Dimension Reduction of Distributionally Robust Optimization Problems
By: Brandon Tam, Silvana M. Pesenti
Potential Business Impact:
Simplifies hard math problems with uncertain numbers.
Plain English Summary
This helps companies make smarter decisions when they face unpredictable situations, like supply chain disruptions or market changes. It simplifies complex calculations so businesses can plan better and reduce risks. This means they can offer more stable prices and reliable services, which benefits everyone in their daily lives.
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 contained in uncertainty sets of univariate random variables defined via a univariate Wasserstein distance. This allows 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.
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