performing solutions with worse in-sample and out-of-sample performances under various settings. We further demonstrate the speed-ups obtained by the solution algorithm against the off-the-shelf solver over various instances.

Figures (6)

• Figure 1: The equivalence between multimodal SAA (left) and single-modal SAA (right).
• Figure 2: Locations of customer sites and potential facilities on a 100$\times$100 grid
• Figure 3: In-sample and out-of-sample cost comparison between multimodal $\rm{D^3RO}$ model and its single-modal counterpart with different robustness level $\rho$ and support size $K$.
• Figure 4: In-sample and out-of-sample cost comparison between multimodal $\rm{D^3RO}$ model and its decision-independent counterpart with different robustness level $\rho$.
• Figure 5: Computational time comparison with different in-sample scenarios $\sum_{l=1}^LK_l$ and support size $K$.

Theorems & Definitions (42)

• Theorem 1: $\phi$-Divergence
• Theorem 2: Variation Distance
• Remark 1
• Theorem 3: $\chi^2$-Distance
• Remark 2
• Remark 3
• Theorem 4: Variation Distance + Moment-based
• proof
• Theorem 5: $\chi^2$-Distance + Moment-based
• proof