Tractable reformulations of DRO problems over structured optimal transport ambiguity sets

TL;DR

This work develops tractable reformulations for distributionally robust stochastic optimization where uncertainty decomposes into independent components. It introduces multi-transport hyperrectangles (MTHs) as structured Wasserstein-based ambiguity sets and leverages duality to derive finite-dimensional reformulations for DRO, uncertainty quantification, and distributionally robust chance constraints across piecewise affine, convex, and quadratic cost/constraint classes. The authors prove fundamental properties of MTHs, including weak compactness and finiteness of the DRO value, and address computational complexity via clustering of the product empirical reference distribution while preserving probabilistic guarantees. A key contribution is the systematic derivation of tractable convex/SDP reformulations for a range of problems and the demonstration that clustering can dramatically reduce problem size with limited loss in performance. A numerical power-dispatch example shows that MTHs outperform traditional Wasserstein balls in both conservatism and cost, with clustering enabling scalable solutions for high-dimensional uncertainty.

Abstract

Structuring ambiguity sets in Wasserstein-based distributionally robust optimization (DRO) can improve their statistical properties when the uncertainty consists of multiple independent components. The aim of this paper is to solve stochastic optimization problems with unknown uncertainty when we only have access to a finite set of samples from it. Exploiting strong duality of DRO problems over structured ambiguity sets, we derive tractable reformulations for certain classes of DRO and uncertainty quantification problems. We also derive tractable reformulations for distributionally robust chance-constrained problems. As the complexity of the reformulations may grow exponentially with the number of independent uncertainty components, we employ clustering strategies to obtain informative estimators, which yield problems of manageable complexity. We demonstrate the effectiveness of the theoretical results in a numerical simulation example.

Figures (4)

• Figure 1: The figure shows the empirical distribution $P_\xi^N$ of $N=6$ samples, represented by the thick red impulses, and the product empirical distribution $\bm P_\xi^N$, represented by the thin purple impulses. The product empirical distribution is the product of the marginal empirical distributions $P_{\xi_1}^N$ and $P_{\xi_2}^N$ of the independent components of $\xi$, which are depicted by the thin red impulses.
• Figure 2: Illustration of the considered power dispatch problem. The goal is to cover the power demand $d+\xi_2$ with probability at least $1-\alpha$, using the renewable power $\xi_1$ and the smaller possible amount $x$ of power from the market.
• Figure 3: Satisfaction probability of the ${\rm CVaR}$ constraint for each ambiguity set with respect to its radius $\varepsilon$ for the parameters $d=4.5$, $\alpha = 0.2$, $N=20$ and $M=72$. For each hyperrectangle, $\varepsilon$ represents the radius of the smallest ball enclosing it. The minimum radii ensuring the desired 0.9 confidence level for each ambiguity set are indicated by the vertical lines. For every $\varepsilon$, MTHs satisfy the chance-constraint with confidence levels that are always greater than that of the corresponding ambiguity ball, which turns out to be much more conservative. We also observe that despite the clear complexity reduction of clustered distribution compared to the product empirical distribution, the confidence levels of their associated MTHs are very close for every $\varepsilon$.
• Figure 4: Cumulative distributions of $\widehat{x}_{\mathcal{B}}$ in red, $\widehat{x}_{\mathcal{T}_{\rm cl}}$ in green and $\widehat{x}_{\mathcal{T}}$ in magenta for the smallest size of each ambiguity set that ensures the satisfaction of the ${\rm CVaR }$ constraint with a confidence level of 0.9 for the parameters $d=4.5$, $\alpha=0.2$, $N=20$ and $M=72$. The mean values of the distributionally robust solutions $\widehat{x}_{\mathcal{B}}$ in red, $\widehat{x}_{\mathcal{T}_{\rm cl}}$ in green, and $\widehat{x}_{\mathcal{T}}$ in magenta are depicted with vertical lines. The obtained DRO values with the MTHs are significantly smaller than the ones with the traditional ambiguity balls. Further, the average solution when the MTH is centered at the clustered distribution is very close to the one where the MTH is centered at the product empirical distribution, despite the considerable difference in the number of atoms of the two centers.

Theorems & Definitions (46)

• Theorem 4.1
• Proposition 4.2
• proof
• Theorem 4.3
• Proposition 4.4
• Proposition 5.1
• Proposition 5.2
• Proposition 5.3
• Theorem 6.1
• proof