Dimension Reduction of Distributionally Robust Optimization Problems

TL;DR

This work develops a dimension-reduction framework for distributionally robust optimization (DRO) problems where uncertainty sets are high-dimensional and defined via multivariate Wasserstein or Bregman-Wasserstein divergences. It proves that for a Lipschitz aggregation function $g$, the multivariate uncertainty can be bounded or exactly represented by a univariate Wasserstein ball on the aggregated output, enabling tractable univariate DRO formulations. The authors extend these results to BW divergences (including Mahalanobis and separable/composable generators) and introduce max-sliced variants to model joint uncertainty in risk factors and the aggregation function, accompanied by explicit bounds for the class of signed Choquet integrals. Numerical experiments illustrate the bounds for risk measures such as Expected Shortfall and the Inter-Expected Shortfall Range, highlighting the practical impact for risk management and portfolio optimization.

Abstract

We study distributionally robust optimization (DRO) problems with uncertainty sets consisting of high-dimensional random vectors that are close in the multivariate Wasserstein distance to a reference random vector. We give conditions under which the images of these sets under scalar-valued aggregation functions are equal to or bounded by uncertainty sets of univariate random variables defined via a univariate Wasserstein distance. This allows us to rewrite or bound high-dimensional DRO problems with simpler DRO problems over the space of univariate random variables. We generalize the results to uncertainty sets defined via the Bregman-Wasserstein divergence and the max-sliced Wasserstein and Bregman-Wasserstein divergence. The max-sliced divergences allow us to jointly model distributional uncertainty around the reference random vector and uncertainty in the aggregation function. Finally, we derive explicit bounds for worst-case risk measures that belong to the class of signed Choquet integrals.

Figures (3)

• Figure 1: Worst-case quantile functions for the ES (with $\alpha=0.95$). The reference distribution is plotted in blue. The red and orange lines correspond to the quantile functions attaining the upper and lower bounds for ${\varepsilon}\in\{0.3, 1\}$, respectively.
• Figure 2: Worst-case quantile functions for the IER (with $\alpha=0.75$). The reference distribution is plotted in blue. The red and orange lines correspond to the quantile functions attaining the upper and lower bounds for ${\varepsilon}\in\{0.3, 1\}$, respectively. The left plot shows the left tail $u\in[0.24, 0.26]$ and the right plot shows the right tail $u\in [0.74, 0.76]$.
• Figure 3: Left: inverse-S shaped distortion weight function $\gamma$. Right: upper bound for worst-case quantile function $G_{\mkern 1.5mu\overline{\mkern-1.5mu\lambda\mkern-1.5mu}\mkern 1.5mu}^{-1}$ (red, dashed) for ${\varepsilon}=\sqrt{8.1}$, reference quantile function $F^{-1}_{g({\boldsymbol{X}})}$ (black, solid), and the function $H_{\mkern 1.5mu\overline{\mkern-1.5mu\lambda\mkern-1.5mu}\mkern 1.5mu}(u) = F^{-1}_{g({\boldsymbol{X}})}(u)+ \frac{\gamma(u)}{2\mkern 1.5mu\overline{\mkern-1.5mu\lambda\mkern-1.5mu}\mkern 1.5mu}$ (green, dashed-dotted). Recall that $G_{\mkern 1.5mu\overline{\mkern-1.5mu\lambda\mkern-1.5mu}\mkern 1.5mu}^{-1} =(H_{\mkern 1.5mu\overline{\mkern-1.5mu\lambda\mkern-1.5mu}\mkern 1.5mu} )^\uparrow$.

Theorems & Definitions (47)

• Definition 1: Wasserstein Uncertainty Sets for cdfs
• Definition 2: Wasserstein Uncertainty Sets for RvS
• Theorem 1: Lipschitz aggregation
• Remark 1
• Proposition 1
• Lemma 1
• Corollary 1
• Proposition 2: Worst-case risks under Lipschitz aggregation
• Definition 3: Wasserstein Uncertainty
• Lemma 2